Hermitian K-theory for stable ∞ -categories I: Foundations
Baptiste Calmès, Emanuele Dotto, Yonatan Harpaz, Fabian Hebestreit, Markus Land, Kristian Moi, Denis Nardin, Thomas Nikolaus, Wolfgang Steimle
aa r X i v : . [ m a t h . K T ] S e p HERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I:FOUNDATIONS BAPTISTE CALMÈS, EMANUELE DOTTO, YONATAN HARPAZ, FABIAN HEBESTREIT, MARKUS LAND,KRISTIAN MOI, DENIS NARDIN, THOMAS NIKOLAUS, AND WOLFGANG STEIMLE
To Andrew Ranicki. A BSTRACT . This paper is the first in a series in which we offer a new framework for hermitian K -theory inthe realm of stable ∞ -categories. Our perspective yields solutions to a variety of classical problems involvingGrothendieck-Witt groups of rings and clarifies the behaviour of these invariants when is not invertible.In the present article we lay the foundations of our approach by considering Lurie’s notion of a Poincaré ∞ -category, which permits an abstract counterpart of unimodular forms called Poincaré objects. We analysethe special cases of hyperbolic and metabolic Poincaré objects, and establish a version of Ranicki’s algebraicThom construction. For derived ∞ -categories of rings, we classify all Poincaré structures and study in detailthe process of deriving them from classical input, thereby locating the usual setting of forms over rings withinour framework. We also develop the example of visible Poincaré structures on ∞ -categories of parametrisedspectra, recovering the visible signature of a Poincaré duality space.We conduct a thorough investigation of the global structural properties of Poincaré ∞ -categories, showingin particular that they form a bicomplete, closed symmetric monoidal ∞ -category. We also study the processof tensoring and cotensoring a Poincaré ∞ -category over a finite simplicial complex, a construction featuringprominently in the definition of the L - and Grothendieck-Witt spectra that we consider in the next instalment.Finally, we define already here the 0-th Grothendieck-Witt group of a Poincaré ∞ -category using generatorsand relations. We extract its basic properties, relating it in particular to the 0-th L - and algebraic K -groups, arelation upgraded in the second instalment to a fibre sequence of spectra which plays a key role in our applica-tions. C ONTENTS
Introduction ∞ -categories 171.3 Classification of hermitian structures 221.4 Functoriality of hermitian structures 26 L -groups 352.4 The Grothendieck-Witt group 41 Date : September 16, 2020. ∞ -categories 785.2 Construction of the symmetric monoidal structure 825.3 Day convolution of hermitian structures 885.4 Examples 95 ∞ -categories 1297.3 The categorical Thom isomorphism 1357.4 Genuine semi-additivity and spectral Mackey functors 1467.5 Multiplicativity of Grothendieck-Witt and L -groups 152 References
NTRODUCTION
Quadratic forms are among the most ubiquitus notions in mathematics. In his pioneering paper [Wit37],Witt suggested a way to understand quadratic forms over a field 𝑘 in terms of an abelian group W q ( 𝑘 ) , nowknown as the Witt group of quadratic forms. By definition, the Witt group is generated by isomorphismclasses [ 𝑉 , 𝑞 ] of finite dimensional 𝑘 -vector spaces equipped with a unimodular quadratic form 𝑞 , wherewe impose the relations [ 𝑉 ⊕ 𝑉 ′ , 𝑞 ⟂ 𝑞 ′ ] = [ 𝑉 , 𝑞 ] + [ 𝑉 ′ , 𝑞 ′ ] and declare as trivial the classes of hyperbolicforms [ 𝑉 ⊕ 𝑉 ∗ , ℎ ] given by the canonical pairing between 𝑉 and its dual 𝑉 ∗ . In arithmetic geometry theWitt group became an important invariant of fields, related to their Milnor K -theory and Galois cohomologyvia the famous Milnor conjecture.The definition of the Witt group naturally extends from fields to commutative rings 𝑅 , where one re-places vectors spaces by finitely generated projective 𝑅 -modules. More generally, instead of starting witha commutative ring 𝑅 and taking 𝑅 -valued forms, one can study unimodular hermitian forms valued inan invertible ( 𝑅 ⊗ 𝑅 ) -module 𝑀 equipped with an involution, a notion which makes sense also for non-commutative 𝑅 . This includes for example the case of a ring 𝑅 with anti-involution by considering 𝑀 = 𝑅 ,and also allows to consider skew-quadratic forms by changing the involution on 𝑀 by a sign. Quadraticforms at this level of generality also show up naturally in the purely geometric context of surgery theory through the quadratic L -groups of the group ring ℤ [ 𝜋 ( 𝑋 )] for a topological space 𝑋 . The latter groups,whose name, coined by Wall, suggests their relation with algebraic K -theory, are a sequence of groups L q 𝑖 associated to a ring with anti-involution 𝑅 , or more generally, a ring equipped with an invertible ( 𝑅 ⊗ 𝑅 ) -module with involution 𝑀 as above, with L q0 ( 𝑅, 𝑀 ) being the Witt group of 𝑀 -valued quadratic forms over 𝑅 . They are -periodic, or more precisely, satisfy the skew-periodic relation L q 𝑛 +2 ( 𝑅, 𝑀 ) ≅ L q 𝑛 ( 𝑅, − 𝑀 ) ,where − 𝑀 is obtained from 𝑀 by twisting the involution by a sign. In particular, for a ring with anti-involution 𝑅 the even quadratic L -groups consist of the Witt groups of quadratic and skew-quadratic forms.To obtained richer information about quadratic forms over a given 𝑅 , the Witt group W q ( 𝑅, 𝑀 ) wasoften compared to the larger group generated by the isomorphism classes of unimodular quadratic 𝑀 -valuedforms [ 𝑃 , 𝑞 ] over 𝑅 under the relation [ 𝑃 ⊕𝑃 ′ , 𝑞 ⟂ 𝑞 ′ ] = [ 𝑃 , 𝑞 ]+[ 𝑃 ′ , 𝑞 ′ ] , but without taking the quotient byhyperbolic forms. The latter construction leads to the notion of the Grothendieck-Witt group GW q0 ( 𝑅, 𝑀 ) of quadratic forms. The Witt and Grothendieck-Witt groups are then related by an exact sequence(1) K ( 𝑅 ) C hyp ←←←←←←←←←←←←←←→ GW q0 ( 𝑅, 𝑀 ) → W q ( 𝑅 ) → , ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 3 where the first term denotes the orbits for the C -action on the K -theory group K ( 𝑅 ) which sends the classof a finitely generated projective 𝑅 -module 𝑃 to the class of its 𝑀 -dual Hom 𝑅 ( 𝑃 , 𝑀 ) . The left hand mapthen sends [ 𝑃 ] to the class of the associated hyperbolic form on 𝑃 ⊕
Hom(
𝑃 , 𝑀 ) , and is invariant underthis C -action. The sequence (1) can often be used to compute GW q0 ( 𝑅, 𝑀 ) from the two outer groups,and consequently obtain more complete information about quadratic forms. For example, in the case of theintegers this sequence is split short exact and we have an isomorphism W q ( ℤ ) ≅ ℤ given by taking thesignature divided by and an isomorphism K ( ℤ ) C ≅ ℤ given by the dimension.In this paper we begin a four-part investigation revisiting classical questions about Witt, Grothendieck-Witt, and L -groups of rings from a new perspective. One of our main motivating applications is to extendthe short exact sequence (1) to a long exact sequence involving Quillen’s higher K -theory and the higherGrothendieck-Witt groups GW q 𝑖 ( 𝑅, 𝑀 ) introduced by Karoubi and Villamayor [KV71], see below for moredetails. In this paper we will, among many other things, define abelian groups L gq 𝑖 ( 𝑅, 𝑀 ) , called genuinequadratic L -groups , which are the correct higher Witt groups from this point of view: we will show inPaper [III] that we have L gq0 ( 𝑅, 𝑀 ) = W q ( 𝑅, 𝑀 ) and that the sequence (1) can be extended to a long exactsequence involving the groups L gq 𝑖 ( 𝑅, 𝑀 ) which starts off as … → GW q1 ( 𝑅, 𝑀 ) → L gq1 ( 𝑅, 𝑀 ) → K ( 𝑅, 𝑀 ) C hyp ←←←←←←←←←←←←←←→ GW q0 ( 𝑅, 𝑀 ) → L gq0 ( 𝑅, 𝑀 ) → . The groups L gq 𝑖 ( 𝑅, 𝑀 ) are generally different from Wall’s quadratic L -groups, and in particular are usuallynot -periodic. They are however relatively accessible for study by means of algebraic surgery . Combiningthis with the above long exact sequence will allow us in Paper [III] to obtain many new results about theGrothendieck-Witt groups GW q 𝑖 ( 𝑅 ) of rings. For example, we will obtain an essentially complete calcula-tion of these groups in the case of the integers 𝑅 = ℤ . In what follows we give more background, outlineour approach and its main applications, and elaborate more on what is done in the present paper. Background.
The higher Grothendieck-Witt groups GW q 𝑖 ( 𝑅, 𝑀 ) mentioned above were first defined byKaroubi and Villamayor [KV71] by applying Quillen’s foundational techniques from algebraic K -theory.This is done by producing a homotopy-theoretical refinement of the 0-th Grothendieck-Witt group into a Grothendieck-Witt space and then defining GW 𝑖 ( 𝑅, 𝑀 ) as the 𝑖 -th homotopy group of this space. Given 𝑅 and 𝑀 as above, one organizes the collection of unimodular quadratic 𝑀 -valued forms ( 𝑃 , 𝑞 ) into agroupoid Unimod q ( 𝑅, 𝑀 ) , which may be viewed as an E ∞ -space using the symmetric monoidal structureon Unimod q ( 𝑅 ) arising from the orthogonal sum. One can then take its group completion to obtain an E ∞ -group GW qcl ( 𝑅, 𝑀 ) ∶= Unimod q ( 𝑅, 𝑀 ) grp , whose group of components is the Grothendieck-Witt group described above. Here the subscript cl standsfor classical, and is meant to avoid confusion with the constructions of the present paper series. This con-struction can equally well be applied for other interesting types of forms, such as symmetric bilinear, orsymmetric bilinear forms which admit a quadratic refinement, also known as even forms, and these can betaken with values in an arbitrary invertible module with involution 𝑀 as above. Taking the polarisation ofquadratic form determines maps GW qcl ( 𝑅, 𝑀 ) ⟶ GW evcl ( 𝑅, 𝑀 ) ⟶ GW scl ( 𝑅, 𝑀 ) , which are equivalences if is a unit in 𝑅 . In this latter case Grothendieck-Witt groups are generally muchmore accessible. For example, when is invertible Schlichting [Sch17] has produced a (generally non-connective) delooping of the Grothendieck-Witt space to a Grothendieck-Witt spectrum GW cl ( 𝑅, 𝑀 ) , inwhich case the forgetful and hyperbolic maps can be refined to a spectrum level C -equivariant maps K( 𝑅 ) hyp ←←←←←←←←←←←←←←→ GW cl ( 𝑅, 𝑀 ) fgt ←←←←←←←←←←←←→ K( 𝑅 ) . He then showed in loc. cit. that the cofibre of the induced map(2) K( 𝑅 ) hC → GW cl ( 𝑅, 𝑀 ) has -periodic homotopy groups, whose even values are given by the Witt groups W( 𝑅, 𝑀 ) and W( 𝑅, − 𝑀 ) .More precisely, Schlichting’s identification of these homotopy groups matches the L -groups of Wall-Ranicki, CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE that which has lead to the folk theorem that if is a unit in 𝑅 then the cofibre of (2) is naturally equivalentto Ranicki’s L -theory spectum L( 𝑅, 𝑀 ) from [Ran92]. This allows one, when is invertible, to producean extension of (1) to a long exact sequence and obtain information about higher Grothendieck-Witt groupsfrom information about higher K -theory and L -groups. A closely related connection between Grothendieck-Witt spaces with coefficients in ± 𝑀 when is invertible was established by Karoubi in his influential pa-per [Kar80], where he proved what is now known as Karoubi’s fundamental theorem , forming one of theconceptual pillars of hermitian K -theory, as well as part of its standard tool kit. It permits, for example, toinductively deduce results on higher Grothendieck-Witt groups from information about algebraic K -theoryand about the low order Grothendieck-Witt groups GW ( 𝑅, ± 𝑀 ) and GW ( 𝑅, ± 𝑀 ) .By contrast, when is not invertible non of these assertions hold as stated. In particular, the rela-tion between Grothendieck-Witt theory and L -theory remained, in this generality, completely mysterious.Karoubi, in turn, conjectured in [Kar09] that his fundamental theorem should have an extension to generalrings, relating Grothendieck-Witt spaces for two different form parameters, as was also suggested earlierby Giffen [Wil05]. In the context of motivic homotopy theory, crucial properties such as devissage and 𝐀 -invariance of Grothendieck-Witt theory were only known to hold when is invertible by the work ofSchlichting and Hornbostel [Hor02], [HS04]. Consequently, hermitian K -theory was available to studyas a motivic spectrum exclusively over ℤ [ ] , see [Hor05]. Finally, while all the above tools could beused to calculate Grothendieck-Witt groups of rings in which is invertible, such as the ring ℤ [ ] whoseGrothendieck-Witt groups were calculated by Berrick and Karoubi in [BK05], higher Grothendieck-Wittgroups of general rings remain largely unknown. Hermitian K -theory of Poincaré ∞ -categories. The goal of the present paper series is to offer new foun-dations for hermitian K -theory in a framework that unites its algebraic and surgery theoretic incarnationsand that is robustly adapt to handle the subtleties involved when is not invertible. We begin by situatinghermitian K -theory in the general framework of Poincaré ∞ -categories , a notion suggested by Lurie inhis treatise of L -theory [Lur11]. A Poincaré ∞ -category consists of a stable ∞ -category C together with afunctor Ϙ ∶ C op → Sp which is quadratic in the sense of Goodwillie calculus and satisfies a suitable uni-modularity condition, the latter determining in particular a duality D Ϙ ∶ C op ≃ ←←←←←←←→ C on C . We refer to such a Ϙ as a Poincaré structure on C . Roughly speaking, the role of the Poincaré structure Ϙ is to encode the flavourof forms that we want to consider. For example, for a commutative ring 𝑅 one may take C = D p ( 𝑅 ) to bethe perfect derived category of 𝑅 . One should then think of the mapping spectrum hom D p ( 𝑅 ) ( 𝑋 ⊗ 𝑅 𝑋, 𝑅 ) as the spectrum of bilinear forms on the chain complex 𝑋 , which acquires a natural C -action by flippingthe components in the domain term. In this case the Poincaré structure Ϙ s 𝑅 ( 𝑋 ) = hom D p ( 𝑅 ) ( 𝑋 ⊗ 𝑅 𝑋, 𝑅 ) hC encodes a homotopy coherent version of the notion of symmetric bilinear forms, while Ϙ q 𝑅 ( 𝑋 ) = hom D p ( 𝑅 ) ( 𝑋 ⊗ 𝑅 𝑋, 𝑅 ) hC encodes a homotopy coherent version of quadratic forms. Both these Poincaré structures have the sameunderlying duality, given by 𝑋 ↦ Hom cx 𝑅 ( 𝑋, 𝑅 ) .Alternatively, as we will develop in the present paper, one may also obtain Poincaré structures on D p ( 𝑅 ) by taking a non-abelian derived functor associated to a quadratic functor Proj( 𝑅 ) op → A 𝑏 from finitelygenerated projective modules to abelian groups. For example, taking the functors which associate to aprojective module 𝑃 the abelian groups of quadratic, even and symmetric forms on 𝑃 one obtained Poincaréstructures Ϙ gq 𝑅 , Ϙ ge 𝑅 and Ϙ gs 𝑅 on D p ( 𝑅 ) , respectively. We call these the genuine quadratic, even and symmetricfunctors, and consider them as encoding the classical, rigid notions of hermitian forms in the present setting,where as Ϙ q 𝑅 and Ϙ s 𝑅 encode their homotopy coherent counterparts. More generally, one can apply theseconstruction to any associative rings equipped with an invertible ( 𝑅 ⊗ 𝑅 ) -module with involution 𝑀 asabove. The resulting Poincaré structures are then all related by a sequence of natural transformations Ϙ q 𝑀 ⇒ Ϙ gq 𝑀 ⇒ Ϙ ge 𝑀 ⇒ Ϙ gs 𝑀 ⇒ Ϙ s 𝑀 , which encode the polarisation map between the quadratic, even and symmetric flavours of hermitian formsand at the same time the comparison between homotopy coherent and rigid variants of such forms. The ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 5 fact that these two types of distinctions are not entirely unrelated leads to some of the more surprisingapplications of our approach. When is invertible in 𝑅 , all these maps are equivalences.The fundamental invariant of a Poincaré ∞ -category is its space Pn( C , Ϙ ) of Poincaré objects , which arepairs ( 𝑥, 𝑞 ) consisting of an object 𝑥 ∈ C and a point 𝑞 ∈ Ω ∞ Ϙ ( 𝑥 ) whose associated map 𝑞 ♯ ∶ 𝑥 → D Ϙ ( 𝑥 ) isan equivalence. These are the avatars in the present context of the notion of a unimodular hermitian form.From this raw invariant one may produce two principal spectrum valued invariants - the Grothendieck-Wittspectrum
GW( C , Ϙ ) and L -theory spectrum L( C , Ϙ ) . The L -theory spectrum was transported by Lurie fromthe classical work of Wall-Ranicki to the context of Poincaré ∞ -categories in [Lur11]. In particular, the L -theory spectra L q ( 𝑅, 𝑀 ) ∶= L( D p ( 𝑅 ) , Ϙ q 𝑀 ) and L s ( 𝑅, 𝑀 ) ∶= L( D p ( 𝑅 ) , Ϙ s 𝑀 ) coincide with Ranicki’s -periodic quadratic and symmetric L -theory spectra, respectively. When applied to the genuine Poincaréstructures this yields new types of L -theory spectra L gq ( 𝑅, 𝑀 ) , L ge ( 𝑅, 𝑀 ) and L gs ( 𝑅, 𝑀 ) . It turns out thatthese are in fact not entirely new: we will show in Paper [III] that for the genuine symmetric structurethe homotopy groups of L gs ( 𝑅, 𝑀 ) coincide with Ranicki’s original non-periodic variant of symmetric L -groups, as defined in [Ran80]. Somewhat surprisingly, the genuine quadratic L -theory spectrum L gq ( 𝑅, 𝑀 ) is a -fold shift of L gs ( 𝑅 ) .The Grothendieck-Witt spectrum GW( C , Ϙ ) of a Poincaré ∞ -category will be defined in Paper [II],though in the present paper we will already introduce its zero’th homotopy group GW ( C , Ϙ ) , namely, theGrothendieck-Witt group. The underlying infinite loop space GW ( C , Ϙ ) ∶= Ω ∞ GW( C , Ϙ ) is then called the Grothendieck-Witt space of ( C , Ϙ ) . When is invertible in 𝑅 we will show in Paper [II]that GW(
𝑅, 𝑀 ) ∶= GW( D p ( 𝑅 ) , Ϙ 𝑀 ) is equivalent to the Grothendieck-Witt spectrum defined by Schlicht-ing in [Sch17] (where Ϙ 𝑀 is any of the Poincaré structures considered above, which coincide due to theinvertibility condition on ). When is not invertible, the second and ninth author show in the compan-ion paper [HS20] that the Grothendieck-Witt spaces of D p ( 𝑅 ) with respect to the genuine Poincaré struc-tures Ϙ gq 𝑀 , Ϙ ge 𝑀 and Ϙ gs 𝑀 coincide with the classical Grothendieck-Witt spaces of quadratic, even and symmet-ric 𝑀 -valued forms, respectively. On the other hand, the Grothendieck-Witt spectra of ( D p ( 𝑅 ) , Ϙ q 𝑀 ) and ( D p ( 𝑅 ) , Ϙ s 𝑀 ) are actually new invariants of rings, which are based on the homotopy coherent avatars of qua-dratic and symmetric forms. These sometimes have better formal properties. For example, in the up comingwork [CHN], the first, third and seventh authors show that the GW - and L -theory spectra associated to thesymmetric Poincaré structures Ϙ s 𝑅 satisfy 𝐀 -invariance, and can further be encoded via motivic spectraover the integers. This statement does not hold for any of the other Poincaré structures above, including thegenuine symmetric one.One of the principal results we will prove in Paper [II] is that the relation between Grothendieck-Witt-, L - and algebraic K -theory is governed by the fundamental fibre sequence(3) K( C ) hC → GW( C , Ϙ ) → L( C , Ϙ ) , where the first term is the homotopy orbits of the algebraic K -theory spectra of C with respect to the C -action induced by the duality of Ϙ . In the case of the genuine symmetric Poincaré structure Ϙ gs 𝑀 , this givesa relation between classical symmetric Grothendieck-Witt groups and Ranicki’s non-periodic symmetric L -groups, which to our knowledge is completely new. In the case of the genuine quadratic structure theconsequence is even more surprising: the resulting long exact sequence in homotopy groups extends theclassical exact sequence (1) to a long exact sequence involving a shifted copy of Ranicki’s non-periodic L -groups.The main role of the present instalment is to lay down the mathematical foundations that enable thearguments of the next three papers, and eventually their fruits, to take place. In particular, we carefullydevelop the main concepts of Poincaré ∞ -categories and Poincaré objects, discuss hyperbolic objects andLagrangians, and prove a version of Ranicki’s algebraic Thom construction in the present setting. We alsodefine the L -groups and zero’th Grothendieck-Witt group of a Poincaré ∞ -category, and conduct a thoroughinvestigation of the global structure properties enjoyed by the ∞ -category of ∞ -categories. In addition tothe general framework, we will also introduce and study important constructions of Poincaré ∞ -categories,which give rise to our motivating examples of interest. In particular: CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE i) We classify all Poincaré structures in the case where C is the ∞ -category of perfect modules over aring spectrum, and show that they can be efficiently encoded by the notion of a module with genuineinvolution.ii) When C is the perfect derived category of a discrete ring, we develop the procedure of deriving Poincaré structures used to produce the genuine Poincaré structures above. Here we will pick up onsome recent ideas of Glasman, Mathew and Illusie, and show that Poincaré structures on C are in factuniquely determined by their values on projective modules. This allows for the connection betweenthe present setup and Grothendieck-Witt theory of rings, through which the applications of Paper [II]and Paper [III] to classical problems can be carried out.iii) We develop in some detail the example of visible Poincaré structures on ∞ -categories of parameterizedspectra, which allow us to reproduce visible L -theory as well as LA -theory of Weiss-Wiliams in thepresent setting. This leads to applications in surgery theory which will be pursued by the second andninth author in future work.iv) Following Lurie’s treatment of L -theory we study the process of tensoring and cotensoring a Poincaré ∞ -category over a finite simplicial complex. This construction is later exploited in Paper [II] to defineand study the Grothendieck-Witt spectrum.v) We show that the ∞ -category of Poincaré ∞ -categories has all limits and colimits. This enables one,for example, to produce new Poincaré ∞ -categories by taking fibres and cofibres of Poincaré functors,and enables the notion of additivity , which lies at the heart of Grothendieck-Witt theory, to be properlyset up in Paper [II].vi) We show that Poincaré ∞ -categories can be tensored with each other. This can be used to produce newPoincaré ∞ -categories from old, but also to identity additional important structures, such as a Poincarésymmetric monoidal structure, which arises in many examples of interest and entails the refinementof their Grothendieck-Witt and L -theory spectra to E ∞ -rings. This last claim will be proven in Paper[IV], though we will prove it for the Grothendieck-Witt and L -groups already in the present paper. Applications.
Our framework of Poincaré ∞ -categories is motivated by a series of applications whichwill be extracted in the following instalments, many of which pertain to classical questions in hermitian K -theory. To give a brief overview of what’s ahead, we first mention that a key feature of the Grothendieck-Witt spectrum we will construct in Paper [II] is its additivity. In the setting of Poincaré ∞ -categories, thiscan be neatly phrased by saying that the functor ( C , Ϙ ) ↦ GW( C , Ϙ ) sends split bifibre sequences ( C , Ϙ ) → ( C ′ , Ϙ ′ ) → ( C ′′ , Ϙ ′′ ) of Poincaré ∞ -categories to bifibre sequences of spectra, where by split we mean that C ′ → C ′′ admits botha left and a right adjoint. One of the main results of Paper [II] is that GW is additive, and is furthermoreuniversally characterized by this property as initial among additive functors from Poincaré ∞ -categoriesto spectra equipped with a natural transformation from Σ ∞ Pn . This is analogous to the universal propertycharacterizing algebraic K -theory of stable ∞ -categories established in [BGT13]. In fact, we will show inPaper [II] that GW is not only additive but also Verdier localising , a property formulated as above but withthe splitness condition removed. This will be used in [CHN] by the first, third and seventh author in orderto show that the GW -spectrum satisfies Nisnevich descent over smooth schemes. It will also play a key rolein the study of Grothendieck-Witt theory of Dedekind rings in Paper [III].One major consequence of additivity is that the hyperbolic and forgetful maps fit to form the
Bott-Genauer sequence
GW( C , Ϙ [−1] ) fgt ←←←←←←←←←←←←→ K( C ) hyp ←←←←←←←←←←←←←←→ GW( C , Ϙ ) , where Ϙ [ 𝑛 ] = Σ 𝑛 Ϙ is the shifting operation on Poincaré functors. Such a sequence was established in thesetting of rings in which is invertible by Schlichting [Sch17], who used it to produce another proof ofKaroubi’s fundamental theorem. The same argument then yields a version of Karoubi’s fundamental theo-rem in the setting of Poincaré ∞ -categories. When applied to the genuine Poincaré structures we constructin the present paper, this yields an extension of Karoubi’s fundamental theorem to rings in which is notassumed invertible, establishing, in particular, a conjecture of Karoubi and Giffen.The fundamental fibre sequence (3) will be heavily exploited in Paper [III] to obtain applications forclassical Grothendieck-Witt groups of rings. In particular, improving a comparison bound of Ranicki we ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 7 will show in Paper [III] that if 𝑅 is Noetherian of global dimension 𝑑 the maps L gq ( 𝑅, 𝑀 ) ⟶ L ge ( 𝑅, 𝑀 ) ⟶ L gs ( 𝑅, 𝑀 ) ⟶ L s ( 𝑅, 𝑀 ) are equivalences in degrees past 𝑑 + 2 , 𝑑 and 𝑑 − 2 , respectively. Thus, even though the genuine L -theoryspectra are not -periodic, they become so in degrees sufficiently large compared to the global dimension.In addition, when combined with the fundamental fiber sequence (3) one deduces that the maps of classicalGrothendieck-Witt spaces GW qcl ( 𝑅, 𝑀 ) → GW evcl ( 𝑅, 𝑀 ) → GW scl ( 𝑅, 𝑀 ) are isomorphisms on homotopy groups in sufficiently high degrees. This is a new and quite unexpected resultabout classical Grothendieck-Witt groups, and to our knowledge is the first time that the global dimension ofa ring has been related in any way to the gap between its quadratic and symmetric GW -groups. Combinedwith our extension of Karoubi’s fundamental theorem this implies that in the case of finite global dimensionKaroubi’s fundamental theorem holds in its classical form in sufficiently high degrees, allowing for manyof the associated arguments to be picked up in this context. In a different direction, for such rings onecan eventually deduce results about classical symmetric GW -groups from results on the correspondinghomotopy coherent symmetric GW -groups, allowing one to exploit some of the useful properties of thelatter, such as a devissage property we will prove in Paper [III] and the 𝐀 -invariance established in [CHN],for the benefit of the former. We will exploit these ideas in Paper [III] to solve the homotopy limit problemfor number rings, show that their Grothendieck-Witt groups are finitely generated, and produce an essentiallycomplete calculation of the quadratic and symmetric Grothendieck-Witt groups (in both the skew and non-skew cases) of the integers, affirming, in particular, a conjecture of Berrick and Karoubi from [BK05]. Organization of the paper.
Let us now describe the structure and the content of the present paper in moredetail. In §1 we define
Poincaré ∞ -categories . As indicated before, a Poincaré ∞ -category is a stable ∞ -category C equipped with a quadratic functor Ϙ ∶ C op → S 𝑝 which is perfect in a suitable sense. We willgive the precise definition in §1.2, after a discussion of quadratic functors in §1.1. We will also considerthe weaker notion of a hermitian ∞ -category , obtained by removing the perfectness condition on Ϙ , andexplain how to extract from a Poincaré structure Ϙ a duality Ϙ Ϙ ∶ C op ≃ ←←←←←←←→ C . In §1.3 we will describe howone can classify hermitian and Poincaré structures on a given stable ∞ -category in terms of their linear and bilinear parts. Finally, in §1.4 we will discuss the functorial dependence of hermitian structures on theunderlying stable ∞ -category, and relate it to the classification discussed in §1.3.In §2 we define the notion of a Poincaré object in a given Poincaré ∞ -category ( C , Ϙ ) . Such a Poincaréobject consists of an object 𝑥 ∈ C together with a map 𝑞 ∶ 𝕊 → Ϙ ( 𝑥 ) , to be though of a form in 𝑋 ,such that a certain induced map 𝑞 ♯ ∶ 𝑥 → D Ϙ 𝑥 is an equivalence. The precise definition will be given in§2.1. We will then proceed to discuss hyperbolic Poincaré objects in §2.2, and in §2.3 the slightly moregenereal notion of metabolic Poincaré objects, that is, Poincaré objects that admit a Lagrangian. We willshow how one can understand metabolic Poincaré objects via Poincaré objects in a certain Poincaré ∞ -category Met( C , Ϙ ) constructed from ( C , Ϙ ) . The notion of metabolic Poincaré objects is the main inputin the definition of the L -groups of a given Poincaré ∞ -category (Definition 2.3.11). Finally, in §2.4 weshall define the Grothendieck-Witt group GW ( C , Ϙ ) of a given Poincaré ∞ -category and develop its basicproperties.In §3 we study Poincaré structures on the ∞ -category Mod 𝜔𝐴 of perfect modules over a ring spectrum 𝐴 .To this end, we introduce the notion of a module with involution in §3.1 and show how it can be used tomodel bilinear functors on module ∞ -categories. We then refine this notion §3.2 to a module with genuine involution, that which will allow us to encode not only bilinear functors but also hermitian and Poincaréstructures. Then, in §3.3 we will discuss the basic operations of restriction and induction of modules withgenuine involution along maps of ring spectra.In §4 we will discuss several examples of interest of Poincaré ∞ -categories in further detail. We willbegin in §4.1 with the important example of the universal Poincaré ∞ -category ( S 𝑝 f , Ϙ u ) , which is charac-terized by the property that Poincaré functors out of it pick out Poincaré objects in the codomain. In §4.2we will consider perfect derived ∞ -categories of ordinary rings and show how to translate the classicallanguage of forms on projective modules into that of the present paper via the process of deriving quadratic CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE functors. In §4.3 and §4.4 we explain how to construct Poincaré structures producing visible L -theory asstudied by Weiss [Wei92], Ranicki [Ran92], and more recenetly Weiss-Wiliams [WW14].In §5 we will show that the tensor product of stable ∞ -categories refines to give a symmetric monoidalstructure on the ∞ -category Cat p∞ of Poincaré ∞ -categories. The precise definition and main properties ofthis monoidal product will be elaborated in §5.1 and §5.2. In §5.3 we analyse what it means for a Poincaré ∞ -category to be an algebra with respect to this structure, and use this analysis in §5.4 in order to identifyvarious examples of interest of symmetric monoidal Poincaré ∞ -categories.In §6 we study the global structural properties of the ∞ -categories Cat p∞ and Cat h∞ of Poincaré andhermitian ∞ -categories respectively. We will begin in §6.1 by showing that these two ∞ -categories haveall small limits and colimits, and describe how these can be computed. In §6.2 we will prove that thesymmetric monoidal structures on Cat p∞ and Cat h∞ constructed in §5.2 are closed , that is, admit internalmapping objects. We will then show in §6.3 and §6.4 that Cat h∞ is tensored and cotensored over Cat ∞ . Aspecial role is played by indexing diagrams coming from the poset of faces of a finite simplicial complex,which we study in §6.5 and §6.6, showing in particular that in this case this procedure preserves Poincaré ∞ -categories. The cotensor construction will be used in Paper [II] to define the hermitian Q -constructionand eventually Grothendieck-Witt theory, while the tensor construction plays a role in proving the universalproperty of Grothendieck-Witt theory.In §7 we consider the relationship between Cat p∞ and Cat h∞ , and between both of them and various coarservariants, such as bilinear and symmetric bilinear ∞ -categories. By categorifying the relationship betweenPoincaré forms, hermitian forms and bilinear forms we construct in §7.2 and §7.3 left and right adjoints to allrelevant forgetful functors. In §7.3 we also prove a generalized version of the algebraic Thom construction,which will be used in Paper [II] for the formation of algebraic surgery. In §7.4 we use this to study Cat p∞ and Cat h∞ from the perspective of C -category theory as developed by Barwick and collaborators, and setup some of the foundations leading to the genuine C -refinement of the Grothendieck-Witt spectrum we willconstruct in Paper [II]. Finally, in §7.5 we show that the Grothendieck-Witt group and the L -groups are laxsymmetric monoidal functors with respect to the tensor product of Poincaré ∞ -categories. Acknowledgements.
For useful discussions about our project, we heartily thank Tobias Barthel, LukasBrantner, Mauricio Bustamante, Denis-Charles Cisinski, Dustin Clausen, Uriya First, Søren Galatius, RuneHaugseng, André Henriques, Lars Hesselholt, Gijs Heuts, Geoffroy Horel, Marc Hoyois, Max Karoubi,Daniel Kasprowski, Ben Knudsen, Manuel Krannich, Achim Krause, Henning Krause, Sander Kupers,Wolfgang Lück, Ib Madsen, Cary Malkiewich, Mike Mandell, Akhil Matthew, Lennart Meier, Irakli Patchko-ria, Nathan Perlmutter, Andrew Ranicki, Oscar Randal-Williams, George Raptis, Marco Schlichting, PeterScholze, Stefan Schwede, Graeme Segal, Markus Spitzweck, Jan Steinebrunner, Georg Tamme, UlrikeTillmann, Maria Yakerson, Michael Weiss, and Christoph Winges.Besides these discussions, we owe a tremendous intellectual debt to Jacob Lurie.The authors would also like to thank the Hausdorff Center for Mathematics at the University of Bonn,the Newton Institute at the University of Cambridge, the University of Copenhagen and the MathematicalResearch Institute Oberwolfach for hospitality and support while parts of this project were undertaken.BC was supported by the French National Centre for Scientific Research (CNRS) through a “délégation”at LAGA, University Paris 13. ED was supported by the German Research Foundation (DFG) through thepriority program “Homotopy theory and Algebraic Geometry” (DFG grant no. SPP 1786) at the Universityof Bonn and WS by the priority program “Geometry at Infinity” (DFG grant no. SPP 2026) at the Universityof Augsburg. YH and DN were supported by the French National Research Agency (ANR) through thegrant “Chromatic Homotopy and K-theory” (ANR grant no. 16-CE40-0003) at LAGA, University of Paris13. FH is a member of the Hausdorff Center for Mathematics at the University of Bonn (DFG grant no.EXC 2047 390685813) and TN of the cluster “Mathematics Münster: Dynamics-Geometry-Structure” atthe University of Münster (DFG grant no. EXC 2044 390685587). FH, TN and WS were further supportedby the Engineering and Physical Sciences Research Council (EPSRC) through the program “Homotopyharnessing higher structures” at the Isaac Newton Institute for Mathematical Sciences (EPSRC grants no.EP/K032208/1 and EP/R014604/1). FH was also supported by the European Research Council (ERC)through the grant “Moduli spaces, Manifolds and Arithmetic” (ERC grant no. 682922) and KM by thegrant “ K -theory, L -invariants, manifolds, groups and their interactions” (ERC grant no. 662400). ML and ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 9 DN were supported by the collaborative research centre “Higher Invariants” (DFG grant no. SFB 1085)at the University of Regensburg. ML was further supported by the research fellowship “New methods inalgebraic K -theory” (DFG grant no. 424239956) and by the Danish National Research Foundation (DNRF)through the Center for Symmetry and Deformation (DNRF grant no. 92) and the Copenhagen Centre forGeometry and Topology (DNRF grant GeoTop) at the University of Copenhagen. KM was also supportedby the K&A Wallenberg Foundation. 1. P OINCARÉ CATEGORIES
In this section we introduce the principal notion of this paper, namely that of
Poincaré ∞ -categories .These were first defined by Lurie in [Lur11], though no name was chosen there. Succinctly stated, Poincaré ∞ -categories are stable ∞ -categories C equipped with a quadratic functor Ϙ ∶ C → S 𝑝 to spectra, whichis perfect in a sense we will explain below. We will then refer to Ϙ as a Poincaré structure on C . It willbe convenient to consider also the more general setting where Ϙ is not necessarily perfect, leading to anotion that we will call a hermitian ∞ -category . We will present both of these in §1.2, after devoting §1.1to surveying quadratic functors and their basic properties. In §1.3 we will describe how one can classifyhermitian and Poincaré structures on a given stable ∞ -category in terms of their linear and bilinear parts.This is a particular case of the general structure theory of Goodwillie calculus, but we will take the time toelaborate the details relevant to the case at hand, as we will rely on this classification very frequently, both inexplicit constructions of examples and in general arguments. Finally, in §1.4 we will discuss the functorialdependence of hermitian structures on the underlying stable ∞ -category, and relate it to the classificationdiscussed in §1.3.1.1. Quadratic and bilinear functors.
In this subsection we will recall the notions of quadratic and bilin-ear functors, and survey their basic properties. These notions fit most naturally in the context of
Goodwilliecalculus , as adapted to the ∞ -categorical setting in [Lur17, §6]. Our scope of interest here specializes thatof loc. cit. in two ways: first, we will only consider the Goodwillie calculus up to degree , and second,we will focus our attention on functors from a stable ∞ -category C to the stable ∞ -category S 𝑝 of spectra.This highly simplifies the general theory, and will allow us to give direct arguments for most claims, insteadof quoting [Lur17, §6]. The reader should however keep in mind that the discussion below is simply aparticular case of Goodwillie calculus, to which we make no claim of originality.Recall that an ∞ -category C is said to be pointed if it admits an object which is both initial and terminal.Such objects are then called zero objects . A functor 𝑓 ∶ C → D between two pointed ∞ -categories is called reduced if it preserves zero objects. Given two pointed ∞ -categories C , D we will denote by Fun ∗ ( C , D ) ⊆ Fun( C , D ) the full subcategory spanned by the reduced functors. A stable ∞ -category is by definition apointed ∞ -category which admits pushouts and pullbacks and in which a square is a pushout square if andonly if it is a pullback square. To avoid breaking the symmetry one then refers to such squares as exact .A functor 𝑓 ∶ C → D between two stable ∞ -categories is called exact if it preserves zero objects andexact squares. We note that stable ∞ -categories automatically admit all finite limits and colimits, and thata functor between stable ∞ -categories is exact if and only if it preserves finite colimits, and if and only if itpreserves finite limits. If D is a stable ∞ -category and C ⊆ D is a full subcategory which is closed underfinite limits and finite colimits then C is also stable and the inclusion C ⊆ D is an exact functor. In thiscase we will say that C is a stable subcategory of D . Given two stable ∞ -categories C , D with C smallwe will denote by Fun ex ( C , D ) ⊆ Fun( C , D ) the full subcategory spanned by the exact functors. We notethat when C and D are stable one has that Fun( C , D ) is also stable and Fun ∗ ( C , D ) and Fun ex ( C , D ) arestable subcategories. We will denote by Cat ex∞ the (non-full) subcategory of
Cat ∞ spanned by the stable ∞ -categories and exact functors between them.If one considers stable ∞ -categories as a categorified version of a vector space, then reduced functorscorrespond to zero-preserving maps, while exact functors correspond to linear maps . If a functor 𝑓 ∶ C → D is only required to preserves exact squares, but is not necessarily reduced, then one says that 𝑓 is -excisive . More generally, if C is an ∞ -category with finite colimits and D and ∞ -category with finitelimits, then 𝑓 ∶ C → D is said to be -excisive if it sends pushout squares to pullback squares. In the aboveanalogy with linear algebra, these correspond to affine maps, that is, maps which contain a linear part anda constant term, or said differently: polynomial maps of degree 1. In the theory of Goodwillie calculus thispoint view is generalized to higher degrees as follows:
Definition. A -cube 𝜌 ∶ (Δ ) → C is said to be cartesian if it exhibits 𝜌 (0 , , as the limit ofthe restriction of 𝜌 to the subsimplicial set (Δ ) spanned by the complement of (0 , , . Such a -cube 𝜌 is called strongly cartesian if its restriction to each -dimensional face of (Δ ) is a cartesian square. Inparticular, strongly cartesian -cubes are cartesian. Dually, 𝜌 is said to be (strongly) cocartesian if 𝜌 op isa (strongly) cartesian cube in C op . A functor 𝑓 ∶ C → D whose domain admits finite colimits and whosetarget admits finite limits is called -excisive if it sends strongly cocartesian -cubes to cartesian -cubes.If C is stable then a -cube is (strongly) cartesian if and only if it is (strongly) cocartesian, in whichcase we simply say that 𝜌 is (strongly) exact. A functor 𝑓 ∶ C → D between stable ∞ -categories is then -excisive if it sends strongly exact -cubes to exact -cubes.1.1.2. Remark.
Though in the present paper we will focus almost entirely on the case of stable ∞ -categories,we chose to formulate the above definition in the slightly more general setting where 𝑓 ∶ C → D is a functorfrom an ∞ -category with finite colimits to an ∞ -category with finite limits. This level of generality, in whichmost of Goodwillie calculus can be carried out, will be used in §4.2, but will otherwise not be needed inthe present paper.We note that every -excisive functor is in particular -excisive. If the former are analogous to affine mapsbetween vector spaces, the latter are then analogous to maps between vector spaces which are polynomialof degree , that is, contain a homogeneous quadratic part, a linear part, and a constant term. If we restrictattention to -excisive functors which are reduced, then we get the analogue of maps with terms in degrees and , but no constant term. These are going to be the functors we consider in this paper.In the present work it will be convenient to take a slightly different route to the definition of reduced -excisive functors, which proceeds as follows. Given a small stable ∞ -category C , let us denote by BiFun( C ) ⊆ Fun ∗ ( C op × C op , S 𝑝 ) the full subcategory spanned by those reduced functors B ∶ C op × C op → S 𝑝 such that B( 𝑥, 𝑦 ) ≃ 0 if either 𝑥 or 𝑦 is a zero object. Such functors may be referred to as bi-reduced . Then BiFun( C ) is closed under all limits and colimits in Fun ∗ ( C op × C op , S 𝑝 ) , and hence the inclusion of the formerin the latter admits both a left and a right adjoint. These left and right adjoints are in fact canonically equiv-alent, and can be described by the following explicit formula: given a reduced functor B ∶ C op × C op → S 𝑝 we have a canonically associated retract diagram(4) B( 𝑥, ⊕ B(0 , 𝑦 ) → B( 𝑥, 𝑦 ) → B( 𝑥, ⊕ B(0 , 𝑦 ) , where C is a chosen zero object, and all the maps are induced by the essentially unique maps → 𝑥 → and → 𝑦 → . The composition of these two maps is the identity thanks to the assumption that B isreduced, that is, B(0 ,
0) ≃ 0 . The above retract diagram then induces a canonical splitting B( 𝑥, 𝑦 ) ≃ B red ( 𝑥, 𝑦 ) ⊕ B( 𝑥, ⊕ B(0 , 𝑦 ) , where B red ( 𝑥, 𝑦 ) can be identified with both the cofibre of the left map in (4) and the fibre of the right mapin (4). We note that by construction the resulting functor B red (− , −) ∶ C op × C op → S 𝑝 is bi-reduced. The following lemma records the fact that the association B ↦ B red yields both a left and aright adjoint to the inclusion BiFun( C ) ⊆ Fun ∗ ( C op × C op , S 𝑝 ) .1.1.3. Lemma.
The split inclusion B red (− , −) ⇒ B(− , −) is universal among natural transformations to B from a bi-reduced functor, while the projection B(− , −) ⇒ B red (− , −) is universal among natural trans-formations from B to a bi-reduced functor. In particular, the association B ↦ B red is both left and rightadjoint to the full inclusion BiFun( C ) ⊆ Fun ∗ ( C op × C op , S 𝑝 ) .Proof. Given that
Fun ∗ ( C op × C op , S 𝑝 ) is stable and BiFun( C ) is a stable full subcategory, to prove bothclaims it suffices to show that for B ∈ Fun ∗ ( C op × C op , S 𝑝 ) , the associated functors ( 𝑥, 𝑦 ) ↦ 𝐵 ( 𝑥, 𝑎𝑛𝑑 ( 𝑥, 𝑦 ) ↦ 𝐵 (0 , 𝑦 ) considered as functors in Fun ∗ ( C op × C op , S 𝑝 ) have a trivial mapping spectrum to any and from any bi-reducedfunctor. Indeed, since C op is both final and initial it follows that the inclusion C op × {0} ⊆ C op × C op isboth left and right adjoint to the projection C op × C op → C op ×{0} , and hence restricting along this inclusionis both left and right adjoint to restricting along this projection. The same statement holds for the inclusion ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 11 {0} × C op ⊆ C op × C op of the second factor. The mapping spectrum between any bi-reduced functor and afunctor restricted along either projection is consequently trivial. (cid:3) Definition.
Let C be a stable ∞ -category and Ϙ ∶ C op → S 𝑝 a reduced functor. We will denote by B Ϙ ∈ BiFun( C ) the functor B Ϙ (− , −) ∶= Ϙ ((−) ⊕ (−)) red ∶ C op × C op → S 𝑝 obtained by taking the universal bi-reduced replacement described above of the reduced functor ( 𝑥, 𝑦 ) ↦ Ϙ ( 𝑥 ⊕ 𝑦 ) . Following the terminology of Goodwillie calculus we will refer to B Ϙ (− , −) as the cross effect of Ϙ . The formation of cross effects then yields a functor(5) B (−) ∶ Fun ∗ ( C op , S 𝑝 ) → BiFun( C ) sending Ϙ to B Ϙ .1.1.5. Remark.
In [Lur11] the term polarization is used for what we called above cross effect, thoughin [Lur17, §6] the term cross effect is employed.1.1.6.
Remark. If 𝑓 , 𝑔 ∶ C → D are reduced functors then the associated restriction functor ( 𝑓 × 𝑔 ) ∗ ∶ Fun ∗ ( D op × D op , S 𝑝 ) → Fun ∗ ( C op × C op , S 𝑝 ) along ( 𝑓 × 𝑔 ) op ∶ C op × C op → D op × D op sends the retract diagram B( 𝑥, ⊕ B(0 , 𝑦 ) → B( 𝑥, 𝑦 ) → B( 𝑥, ⊕ B(0 , 𝑦 ) to the retract diagram B( 𝑓 ( 𝑥 ) , ⊕ B(0 , 𝑔 ( 𝑦 )) → B( 𝑓 ( 𝑥 ) , 𝑔 ( 𝑦 )) → B( 𝑓 ( 𝑥 ) , ⊕ B(0 , 𝑔 ( 𝑦 )) , where we have used the symbols 𝑥 and 𝑦 to distinguish the two entries. It then follows that the universal bi-reduction procedure described above commutes with restriction (along pairs of reduced functors). Similarly,if 𝑓 ∶ C → D furthermore preserves direct sums, then the formation of cross effects is compatible withrestriction along 𝑓 , that is, the square Fun ∗ ( D op , S 𝑝 ) Fun ∗ ( C op , S 𝑝 )BiFun( D ) BiFun( C ) 𝑓 ∗ ( 𝑓 × 𝑓 ) ∗ naturally commutes.Given a stable ∞ -category C , the diagonal functor Δ ∶ C op → C op × C op induces a pullback functor Δ ∗ ∶ BiFun( C ) → Fun ∗ ( C op , S 𝑝 ) . In what follows, for any
B ∶ C op × C op → S 𝑝 , we will denote by B Δ ∶= Δ ∗ B the restriction of B along thediagonal. Now the maps Ϙ ( 𝑥 ⊕ 𝑥 ) → Ϙ ( 𝑥 ) and Ϙ ( 𝑥 ) → Ϙ ( 𝑥 ⊕ 𝑥 ) induced by the diagonal Δ 𝑥 ∶ 𝑥 → 𝑥 ⊕ 𝑥 and collapse map ∇ 𝑥 ∶ 𝑥 ⊕ 𝑥 → 𝑥 induce natural maps(6) B Ϙ ( 𝑥, 𝑥 ) → Ϙ ( 𝑥 ) → B Ϙ ( 𝑥, 𝑥 ) , which can be considered as natural transformations(7) B Δ Ϙ ⇒ Ϙ ⇒ B Δ Ϙ The formation of cross effects then enjoys the following universal property:1.1.7.
Lemma.
The two natural transformations in (7) act as a unit and counit exhibiting the cross effectfunctor (5) as left and right adjoint respectively to the restriction functor Δ ∗ ∶ BiFun( C ) → Fun ∗ ( C op , S 𝑝 ) .Proof. The direct sum functor C op × C op → C op realizes both the product and coproduct (since C op is stable)and is hence both left and right adjoint to Δ ∶ C op → C op × C op , with units an counits given by the diagonaland collapse maps of the objects in C . It then follows that restriction along the direct sum functor is bothright and left adjoint to restriction along Δ , with unit and counit induced by the diagonal and collapse maps.The desired result now follows from Lemma 1.1.3. (cid:3) Remark.
The two sided adjunction of Lemma 1.1.7 is obtained by composing a pair of two-sidedadjunctions
Fun ∗ ( C op , S 𝑝 ) ⇆ Fun ∗ ( C op × C op , S 𝑝 ) ⇆ BiFun( C ) , where the one on the left is induced by the two sided adjunction C op Δ ⇆ ⊕ C op × C op witnessing the existenceof biproducts in C op , and the one of the right exhibits the full subcategory BiFun( C ) ⊆ Fun ∗ ( C op × C op , S 𝑝 ) as reflective and coreflective (Lemma 1.1.3). In particular, we may express those unit and counit of the twosided adjunction Fun ∗ ( C op , S 𝑝 ) ⇆ BiFun( C ) which are not specified in Lemma 1.1.7 via the unit ( 𝑥, 𝑦 ) → ( 𝑥 ⊕ 𝑦, 𝑥 ⊕ 𝑦 ) of the adjunction C op × C op ⟂ C and counit ( 𝑥 ⊕ 𝑦, 𝑥 ⊕ 𝑦 ) → ( 𝑥, 𝑦 ) of the adjunction C op ⟂ C op × C op , which are all given by the corresponding component inclusions and projections. Unwinding thedefinitions, we get that the unit of the adjunction BiFun( C ) ⟂ Fun ∗ ( C op , S 𝑝 ) is given by the induced map B( 𝑥, 𝑦 ) → f ib[B( 𝑥 ⊕ 𝑦, 𝑥 ⊕ 𝑦 ) → B( 𝑥, 𝑥 ) ⊕ B( 𝑦, 𝑦 )] and the counit of the adjunction Fun ∗ ( C op , S 𝑝 ) ⟂ BiFun( C ) is given by the induced map cof [B( 𝑥, 𝑥 ) ⊕ B( 𝑦, 𝑦 ) → B( 𝑥 ⊕ 𝑦, 𝑥 ⊕ 𝑦 )] → B( 𝑥, 𝑦 ) . Lemma.
Let Ϙ ∶ C op → S 𝑝 be a reduced functor. Then the cross effect B Ϙ is symmetric, i.e. it canon-ically refines to an element of Fun( C op × C op , S 𝑝 ) hC , where the cyclic group with two elements C acts byflipping the two input variables.Proof. By [Lur17, Proposition 6.1.4.3, Remark 6.1.4.4] the bi-reduction functor (−) red ∶ Fun ∗ ( C op × C op , S 𝑝 ) → BiFun( C ) discussed above refines to a compatible functor Fun ∗ ( C op × C op , S 𝑝 ) hC → BiFun( C ) hC on C -equivariant objects. It will hence suffice to show that the functor ( 𝑥, 𝑦 ) ↦ Ϙ ( 𝑥⊕𝑦 ) naturally refines toa C -equivariant object. For this, it suffices to note that the direct sum functor C op × C op → C op is equippedwith a C -equivariant structure with respect to the flip action on C op × C op and the trivial action on C op .Indeed, this is part of the symmetric monoidal structure afforded to the direct sum, canonically determinedby its universal description as the coproduct in C op . (cid:3) Keeping in mind the proofs of Lemma 1.1.7 and Lemma 1.1.9, we now note that the diagonal functor
Δ ∶ C op → C op × C op , which is both left and right adjoint to the C -equivariant direct sum functor, is alsocanonically invariant under the C -action on the right hand side switching the two components. This meansthat the associated restriction functor Δ ∗ ∶ BiFun( C ) → Fun ∗ ( C op , S 𝑝 ) is equivariant for the trivial C -action on the target, and so the restricted functor B Δ Ϙ = Δ ∗ B Ϙ becomes a C -object of Fun( C op , S 𝑝 ) . In particular, B Ϙ ( 𝑥, 𝑥 ) is naturally a spectrum with a C -action for every 𝑥 ∈ C .Explicitly, this action is induced by the canonical action of C on 𝑥 ⊕ 𝑥 by swapping the components.1.1.10. Lemma.
The natural transformations in (7) both naturally refine to C -equivariant maps with re-spect to the above C -action on B Δ . In particular, the maps (7) induces natural transformations (8) [B Δ Ϙ ] hC ⇒ Ϙ ⇒ [B Δ Ϙ ] hC . Proof.
Inspecting the construction of the natural transformations in (7) we see that it will suffice to put a C -equivariant structure on the diagonal and collapse natural transformations Δ ∶ id ⇒ id ⊕ id and ∇ ∶ id ⊕ id ⇒ id of functors C → C . This in turn follows from the fact that the direct sum monoidal structure is both cartesianand cocartesian and every object is canonically a commutative algebra object with respect to coproducts([Lur17, Proposition 2.4.3.8]). (cid:3) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 13 Definition.
For C , D and E stable ∞ -categories, we will say that a functor 𝑏 ∶ C × D → E is bilinear if it is exact in each variable separately. For a stable ∞ -category C we will denote by Fun b ( C ) ⊆ Fun( C op × C op , S 𝑝 ) the full subcategory spanned by the bilinear functors. We note that this full subcategory is closedunder, and hence inherits, the flip action of C . We will then denote by Fun s ( C ) ∶= [Fun b ( C )] hC the ∞ -category of C -equivariant objects in Fun b ( C ) with respect to the flip action in the entries, and refer to themas symmetric bilinear functors on C .1.1.12. Example.
Suppose that C is a stable ∞ -category equipped with a monoidal structure which is exactin each variable separately. Then for every object 𝑎 ∈ C we have an associated bilinear functor B 𝑎 ∶ C op × C op → S 𝑝 defined by B 𝑎 ( 𝑥, 𝑦 ) ∶= hom C ( 𝑥 ⊗ 𝑦, 𝑎 ) . If the monoidal structure refines to a symmetric one then B 𝑎 refines to a symmetric bilinear functor. Naturalexamples of interest to keep in mind are when C is the perfect derived category of a commutative ring (or,more generally, an E ∞ -ring spectrum), or the ∞ -category of perfect quasi-coherent sheaves on a scheme.1.1.13. Proposition.
Let Ϙ ∶ C op → S 𝑝 be a functor. Then the following are equivalent:i) Ϙ is reduced and -excisive;ii) the cross effect B Ϙ is bilinear and the fibre of the natural transformation Ϙ ( 𝑥 ) → B Ϙ ( 𝑥, 𝑥 ) hC from (8) is an exact functor in 𝑥 ;iii) the cross effect B Ϙ is bilinear and the cofibre of the natural transformation B Ϙ ( 𝑥, 𝑥 ) hC → Ϙ ( 𝑥 ) from (8) is an exact functor in 𝑥 .Proof. Since S 𝑝 is stable the property of being reduced and -excisive is preserved under limits and colimitsof functors C op → S 𝑝 . It then follows that both ii) and iii) imply i), since exact functors and diagonalrestrictions of bilinear functors are in particular reduced and -excisive (see [Lur17, Cor. 6.1.3.5]).In the other direction, if Ϙ is -excisive then its cross effect is bilinear by [Lur17, Pr. 6.1.3.22]. Moreover,since taking the cross effect commutes with fibres and cofibres, the functors in the statement of ii) and iii)have trivial cross effect. But they are also reduced and -excisive by the first part of the argument, and arehence exact by [Lur17, Pr. 6.1.4.10]. (cid:3) Definition.
We will say that Ϙ ∶ C op → S 𝑝 is quadratic if it satisfies the equivalent conditions ofProposition 1.1.13. For a small stable ∞ -category C we will then denote by Fun q ( C ) ⊆ Fun( C op , S 𝑝 ) thefull subcategory spanned by the quadratic functors.1.1.15. Remark.
It follows from the first criterion in Proposition 1.1.13 that
Fun q ( C ) is closed under limitsand colimits in Fun( C op , S 𝑝 ) . Since the latter is stable it follows that Fun q ( C ) is stable as well.In light of Lemma 1.1.9 and Proposition 1.1.13, the cross effect functor refines to a functor B (−) ∶ Fun q ( C ) ⟶ Fun s ( C ) . We will then refer to B Ϙ ∈ Fun s ( C ) as the symmetric bilinear part of Ϙ ∈ Fun q ( C ) , and refer to the underlyingbilinear functor of B Ϙ as the bilinear part of Ϙ .1.1.16. Examples. i) Any exact functor C op → S 𝑝 is quadratic. These are exactly the quadratic functors whose bilinearpart vanishes. In particular, we have an exact full inclusion of stable ∞ -categories Fun ex ( C op , S 𝑝 ) ⊆ Fun q ( C ) .ii) If B ∶ C × C op → S 𝑝 is a bilinear functor then the functor B Δ ( 𝑥 ) = B( 𝑥, 𝑥 ) is a quadratic functor([Lur17, Cor. 6.1.3.5]). Its symmetric bilinear part is given by the symmetrization ( 𝑥, 𝑦 ) ↦ B( 𝑥, 𝑦 ) ⊕ B( 𝑦, 𝑥 ) of B , equipped with its canonical symmetric structure.1.1.17. Example. If B ∈ Fun s ( C ) is a symmetric bilinear functor then the functors Ϙ qB ( 𝑥 ) ∶= B ΔhC ( 𝑥 ) = B( 𝑥, 𝑥 ) hC and Ϙ sB ( 𝑥 ) ∶= (B Δ ) hC ( 𝑥 ) = B( 𝑥, 𝑥 ) hC are both quadratic functors. Indeed, this follows from the previous example by noting that the symmetryinduces a C -action on B Δ and invoking Remark 1.1.15. Since taking cross-effects commutes with all limitsand colimits the symmetric bilinear parts of these functors are given respectively by [B( 𝑥, 𝑦 ) ⊕ B( 𝑦, 𝑥 )] hC and [B( 𝑥, 𝑦 ) ⊕ B( 𝑦, 𝑥 )] hC , which are both canonically equivalent to B itself: indeed, when B is symmetric its symmetrization canon-ically identifies with B[C ] = B ⊕ B as a C -object in Fun s ( C ) , which, since the latter is stable, is the C -object both induced and coinduced from B .The superscript (−) q and (−) s above refer to the relation between these constructions and the notionsof quadratic and symmetric forms in algebra. To see this, consider the case where C ∶= D p ( 𝑅 ) is theperfect derived category of a commutative ring 𝑅 . We then have a natural choice of a bilinear functor B 𝑅 ∶ C op × C op → S 𝑝 given by B 𝑅 ( 𝑋, 𝑌 ) = hom 𝑅 ( 𝑋 ⊗ 𝑅 𝑌 , 𝑅 ) where ⊗ 𝑅 denotes the (derived) tensor product over 𝑅 . A point 𝛽 ∈ Ω ∞ B( 𝑋, 𝑌 ) then corresponds to a map 𝑋 ⊗ 𝑅 𝑌 → 𝑅 , which we can consider as a bilinear form on the pair ( 𝑋, 𝑌 ) . If 𝑋, 𝑌 are ordinary projectivemodules then 𝜋 B 𝑅 ( 𝑋, 𝑌 ) is simply the abelian group of bilinear forms on ( 𝑋, 𝑌 ) in the ordinary sense.For a projective 𝑅 -module 𝑋 we may then identify the C -fixed subgroup 𝜋 B 𝑅 ( 𝑋, 𝑋 ) C with the groupof symmetric bilinear forms on 𝑋 , while the C -quotient group 𝜋 B 𝑅 ( 𝑋, 𝑋 ) C can be identified with thegroup of quadratic forms on 𝑋 via the map sending the orbit of bilinear form 𝑏 ∶ 𝑋 ⊗ 𝑅 𝑋 → 𝑅 to thequadratic form 𝑞 𝑏 ( 𝑥 ) = 𝑏 ( 𝑥, 𝑥 ) . In this case the quadratic functors Ϙ q 𝑅 ∶= Ϙ qB 𝑅 and Ϙ s 𝑅 ∶= Ϙ qB 𝑅 definedas above can be considered as associating to a perfect 𝑅 -complex 𝑋 a suitable spectrum of quadratic andsymmetric forms on 𝑋 , respectively.1.1.18. Remark.
By definition the cross effect of a quadratic functor is bilinear, and on the other handby Example 1.1.16ii) the diagonal restriction of any bilinear functor is quadratic. It then follows fromLemma 1.1.7 that diagonal restriction Δ ∗ ∶ Fun b ( C ) → Fun q ( C ) determines a two-sided adjoint to thebilinear part functor B (−) ∶ Fun q ( C ) → Fun b ( C ) , with unit and counit given by the natural maps B Ϙ ( 𝑥, 𝑥 ) ⇒ Ϙ ( 𝑥 ) ⇒ B Ϙ ( 𝑥, 𝑥 ) . By Remark 1.1.8 the other unit and counit are given by the component inclusion and projections B( 𝑥, 𝑦 ) ⇒ B( 𝑥, 𝑦 ) ⊕ B( 𝑦, 𝑥 ) ⇒ B( 𝑥, 𝑦 ) As quadratic functors are only 2-excisive, but not 1-excisive, they generally don’t preserve exact squares.Their failure to preserve exact squares is however completely controlled by the associated symmetric bilinearparts. More precisely, we have the following:1.1.19.
Lemma.
Let Ϙ ∶ C op → S 𝑝 be a quadratic functor with bilinear part B = B Ϙ and let (9) 𝑥 𝑦𝑧 𝑤 𝛼 ′ 𝛽 ′ 𝛽𝛼 be an exact square in C . Then in the diagram (10) Ϙ ( 𝑤 ) B( 𝑧, 𝑦 ) B(cof( 𝛽 ′ ) , cof( 𝛼 ′ )) Ϙ ( 𝑧 ) × Ϙ ( 𝑥 ) Ϙ ( 𝑦 ) B( 𝑧, 𝑥 ) × B( 𝑥,𝑥 ) B( 𝑥, 𝑦 ) 0 both squares are exact. In particular, there is a natural equivalence cof[ Ϙ ( 𝑤 ) → Ϙ ( 𝑧 )× Ϙ ( 𝑥 ) Ϙ ( 𝑦 )] ≃ cof [B( 𝑧, 𝑦 ) → B( 𝑧, 𝑥 )× B( 𝑥,𝑥 ) B( 𝑥, 𝑦 )] ≃ ΣB(cof( 𝛽 ′ ) , cof( 𝛼 ′ )) ≃ B(f ib( 𝛽 ′ ) , cof( 𝛼 ′ )) . Proof.
Consider the following pair of maps between commutative squares(11) Ϙ ( 𝑤 ) Ϙ ( 𝑦 ) Ϙ ( 𝑧 ) Ϙ ( 𝑥 ) ⇒ Ϙ ( 𝑧 ⊕ 𝑦 ) Ϙ ( 𝑥 ⊕ 𝑦 ) Ϙ ( 𝑧 ⊕ 𝑥 ) Ϙ ( 𝑥 ⊕ 𝑥 ) ⇒ B( 𝑧, 𝑦 ) B( 𝑥, 𝑦 )B( 𝑧, 𝑥 ) B( 𝑥, 𝑥 ) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 15 where the left one is induced by the strongly cocartesian cube(12) 𝑥 ⊕ 𝑥 𝑥 ⊕ 𝑦𝑥 𝑦𝑧 ⊕ 𝑥 𝑧 ⊕ 𝑦.𝑧 𝑤 Here, the map 𝑥 ⊕ 𝑥 → 𝑥 is the collapse map, the map 𝑧 ⊕ 𝑦 → 𝑤 is the one whose components are 𝛼 and 𝛽 , and the maps 𝑥 ⊕ 𝑦 → 𝑦 and 𝑥 ⊕ 𝑧 → 𝑧 have one component the identity and one component 𝛼 ′ or 𝛽 ′ , respectively. Since Ϙ is quadratic it is in particular -excisive by the first characterization in Proposi-tion 1.1.13, and so Ϙ maps (12) to a cartesian cube of spectra. This means that the first map in (11) inducesan equivalence on total fibres. On the other hand, the second map in (11) also induces an equivalence ontotal fibres since its cofibre is the square Ϙ ( 𝑧 ) ⊕ Ϙ ( 𝑦 ) Ϙ ( 𝑥 ) ⊕ Ϙ ( 𝑦 ) Ϙ ( 𝑧 ) ⊕ Ϙ ( 𝑥 ) Ϙ ( 𝑥 ) ⊕ Ϙ ( 𝑥 ) whose total fibre is trivial. We then deduce that the composite of the two maps in (11) induces an equivalenceon total fibres, and hence the left square in (10) is exact. Finally, the right square in (10) is exact because B(− , −) is exact in each variable separately and hence the the total fibre of the right most square in (11)identifies with B(cof ( 𝛽 ′ ) , cof( 𝛼 ′ )) via the natural map B(cof ( 𝛽 ′ ) , cof( 𝛼 ′ )) → B( 𝑧, 𝑦 ) . (cid:3) Remark.
Lemma 1.1.19 admits a natural dual variant. Given a quadratic functor Ϙ ∶ C op → S 𝑝 withbilinear part B = B Ϙ and an exact square as in (9), one may form instead the diagram(13) Ϙ ( 𝑧 ) ⊕ Ϙ ( 𝑤 ) Ϙ ( 𝑦 ) B( 𝑧, 𝑤 ) ⊕ B( 𝑤,𝑤 ) B( 𝑤, 𝑦 ) 0 Ϙ ( 𝑥 ) B( 𝑧, 𝑦 ) B(f ib( 𝛼 ) , f ib( 𝛽 )) obtained using the maps on the left hand side of (6) instead of the right. The dual of the argument in theproof of Lemma 1.1.19 then shows that (13) consists of two exact squares, yielding a natural equivalence f ib[ Ϙ ( 𝑧 ) ⊕ Ϙ ( 𝑤 ) Ϙ ( 𝑦 ) → Ϙ ( 𝑥 )] ≃ f ib[B( 𝑧, 𝑤 ) ⊕ B( 𝑤,𝑤 ) B( 𝑤, 𝑦 ) → B( 𝑧, 𝑦 )] ≃ ΩB(f ib( 𝛼 ) , f ib( 𝛽 )) ≃ B(cof( 𝛼 ) , f ib( 𝛽 )) . Applying Lemma 1.1.19 in the case where 𝑧 = 0 we obtain:1.1.21. Corollary (cf. [Lur11, Lecture 9, Theorem 5]) . For an exact sequence 𝑥 → 𝑦 → 𝑤 in C , the naturalmap Ϙ ( 𝑤 ) totf ib ⎡⎢⎢⎢⎢⎣ Ϙ ( 𝑦 ) Ϙ ( 𝑥 )B Ϙ ( 𝑥, 𝑦 ) B Ϙ ( 𝑥, 𝑥 ) ⎤⎥⎥⎥⎥⎦ from Ϙ ( 𝑤 ) to the total fibre of the square on the right, is an equivalence. Definition.
For a quadratic functor Ϙ we will denote by L Ϙ ∶ C op → S 𝑝 the cofibre of the naturaltransformation (B Δ Ϙ ) hC ⇒ Ϙ , which is exact by Proposition 1.1.13, and refer to it as the linear part of Ϙ . Byconstruction, the linear part L Ϙ sits in an exact sequence(14) B Ϙ ( 𝑥, 𝑥 ) hC → Ϙ ( 𝑥 ) → L Ϙ ( 𝑥 ) . The formation of linear parts can be organized into a functor(15) L (−) ∶ Fun q ( C ) ⟶ Fun ex ( C op , S 𝑝 ) whose post-composition with the inclusion Fun ex ( C op , S 𝑝 ) ⊆ Fun q ( C ) carries a natural transformation fromthe identity Ϙ ⇒ L Ϙ , corresponding to the second arrow in (14). Remark.
It follows from Remark 1.1.6 that the formation of linear parts naturally commutes withrestriction along an exact functor 𝑓 ∶ C → D .1.1.24. Lemma.
The natural map Ϙ ⇒ L Ϙ is a unit exhibiting L (−) as left adjoint to the inclusion Fun ex ( C op , S 𝑝 ) ⊆ Fun q ( C ) .Proof. Since
Fun ex ( C op , S 𝑝 ) ⊆ Fun q ( C ) is a full inclusion it will suffice to show that Ϙ ⇒ L Ϙ induces anequivalence on mapping spectra to every exact functor. Since Fun q ( C ) is stable this is the same as sayingthat the fibre of Ϙ ⇒ L Q maps trivially to any exact functor. This fibre is [B Δ Ϙ ] hC by construction, andso it will hence suffice to show that B Δ Ϙ maps trivially to any exact functor. Indeed, this follows from theadjunction of Remark 1.1.18 since the bilinear part of every linear functor vanishes. (cid:3) Let us also remark that equivalences of quadratic functors can be detected on their connective covers.1.1.25.
Lemma.
Let C be a stable ∞ -category and Ϙ ∶ C op → S 𝑝 be a quadratic functor. Suppose thatfor every 𝑥 ∈ C the spectrum Ϙ ( 𝑥 ) is coconnective. Then Ϙ is the zero functor. In particular, if a naturaltransformation of quadratic functors 𝑓 ∶ Ϙ → Ϙ ′ is an equivalence after applying Ω ∞ , then it is itself anequivalence.Proof. First suppose that Ϙ is exact. Then, for every 𝑥 ∈ C and 𝑛 ∈ ℤ , 𝜋 𝑛 Ϙ ( 𝑥 ) = 𝜋 Ϙ (Σ 𝑛 −1 𝑥 ) = 0 , and so Ϙ = 0 .Let us now prove the general case. For every 𝑥, 𝑦 ∈ C , the spectrum B Ϙ ( 𝑥, 𝑦 ) is a direct summand of Ϙ ( 𝑥 ⊕ 𝑦 ) . In particular it is also coconnective. Hence, if we fix an 𝑥 ∈ C , then B Ϙ ( 𝑥, −) ∶ C op → S 𝑝 is anexact functor taking values in coconnective spectra, and therefore the zero functor by the previous argument.The cross-effect B Ϙ is therefore the zero functor. In particular, Ϙ is exact, and is hence the zero functor bythe same argument.The final statement follows by applying the previous argument to the fibre of 𝑓 . (cid:3) We finish this subsection with a discussion of the left and right adjoints to the inclusion of quadraticfunctors inside reduced functors.1.1.26.
Construction.
Let E be a stable ∞ -category. Given a quadratic functor Ϙ ∶ E op → S 𝑝 , Lemma 1.1.19applied in the case where both 𝑧 and 𝑤 are zero objects implies that the sequence(16) Ϙ ( 𝑤 ) → Ω Ϙ (Ω 𝑤 ) → ΩB Ϙ (Ω 𝑤, Ω 𝑤 ) is exact, and hence that the natural map Ϙ ( 𝑤 ) ≃ ←←←←←←←→ Ω f ib[ Ϙ (Ω 𝑤 ) → B Ϙ (Ω 𝑤, Ω 𝑤 )] is an equivalence. This map itself is however defined for any reduced Ϙ , and is natural in Ϙ . In particular,given a stable ∞ -category we may define a functor T E ∶ Fun ∗ ( E op , S 𝑝 ) → Fun ∗ ( E op , S 𝑝 ) which sends a reduced functor R ∶ E op → S 𝑝 to the reduced functor(17) T E ( R ) ∶= Ω f ib[ R (Ω 𝑤 ) → B R (Ω 𝑤, Ω 𝑤 )] . The operation T E is equipped with a natural map 𝜃 R ∶ R ⇒ T E ( R ) which is an equivalence when R is quadratic by Lemma 1.1.19. Unwinding the definitions, we see thatthe association R ↦ T E ( R ) identifies with the one defined in [Lur17, Construction 6.1.1.22] for C = E op and D = S 𝑝 . Since S 𝑝 is stable and admits small colimits it is in particular differentiable in the senseof [Lur17, Definition 6.1.1.6]. By [Lur17, Theorem 6.1.1.10] we may then conclude that the association P ( R ) ∶= colim[ R 𝜃 𝑅 ←←←←←←←←←←←→ T ( R ) 𝜃 T E R ) ←←←←←←←←←←←←←←←←←←←←←←←←←←→ T E T E ( R ) → ⋯ ] gives a left adjoint to the inclusion Fun q ( E ) ⊆ Fun ∗ ( E , S 𝑝 ) . This procedure is often referred to as -excisiveapproximation . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 17 In a dual manner, if we use Remark 1.1.20 instead of Lemma 1.1.19 then we get that for a quadraticfunctor Ϙ the sequence(18) ΣB Ϙ (Σ 𝑤, Σ 𝑤 ) → Σ Ϙ (Σ 𝑤 ) → Ϙ ( 𝑤 ) is exact, and so the natural map Σ cof[B Ϙ (Σ 𝑤, Σ 𝑤 ) → Ϙ (Σ 𝑤 )] ≃ ←←←←←←←→ Ϙ ( 𝑤 ) is an equivalence. As above, for a general reduced functor R we can define the functor(19) T E ( R ) = Σ cof[B R (Σ 𝑤, Σ 𝑤 ) → R (Σ 𝑤 )] , equipped with a natural map 𝜏 R ∶ T E ( R ) ⇒ R , which is an equivalence when R is quadratic. We may also identify T E with the result of [Lur17, Construc-tion 6.1.1.22] applied to C = E and D = S 𝑝 op . Since S 𝑝 op is also differentiable by the same argument itfollows from [Lur17, Theorem 6.1.1.10] that the association P ( R ) ∶= lim[ ⋯ → T E T E ( R ) 𝜏 T2 E ( R ) ←←←←←←←←←←←←←←←←←←←←←←←←←→ T E ( R ) 𝜏 R ←←←←←←←←←←←←→ R ] provides a right adjoint to the inclusion Fun q ( E ) ⊆ Fun ∗ ( E , S 𝑝 ) .1.2. Hermitian and Poincaré ∞ -categories. In this subsection we introduce the key player in this paper- the notion of a
Poincaré ∞ -category . For this, it will be convenient to pass first through the followingweaker notion:1.2.1. Definition. A hermitian ∞ -category is a pair ( C , Ϙ ) where C is a small stable ∞ -category and Ϙ ∶ C op → S 𝑝 is a quadratic functor in the sense of Definition 1.1.14. We will then also refer to Ϙ as a hermitian struc-ture on C . The collection of hermitian ∞ -categories can be organized into a (large) ∞ -category Cat h∞ ,obtained as the cartesian Grothendieck construction of the functor (Cat ex∞ ) op ⟶ CAT ∞ , C ⟼ Fun q ( C ) . (here CAT ∞ stands for the ∞ -category of possibly large ∞ -categories). We shall also refer to its morphismsas hermitian functors .Unpacking this definition, we find that a hermitian functor from ( C , Ϙ ) to ( C ′ , Ϙ ′ ) consists of an exactfunctor 𝑓 ∶ C → C ′ and a natural transformation 𝜂 ∶ Ϙ ⇒ 𝑓 ∗ Ϙ ′ ∶= Ϙ ′ ◦ 𝑓 op . We will thus generally denotehermitian functors as pairs ( 𝑓 , 𝜂 ) of this form. If ( 𝑓 , 𝜂 ) ∶ ( C , Ϙ ) → ( C ′ , Ϙ ′ ) is a hermitian functor then by Re-mark 1.1.6 we have a natural equivalence ( 𝑓 × 𝑓 ) ∗ B Ϙ ′ ≃ B 𝑓 ∗ Ϙ ′ , and consequently the natural transformation 𝜂 determines a natural transformation(20) 𝛽 𝜂 ∶ B Ϙ ⇒ ( 𝑓 × 𝑓 ) ∗ B Ϙ ′ , which we then denote by 𝛽 𝜂 .The notion of Poincaré ∞ -category is obtained from that of a hermitian ∞ -category ( C , Ϙ ) by requiring Ϙ to satisfy two non-degeneracy conditions. Both of these conditions depend only on the underlying sym-metric bilinear part B Ϙ ∈ Fun s ( C ) . To formulate the first one we first note that the exponential equivalence Fun( C op × C op , S 𝑝 ) ≃ ←←←←←←←→ Fun( C op , Fun( C op , S 𝑝 )) restricts to an equivalence(21) Fun b ( C ) ≃ ←←←←←←←→ Fun ex ( C op , Fun ex ( C op , S 𝑝 )) . We then consider the following condition:1.2.2.
Definition.
We will say that a bilinear functor
B ∈ Fun b ( C ) is right non-degenerate if the associatedexact functor(22) C op → Fun ex ( C op , S 𝑝 ) 𝑦 ↦ B(− , 𝑦 ) takes values in the essential image of the stable Yoneda embedding C ↪ Fun lex ( C op , S ) ≃ Fun ex ( C op , S 𝑝 ) , where Fun lex denotes left exact (that is, finite limit preserving) functors, and the equivalence to the lastterm is by [Lur17, Corollary 1.4.2.23]. In other words, if for each 𝑦 ∈ C the presheaf of spectra B(− , 𝑦 ) isrepresentable by an object in C . In this case we can factor (22) essentially uniquely as a functor D B ∶ C op → C followed by C ↪ Fun ex ( C op , S 𝑝 ) , so that we obtain an equivalence B( 𝑥, 𝑦 ) ≃ hom C ( 𝑥, D 𝑦 ) . Similarly,
B ∈ Fun b ( C ) is called left non-degenerate if the associated exact functor 𝑥 ↦ B( 𝑥, −) takes valuesin the essential image of the stable Yoneda embedding. If B ∈ Fun b ( C ) is left and right non-degenerate,then it is called non-degenerate. In this case the two resulting dualities are, essentially by definition, adjointto each other.We will say that a symmetric bilinear functor is non-degenerate if the underlying bilinear functor is.In this case it of course suffices to check that it is right non-degenerate. The two dualities are in this caseequivalent and we will refer to the representing functor D B as the duality associated to the non-degeneratesymmetric bilinear functor B (though we point out that D B is not in general an equivalence). Given ahermitian structure Ϙ on a stable ∞ -category C , we will say that Ϙ is non-degenerate if its underlying bilinearpart is. In this case we will also say that ( C , Ϙ ) is a non-degenerate hermitian ∞ -category and will denotethe associated duality by D Ϙ .The full subcategories of Fun b ( C ) , Fun s ( C ) and Fun q ( C ) spanned be the non-degenerate functors willbe denoted Fun nb ( C ) , Fun ns ( C ) and Fun nq ( C ) , respectively. The bilinear exponential equivalence (21) thenrestricts to an equivalence Fun nb ( C ) ≃ ←←←←←←←→ Fun R ( C op , C ) , where Fun R denotes the right adjoint functors. To see this it suffices to observe that B ∈ Fun b ( C ) is non-degenerate precisely if it is right non-degenerate and the resulting duality admits a left adjoint. Under thisequivalence the 𝐶 -action on the left correpsonds to the 𝐶 -action on the right given by passing to theadjoint, so that we get an equivalence Fun ns ( C ) ≃ ←←←←←←←→ Fun R ( C op , C ) hC . Both of these equivalences will be denote by B ↦ D B . Similarly, we will also denote the composition Fun nq ( C ) B (−) ←←←←←←←←←←←←←←←←→ Fun ns ( C ) D (−) ←←←←←←←←←←←←←←←←←→ Fun R ( C op , C ) by Ϙ ↦ D Ϙ .Let us make these adjointability statements explicit: if B ∈ Fun s ( C ) is a non-degenerate symmetricbilinear functor with associated duality D = D B ∶ C op → C then the symmetric structure of B determines anatural equivalence(23) hom C ( 𝑥, D( 𝑦 )) ≃ B( 𝑥, 𝑦 ) ≃ B( 𝑦, 𝑥 ) ≃ hom C ( 𝑦, D( 𝑥 )) ≃ hom C op (D op ( 𝑥 ) , 𝑦 ) where D op ∶ C → C op is the functor induced by D upon taking opposites. Such a natural equivalence exhibitsin particular D op as left adjoint to D . We will denote by(24) ev ∶ id ⇒ DD op the unit of this adjunction, and refer it as the evaluation map of D . Its individual components(25) ev 𝑥 ∶ 𝑥 → DD op ( 𝑥 ) are then the maps corresponding to identity D op ( 𝑥 ) → D op ( 𝑥 ) under the equivalence (23). The counit ofthis adjunction is given again by natural transformation (24), but interpreted as an arrow from D op D to theidentity in the ∞ -category Fun( C op , C op ) ≃ Fun( C , C ) op .1.2.3. Remark.
The process of viewing the equivalence (23) as an adjunction between D and D op andextracting its unit as above can be reversed: knowing that ev is a unit of an adjunction we can reproduce theequivalence hom C ( 𝑦, D( 𝑥 )) ≃ hom C ( 𝑥, D( 𝑦 )) as the composite hom C ( 𝑦, D( 𝑥 )) ≃ hom C op (D op ( 𝑥 ) , 𝑦 ) → hom C (DD op ( 𝑥 ) , D( 𝑦 )) → hom C ( 𝑥, D( 𝑦 )) , where the last map is induced by pre-composition with the evaluation map. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 19 Lemma.
Let ( C , Ϙ ) , ( C ′ , Ϙ ′ ) be two non-degenerate hermitian ∞ -categories with associated dualities D Ϙ and D Ϙ ′ , and let 𝑓 , 𝑔 ∶ C → C ′ be two exact functors. Then there is a natural equivalence nat(B Ϙ , ( 𝑓 × 𝑔 ) ∗ B Ϙ ′ ) ≃ nat( 𝑓 D Ϙ , D Ϙ ′ 𝑔 op ) , where nat stands for the the spectrum of (non-symmetric) natural transformations, on the left between twospectrum valued functors on C op × C op , and on the right between two functors C op → D .Proof. Consider the left Kan extension functor ( 𝑓 × id) ! ∶ Fun( C op × C op , S 𝑝 ) → Fun( C ′op × C op , S 𝑝 ) , which is left adjoint to the corresponding restriction functor. Natural transformations B Ϙ ⇒ ( 𝑓 × 𝑔 ) ∗ B Ϙ ′ ≃ ( 𝑓 × id) ∗ (id × 𝑔 ) ∗ B Ϙ ′ then correspond under this adjunction to natural transformations(26) ( 𝑓 × id) ! B Ϙ ⇒ (id × 𝑔 ) ∗ B Ϙ ′ . Now for 𝑦 ∈ C we have (( 𝑓 × id) ! B Ϙ ) | C ′ ×{ 𝑦 } ≃ ( 𝑓 × { 𝑦 }) ! ((B Ϙ ) | C ×{ 𝑦 } ) , as can be seen by the pointwise formula for left Kan extension. Since B Ϙ (− , 𝑦 ) is represented by D Ϙ ( 𝑦 ) andleft Kan extension preserves representable functors it then follows that ( 𝑓 × id) ! B Ϙ ( 𝑥 ′ , 𝑦 ) ≃ hom C ′op ( 𝑥 ′ , 𝑓 (D Ϙ ( 𝑦 ))) for ( 𝑥 ′ , 𝑦 ) ∈ C ′op × C op . On the other hand, we have (id × 𝑔 ) ∗ B Ϙ ′ ( 𝑥 ′ , 𝑦 ) ≃ hom D op ( 𝑥 ′ , D Ϙ ′ ( 𝑔 ( 𝑦 ))) , and so bythe fully-faithfulness of the Yoneda embedding we thus obtain nat(B Ϙ , ( 𝑓 × 𝑔 ) ∗ B Ϙ ′ ) ≃ nat( 𝑓 D Ϙ , D Ϙ ′ 𝑔 op ) , as desired. (cid:3) Definition.
Given a hermitian functor ( 𝑓 , 𝜂 ) ∶ ( C , Ϙ ) → ( C ′ , Ϙ ′ ) , we will denote by 𝜏 𝜂 ∶ 𝑓 D Ϙ ⇒ D Ϙ ′ 𝑓 op the natural transformation corresponding to the natural transformation 𝛽 𝜂 ∶ B Ϙ ⇒ ( 𝑓 × 𝑓 ) ∗ B Ϙ ′ of (20), viaLemma 1.2.4.1.2.6. Remark.
In the situation of Definition 1.2.5, it follows from the triangle identities of the adjunction ( 𝑓 × id) ! ⊣ ( 𝑓 × id) ∗ that the natural transformation 𝛽 𝜂 ∶ B Ϙ ⇒ ( 𝑓 × 𝑓 ) ∗ B Ϙ ′ can be recovered from 𝜏 𝜂 ∶ 𝑓 D Ϙ ⇒ D Ϙ ′ 𝑓 op as the composite B Ϙ ( 𝑥, 𝑦 ) ≃ hom C ( 𝑥, D Ϙ ( 𝑦 )) → hom D ( 𝑓 ( 𝑥 ) , 𝑓 D Ϙ ( 𝑦 )) → hom D ( 𝑓 ( 𝑥 ) , D Ϙ ′ 𝑓 ( 𝑦 )) ≃ B Ϙ ( 𝑓 ( 𝑥 ) , 𝑓 ( 𝑦 )) , where the two middle maps are induced by the action of 𝑓 on mapping spectra and post-composition with 𝜏 𝜂 , respectively.1.2.7. Definition.
A hermitian functor ( 𝑓 , 𝜂 ) ∶ ( C , Ϙ ) → ( C ′ , Ϙ ′ ) between non-degenerate hermitian ∞ -categories is called duality preserving if the transformation 𝜏 𝜂 ∶ 𝑓 D Ϙ ⇒ D Ϙ 𝑓 op constructed above is anequivalence.1.2.8. Definition.
A symmetric bilinear functor B is called perfect if the evaluation map id C ⇒ D B D opB of (24) is an equivalence. An hermitian structure Ϙ is called Poincaré if the underlying bilinear functor of Ϙ is perfect. In this case we will say that ( C , Ϙ ) is a Poincaré ∞ -category . We will denote by Cat p∞ ⊆ Cat h∞ the (non-full) subcategory spanned by the Poincaré ∞ -categories and duality preserving functors, and willgenerally refer to duality-preserving hermitian functors between Poincaré ∞ -category as Poincaré functors .For a stable ∞ -category C , we will denote by Fun p ( C ) ⊆ Fun q ( C ) the subcategory spanned by those hermitian structures which are Poincaré, and those natural transformations 𝜂 ∶ Ϙ ⇒ Ϙ ′ which are duality preserving, that is, for which the associated hermitian functor (id , 𝜂 ) ∶ ( C , Ϙ ) → ( C , Ϙ ′ ) is Poincaré. Remark.
A symmetric bilinear functor B is perfect if and only if it is non-degenerate and D B ∶ C op → C is an equivalence of categories. Indeed, an adjunction consists of a pair of inverse equivalences if andonly if its unit and counit are equivalences.If B is a perfect bilinear functor on C then the duality D B ∶ C op ≃ ←←←←←←←→ C is not just an equivalence of ∞ -categories, but carries a significant amount of extra structure. To make this precise note that there is a C -action on Cat ex∞ given by sending C ↦ C op . This can be seen by using simplicial sets as a model where takingthe opposite gives an action on the nose. Alternatively one can also use that the space of autoequivalencesof Cat ex∞ is equivalent to the discrete group C as shown in [Toë04], see also [Lur09b, Theorem 4.4.1].1.2.10. Definition. A stable ∞ -category with perfect duality is a homotopy fixed point of Cat ex∞ with respectto the C -action given by taking the opposite ∞ -category.We note that a stable ∞ -category with perfect duality consists in particular of a stable ∞ -category C andan equivalence D ∶ C → C op , equipped with additional coherence structure of being a C -fixed point. Forexample, the composition DD op is equipped with a natural equivalence ev ∶ id ≃ DD op , which itself carrieshigher coherence homotopies relating it with its opposite, and so forth. By a perfect duality on a given stable ∞ -category C we will mean a refinement of C to a C -fixed point of Cat ex∞ , that is, a section BC → ̃ Cat ex∞ of the fibration ̃ Cat ex∞ → BC by the 𝐶 -action on the ∞ -category Cat ex∞ given by taking opposites, whichsends the unique object ∗∈ BC to the point of the fibre ( ̃ Cat ex∞ ) ∗ ≃ Cat ex∞ determined by C . We may alsoidentify the notion of a perfect duality with that of a C -fixed equivalence C op ≃ ←←←←←←←→ C , where the C -actionon Fun R ( C op , C ) is obtained via its identification with the ∞ -category Fun nb ( C ) of non-degenerate bilinearfunctors. We will often abuse notation and denote a perfect duality simply by its underlying equivalence D ∶ C op → C .In their work on ∞ -categories with duality, Heine-Lopez-Avila-Spitzweck prove that the duality functor D B associated to a perfect bilinear functor B on a stable ∞ -category C , naturally refines to a perfect dualityon C in the above sense. Furthermore, the association B ↦ D B determines an equivalence between perfectbilinear functors on C and perfect dualities on C see [HLAS16, Corollary 7.3], and [Spi16, Proposition 2.1]for the stable variant. Together with Lemma 1.2.4, this association determines a forgetful functor(27) Cat p∞ → (Cat ex∞ ) hC ( C , Ϙ ) ↦ ( C , D Ϙ ) from Poincaré ∞ -categories to stable ∞ -categories with perfect duality. This provides a key link betweenthe present setup and the existing literature on stable ∞ -categories with duality.1.2.11. Definition.
Given a stable ∞ -category and a symmetric bilinear functor B ∶ C op × C op → S 𝑝 wewill refer to the hermitian structures Ϙ sB , Ϙ qB ∈ Fun q ( C ) of Example 1.1.17 as the symmetric and quadratic hermitian structures associated to B , respectively. As the symmetric bilinear parts of both Ϙ sB and Ϙ qB arecanonically equivalent to B , these hermitian structures are Poincaré if and only if B is perfect.1.2.12. Example.
Let 𝑅 be an ordinary commutative ring and let C = D p ( 𝑅 ) be the perfect derived category of 𝑅 . Similar as in Example 1.1.17 we may then consider the symmetric bilinear functor B 𝑅 ∈ Fun b ( C ) given by B 𝑅 ( 𝑋, 𝑌 ) = hom 𝑅 ( 𝑋 ⊗ 𝑅 𝑌 , 𝑅 ) , with symmetric structure induced by the symmetric structure of the tensor product ⊗ . This bilinear functoris perfect with duality given by D 𝑅 ( 𝑌 ) = Hom cx 𝑅 ( 𝑌 , 𝑅 ) , where the right hand side stands for the internal mapping complex. An element 𝛽 ∈ Ω ∞ B( 𝑋, 𝑌 ) thencorresponds to a map 𝑋 ⊗ 𝑅 𝑌 → 𝑅 , which we can consider as a bilinear form on the pair ( 𝑋, 𝑌 ) . To thisperfect bilinear functor we can associate the corresponding symmetric and quadratic Poincaré structures Ϙ s 𝑅 ( 𝑋 ) ∶= B 𝑅 ( 𝑋, 𝑋 ) hC and Ϙ q 𝑅 ( 𝑋 ) ∶= B 𝑅 ( 𝑋, 𝑋 ) hC , as in Definition 1.2.11. The space Ω ∞ Ϙ s 𝑅 ( 𝑋 ) is then the space of (homotopy) C -fixed points of Ω ∞ B( 𝑋, 𝑋 ) ,which should be viewed as the homotopical counterpart of the notion of a symmetric form on 𝑋 . The space Ω ∞ Ϙ q 𝑅 ( 𝑋 ) , on the other hand, is the space of (homotopy) C -orbits of Ω ∞ B( 𝑋, 𝑋 ) , which we can consideras a homotopical analogue of that a quadratic form on 𝑋 , see Example 1.1.17. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 21 Example.
In the situation of Example 1.2.12 we could also consider the symmetric bilinear functor B − 𝑅 whose underlying bilinear functor is B 𝑅 but whose symmetric structure is twisted by the sign action of C . In other words, the symmetry equivalence B − 𝑅 ( 𝑋 ⊗ 𝑅 𝑌 , 𝑅 ) ≃ ←←←←←←←→ B − 𝑅 ( 𝑌 ⊗ 𝑅 𝑋, 𝑅 ) of B − 𝑅 is minus theone of B 𝑅 . This bilinear functor is again perfect with duality which coincides with D 𝑅 on the level of theunderlying equivalence D p ( 𝑅 ) op → D p ( 𝑅 ) , but which has a different double dual identification. We maythen consider the corresponding symmetric and quadratic Poincaré structures Ϙ s− 𝑅 ( 𝑋 ) ∶= B − 𝑅 ( 𝑋, 𝑋 ) hC and Ϙ q− 𝑅 ( 𝑋 ) ∶= B − 𝑅 ( 𝑋, 𝑋 ) hC , as in Definition 1.2.11. The space Ω ∞ Ϙ s− 𝑅 ( 𝑋 ) is then a homotopical counterpart of the notion of an anti-symmetric form on 𝑋 , while Ω ∞ Ϙ q− 𝑅 ( 𝑋 ) is its quadratic counterpart.1.2.14. Example.
In the spirit of Example 1.2.12, one may also fix a scheme 𝑋 and consider the stable ∞ -category D p ( 𝑋 ) of perfect complexes of quasi-coherent sheaves on 𝑋 . Given a line bundle 𝐿 on 𝑋 wehave an associated bilinear form B 𝐿 on D p ( 𝑋 ) given by B 𝐿 ( F , G ) = hom 𝑋 ( F ⊗ 𝑋 G , 𝐿 ) , which is perfect with duality D 𝐿 ( 𝑋 ) = Hom cx 𝑋 ( F , 𝐿 ) . To this perfect duality we can then associate the corresponding symmetric and quadratic Poincaré structures Ϙ s 𝐿 ( F ) ∶= B 𝐿 ( F , F ) hC and Ϙ q 𝐿 ( F ) ∶= B 𝐿 ( F , F ) hC . Example.
Let S 𝑝 f be the ∞ -category of finite spectra. We define a hermitian structure on S 𝑝 f viathe pullback square Ϙ u ( 𝑋 ) D( 𝑋 )D( 𝑋 ⊗ 𝑋 ) hC D( 𝑋 ⊗ 𝑋 ) tC where D( 𝑋 ) = hom( 𝑋, 𝕊 ) denotes the Spanier-Whitehead dual and the right vertical map is the Tate di-agonal D 𝑋 → (D 𝑋 ⊗ D 𝑋 ) tC of D 𝑋 . This hermitian structure is then Poincaré with duality given bySpanier-Whitehead duality. We note that this Poincaré structure is neither quadratic not symmetric (Defini-tion 1.2.11). The superscript u is suggestive for universal, see §4.1. The Poincaré ∞ -category also functionsas the unit of the symmetric monoidal structure on Poincaré ∞ -categories we will construct in §5.Defining hermitian structures as spectrum valued functors allows to easily implement various usefulmanipulations. One of them, which plays a recurring role in this paper, is the procedure of shifting hermitianstructures:1.2.16. Definition.
Let C be a stable ∞ -category and Ϙ ∶ C op → S 𝑝 a quadratic functor. For 𝑛 ∈ ℤ we willdenote by Ϙ [ 𝑛 ] ∶ C op → S 𝑝 the 𝑛 -fold suspension of Ϙ , given by Ϙ [ 𝑛 ] ( 𝑥 ) = Σ 𝑛 Ϙ ( 𝑥 ) . We note that Ϙ [ 𝑛 ] is again a quadratic functor with bilinear part B Σ 𝑛 Ϙ = Σ 𝑛 B Ϙ and linear part L Σ 𝑛 Ϙ = Σ 𝑛 L Ϙ ;indeed, Fun q ( C ) is a stable subcategory of Fun( C op , S 𝑝 ) and B (−) and L (−) are both exact functors. Inparticular, if Ϙ non-degenerate or perfect then so is Ϙ [ 𝑛 ] with duality D Ϙ [ 𝑛 ] ( 𝑥 ) = Σ 𝑛 D Ϙ ( 𝑥 ) . We will referto Ϙ [ 𝑛 ] is the 𝑛 -fold shift of Ϙ , and to the hermitian ∞ -category ( C , Ϙ [ 𝑛 ] ) as the 𝑛 -fold shift of ( C , Ϙ ) .1.2.17. Remark.
The hermitian ∞ -category ( C , Ϙ [ 𝑛 ] ) is Poincaré if and only if ( C , Ϙ ) is.1.2.18. Example.
In the situation of Example 1.2.12, if we shift the Poincaré structures Ϙ s 𝑅 and Ϙ q 𝑅 on D p ( 𝑅 ) by 𝑛 ∈ ℤ then we get Poincaré structures ( Ϙ s 𝑅 ) [ 𝑛 ] ( 𝑋 ) = hom 𝑅 ( 𝑋 ⊗ 𝑅 𝑋, 𝑅 [ 𝑛 ]) hC and ( Ϙ q 𝑅 ) [ 𝑛 ] ( 𝑋 ) = hom 𝑅 ( 𝑋 ⊗ 𝑅 𝑋, 𝑅 [ 𝑛 ]) hC . respectively, which we consider as encoding 𝑛 -shifted symmetric and quadratic forms. Here, 𝑅 [ 𝑛 ] denotesthe 𝑅 -complex which is 𝑅 in degree 𝑛 and zero everywhere else. We note that the full subcategories of
Fun q ( C ) spanned by non-degenerate and perfect functors respec-tively are not preserved under pullback along exact functors 𝑓 ∶ C → C ′ . For example, the hermitian struc-ture Ϙ s ℚ on D p ( ℚ ) is perfect (see Example 1.2.12), but its pullback to D p ( ℤ ) is not even non-degenerate. Anotable exception to this is however the following:1.2.19. Observation. If ( C , Ϙ ) is a non-degenerate hermitian or Poincaré ∞ -category and D ⊆ C be a fullstable subcategory such that the duality D Ϙ maps D to itself then then ( D , Ϙ | D ) is again non-degeneratewith D Ϙ | D = D Ϙ | D . In particular, if ( C , Ϙ ) is Poincaré then ( D , Ϙ | D ) is again Poincaré. Example. If ( C , Ϙ ) is a non-degenerate hermitian ∞ -category then the full subcategory C ref l ⊆ C spanned by those objects 𝑥 ∈ C for which the evaluation map ev 𝑥 ∶ 𝑥 → D op D( 𝑥 ) is an equivalence ispreserved under the duality by the triangle identities, and hence the hermitian ∞ -category ( C ref l , Ϙ | C refl ) isagain non-degenerate, and even Poincaré, since the evaluation map is now an equivalence by construction.1.3. Classification of hermitian structures.
In this section we will discuss the classification of hermit-ian and Poincaré structures on a fixed stable ∞ -category C , in terms of their linear and bilinear parts.For the hermitian part, this is essentially the 𝑛 = 2 case classification of 𝑛 -excisive functors in Good-willie calculus, and is also a particular instance of the structure theory of stable recollements (see [Lur17,§A.8], [BG16], [QS19]). For the purpose of self containment we will however provide full proofs of thestatements that we need in the present setting. In order to formulate these statements we will first need tobetter understand the role played by the quadratic and symmetric hermitian structures Ϙ qB , Ϙ sB associated toa given symmetric bilinear form B .1.3.1. Lemma.
Let Ϙ ∶ C op → S 𝑝 be a quadratic functor on a small stable ∞ -category C . Then the followingare equivalent:i) The map B Ϙ ( 𝑥, 𝑥 ) hC → Ϙ ( 𝑥 ) of (8) is an equivalence for every 𝑥 ∈ C .ii) Ϙ is equivalent to a quadratic functor of the form Ϙ qB for some symmetric bilinear functor B ∈ Fun s ( C ) (see Example 1.1.17).iii) The spectrum of natural transformations nat( Ϙ , L) is trivial for any exact functor L ∈ Fun ex ( C op , S 𝑝 ) ⊆ Fun q ( C ) . Definition.
Following the conventions of Goodwillie calculus, we will refer to quadratic functors Ϙ ∶ C op → S 𝑝 which satisfy the equivalent conditions of Lemma 1.3.1 as homogeneous . We will denote by Fun hom ( C ) ⊆ Fun q ( C ) the full subcategory spanned by the homogeneous functors. Proof of Lemma 1.3.1.
Clearly i) ⇒ ii). If we assume ii) then iii) follows from Lemma 1.1.24 since thelinear part of Ϙ qB vanishes by definition. Similarly if we assume iii) then i) follows by Lemma 1.1.24 sincethe linear part vanishes. (cid:3) Corollary (cf. [Lur17, Proposition 6.1.4.14]) . The functor (28)
Fun s ( C ) → Fun q ( C ) B ↦ Ϙ qB is fully-faithful and its essential image is spanned by those quadratic functors which are homogeneous inthe above sense.Proof. By Lemma 1.3.1 the functor (28) takes values in homogeneous functors, and hence determines afunctor 𝜑 ∶ Fun s ( C ) → Fun hom ( C ) ⊆ Fun q ( C ) , where the latter denotes the full subcategory spanned by ho-mogeneous quadratic functors. On the other hand, the formation of cross effects determines a functor in theother direction 𝜓 ∶ Fun hom ( C ) → Fun s ( C ) . By Lemma 1.3.1 the composed functor 𝜑 ◦ 𝜓 ∶ Fun hom ( C ) → Fun hom ( C ) is naturally equivalent to the identity, and by Examples 1.1.17 the composite 𝜓 ◦ 𝜑 is naturallyequivalent to the identity Fun s ( C ) → Fun s ( C ) as well. It then follows that 𝜑 is an equivalence from Fun s ( C ) to Fun hom ( C ) ⊆ Fun q ( C ) , as desired. (cid:3) The notion of a homogeneous quadratric functor has a dual counterpart, which consists of the quadraticfunctors which have a trivial mapping spectrum from any exact functor. The argument of Lemma 1.3.1then runs in a completely dual manner to show that this property is equivalent to the canonical map Ϙ ( 𝑥 ) → B Ϙ ( 𝑥, 𝑥 ) hC being an equivalence and is satisfied by quadratic functors of the form Ϙ sB for any B ∈ Fun s ( C ) .We will refer to such functors as cohomogeneous , and denote by Fun coh ( C ) ⊆ Fun q ( C ) the full subcategory ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 23 spanned by the cohomogeneous functors. The argument of Corollary 1.3.3 then runs in a completely dualmanner to show that the functor(29) Fun s ( C ) → Fun q ( C ) B ↦ Ϙ sB is fully-faithful and its essential image is spanned by the cohomogeneous quadratic functors.1.3.4. Remark.
Given a bilinear functor B , the quadratic functor 𝑥 ↦ B( 𝑥, 𝑥 ) is both homogeneous andcohomogeneous.1.3.5. Proposition.
The natural transformation 𝜖 ∶ [B Δ Ϙ ] hC ⇒ Ϙ exhibits [B Δ Ϙ ] hC as final among homo-geneous functors equipped with a map to Ϙ . Dually, the natural transformation 𝜂 ∶ Ϙ ⇒ [B Δ Ϙ ] hC exhibits [B Δ Ϙ ] hC as initial among cohomogeneous functors equipped with a map from Ϙ .Proof. This first statement is equivalent to Lemma 1.1.24 and the second is dual. (cid:3)
Corollary.
For a small stable ∞ -category C the functor B (−) ∶ Fun q ( C ) → Fun s ( C ) admits left and right adjoints, both of which are fully faithful, given by sending B to Ϙ qB and Ϙ sB respectively. Remark.
By Corollary 1.3.6 with Lemma 1.1.24 the pair of fully-faithful inclusions
Fun s ( C ) Ϙ s(−) ←←←←←←←←←←←←←←←←→ Fun q ( C ) ← Fun ex ( C op , S 𝑝 ) form a recollement in the sense of [Lur17, Definition A.8.1], and more precisely a stable recollement inthe sese of [BG16] and [QS19] since all ∞ -categories involved are stable and the all functors involved areexact.Given a symmetric bilinear functor B ∈ Fun s ( C ) , Lemma 1.3.1 implies that the linear part of the quadraticfunctor Ϙ qB is trivial. This however need not be the case for the quadratic functor Ϙ sB . To identify thelinear part of the latter, recall that the symmetric bilinear part of Ϙ qB is canonically identified with B itself(see Example 1.1.17), and so by Corollary 1.3.6 natural transformations Ϙ qB ⇒ Ϙ sB correspond to naturaltransformations B ⇒ B . In particular, there is a distinguished transformation(30) Ϙ qB ⇒ Ϙ sB which corresponds to the identity B ⇒ B . In terms of the adjunctions of Corollary 1.3.6, this map can alsobe identified with the counit of the adjunction (−) ΔhC ⊣ B (−) evaluated at Ϙ sB , or the unit of B (−) ⊣ ((−) Δ ) hC evaluated at Ϙ qB .1.3.8. Lemma.
For
B ∈ Fun s ( C ) the map (30) is canonically equivalent to the norm map associated to the C -action on the object B Δ ∈ Fun q ( C ) .Proof. Given the perfect correspondence between natural transformations Ϙ qB ⇒ Ϙ sB and natural transfor-mations B ⇒ B it will suffice to construct an identification between the map B ⇒ B induced by the normmap of B Δ and the identity on B . For this, note that since the functor B (−) ∶ Fun q ( C ) → Fun s ( C ) preservesall limits and colimits it also sends norm maps to norm maps. In particular, the map B ⇒ B induced onbilinear parts by the norm map Ϙ qB ⇒ Ϙ sB is itself the norm map (B ⊕ B) hC ⇒ (B ⊕ B) hC associated to the C -action on the symmetric bilinear part of B Δ , which we identify with the induced/coinduced C -object B ⊕ B as in Example 1.1.17. The desired result now follows from the following completely generalproperty of norm maps: given a semi-additive ∞ -category D and an object 𝑥 ∈ D , the norm map of the in-duced/coinduced C -object 𝑥 ⊕ 𝑥 identifies with the identity id ∶ 𝑥 → 𝑥 under the canonical identifications ( 𝑥 ⊕ 𝑥 ) hC ≃ 𝑥 ≃ ( 𝑥 ⊕ 𝑥 ) hC . (cid:3) Remark.
Lemma 1.3.8 implies that Ϙ naturally lifts to a functor with values in genuine C -spectra .We will discuss this issue in greater detail and precision in §7.4. Corollary.
For a symmetric bilinear functor
B ∈ Fun s ( C ) the linear part of Ϙ sB ( 𝑥 ) = B( 𝑥, 𝑥 ) hC isnaturally equivalent to the Tate construction (B Δ ) tC ( 𝑥 ) = B( 𝑥, 𝑥 ) tC . In particular, the latter is always anexact functor. By virtue of Lemma 1.3.8 and Corollary 1.3.10, any quadratic functor Ϙ on C determines a diagram ofquadratic functors(31) B Ϙ ( 𝑥, 𝑥 ) hC Ϙ ( 𝑥 ) L Ϙ ( 𝑥 )B Ϙ ( 𝑥, 𝑥 ) hC B Ϙ ( 𝑥, 𝑥 ) hC B Ϙ ( 𝑥, 𝑥 ) tC in which the right square is exact and the right most vertical map is obtained from the middle vertical map bytaking linear parts. Conversely, by Proposition 1.1.13 a symmetric bilinear functor B ∶ C op × C op → S 𝑝 , anexact functor L ∶ C op → S 𝑝 and a natural transformation 𝜏 ∶ L ⟹ (B Δ ) tC together determine a quadraticfunctor Ϙ ∶ C op → S 𝑝 by declaring the square Ϙ ( 𝑥 ) L( 𝑥 )B( 𝑥, 𝑥 ) hC B( 𝑥, 𝑥 ) tC 𝜏 𝑥 cartesian. This observation leads to a well-known classification of quadratic functors, which we now ex-plain. To formulate it, let us first note that for a quadratic functor Ϙ , Lemma 1.1.24 tells us that the naturaltransformation Ϙ ⇒ L Ϙ is universally characterized by the property that it induces an equivalence on map-ping spectra to every exact functor. In particular, if 𝜑 ∶ Ϙ ⇒ L is any map from Ϙ to an exact functor L which induces an equivalence on mapping spectra to any exact functor, then 𝜑 factors through an equiv-alence Ϙ → L Ϙ ≃ ←←←←←←←→ L in an essentially unique manner. In this case, we will also say that 𝜑 exhibits L as the linear part of Ϙ . Similarly, we will say that a map 𝜓 ∶ Ϙ → Ϙ ′ exhibits Ϙ ′ as the cohomogeneouspart of Ϙ if Ϙ ′ is cohomogeneous and 𝜓 induces an equivalence on mapping spectra to any cohomoge-neous functor. In this case, Corollary 1.3.6 tells us that 𝜓 factors through an essentially unique equivalence Ϙ ( 𝑥 ) → B Ϙ ( 𝑥, 𝑥 ) hC ≃ ←←←←←←←→ Ϙ ′ ( 𝑥 ) . Let us now denote by E ⊆ Fun(Δ × Δ , Fun q ( C )) the full subcategoryspanned by those squares of quadratic functors(32) Ϙ L Ϙ ′ L ′ which are exact and for which the top horizontal map exhibits L as the linear part of Ϙ and the left verticalmap exhibits Ϙ ′ as the cohomogeneous part of Ϙ . In particular, any square in E is equivalent to a squareas of the form appearing on the right side of (31) in an essentially unique way. We may consider E as the ∞ -category of quadratic functors equipped with a “cohomogeneous-linear decomposition”.1.3.11. Proposition.
The evaluation at (0 ,
0) ∈ Δ × Δ map E → Fun q ( C ) sending a square as in (32) to Ϙ is an equivalence of ∞ -categories. In particular, every quadratic functor can be written as a pullback ofcohomogeneous and exact functors.Proof. Given Remark 1.3.7 this can be deduced from [BG16, Lemma 9]. We however spell out the detailsfor completeness. Let E ⌜ ⊆ Fun(Λ , Fun q ( C )) be the full subcategory spanned by those Λ -diagrams(33) Ϙ L Ϙ ′ for which the top horizontal map exhibits L as the linear part of Ϙ and the left vertical map exhibits Ϙ ′ the cohomogeneous part of Ϙ . Then the restriction of any square in E to Λ ⊆ Δ × Δ lies in E ⌜ andthe resulting projection E → E ⌜ is a trivial Kan fibration since every square in E is exact and hence a ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 25 left Kan extension of its restriction to Λ . It will hence suffice to show that the projection E ⌜ → Fun q ( C ) sending a diagram as in (33) to Ϙ is an equivalence. Now the ∞ -category E ⌜ can be embedded in the larger ∞ -category E ′ ⊆ Fun(Λ , Fun q ( C )) consisting of those diagrams as in (33) for which L is exact and Ϙ ′ iscohomogeneous. Then the projection E ′ → Fun q ( C ) sending (33) to Ϙ is a cartesian fibration classifiedby the functor sending Ϙ to the product of the comma category of exact functors under Ϙ and the commacategory of cohomogeneous functors under Ϙ . We may then identify E ⌜ with the full subcategory of E ′ spanned by those objects which are initial in their fibres. It then follows that the projection E ⌜ → Fun q ( C ) is a trivial Kan fibration, and so the proof is complete. (cid:3) We may now deduce the classification theorem for hermitian structures (cf. the general classification ofrecollements [Lur17, Proposition A.8.11]):1.3.12.
Corollary (Classification of hermitian structures) . The square (34)
Fun q ( C ) B Ar(Fun ex ( C op , S 𝑝 ))Fun s ( C ) Fun ex ( C op , S 𝑝 ) , 𝜏 t is cartesian. Here the lower horizontal functor sends B to (B Δ ) tC and the right vertical functor sends anarrow to its target.Proof. Let E ⌟ ⊆ Fun(Λ , Fun q ( C )) be the full subcategory spanned by those Λ -diagrams(35) L Ϙ ′ L ′ for which Ϙ ′ is cohomogeneous, L is exact and the bottom horizontal map exhibits L ′ as the linear part of Ϙ ′ . Then restriction along Λ ⊆ Δ × Δ sends every square in E to a square in E ⌟ . On the other hand,if we complete a diagram of the form (35) which belongs to E ⌟ to a cartesian square, then this squarewill belong to E : indeed, this follows from the fact that a map Ϙ ′′ → L ′′ from a quadratic to an exactfunctor exhibits the latter as the linear part of the former if and only if its fibre maps trivially to any exactfunctor, that is, if its fibre is homogeneous. We then conclude that the projection E → E ⌟ induced byrestriction along Λ is a trivial Kan fibration. On the other hand, the ∞ -category E ⌟ is by constructiona fibre product Ar(Fun ex ( C )) × Fun ex ( C ) E ′′ where E ′′ is the full subcategory of Fun(Δ , Fun q ( C )) spannedby those arrows 𝜓 ∶ Ϙ ′ → L ′ such that Ϙ ′ is cohomogeneous and 𝜓 exhibits L ′ as the linear part of Ϙ ′ .As in the proof of Proposition 1.3.11 the projection E ′′ → Fun coh ( C ) sending Ϙ ′′ → L ′′ to Ϙ ′′ is a trivialKan fibration onto the full subcategory Fun coh ( C ) ⊆ Fun q ( C ) spanned by the cohomogeneous functors,and the section is given by sending a cohomogeneous functor Ϙ ′ to the arrow Ϙ ′ → L Ϙ ′ . We hence seethat the projection E ′′ → Fun ex ( C ) is equivalent as an arrow to the functor Fun coh ( C ) → Fun ex ( C ) takinglinear parts. Finally, by Corollary 1.3.6 and Corollary 1.3.10 the latter arrow is also equivalent to the arrow Fun b ( C ) hC → Fun ex ( C ) sending B to (B Δ ) tC . Since E ′′ → Fun ex ( C ) is a categorical fibration the fibreproduct E ⌟ is a model for the homotopy fibre product in the square (34). The desired result now followsfrom Proposition 1.3.11 and the fact that the projection E → E ⌟ is an equivalence. (cid:3) Finally, let us also deduce an analogous classification for Poincaré structure. For this, let us denote by
Fun pb ( C ) ⊆ Fun b ( C ) the non-full subcategory spanned by the perfect bilinear functors and duality pre-serving natural transformations, that is, the natural transformations 𝛽 ∶ B ⇒ B ′ for which the associatedtransformation 𝜏 𝛽 ∶ D B ⇒ D B ′ is an equivalence. We then define Fun ps ( C ) to be the ∞ -category sitting inthe pullback square Fun ps ( C ) Fun s ( C )Fun pb ( C ) Fun b ( C ) . It then follows directly from the definitions that the subcategory inclusion
Fun p ( C ) ⊆ Fun q ( C ) (see Defini-tion 1.2.8) features in a commutative diagram Fun p ( C ) Fun q ( C )Fun ps ( C ) Fun s ( C )Fun pb ( C ) Fun b ( C ) . in which both squares are pullback squares. The following is now a direct consequence of Corollary 1.3.12:1.3.13. Corollary (Classification of Poincaré structures) . The square (36)
Fun p ( C ) B Ar(Fun ex ( C op , S 𝑝 ))Fun ps ( C ) Fun ex ( C op , S 𝑝 ) , 𝜏 t is cartesian. Functoriality of hermitian structures.
In this subsection we will then discuss the functorial depen-dence of
Fun q ( C ) on C from the perspective of the classification described in §1.3, not only contravariantlyvia restriction along exact functors, but also covariantly via left Kan extensions. Recall that in §1.2 wedefined Cat h∞ as the total ∞ -category of the cartesian fibration(37) Cat h∞ → Cat ex∞ which classifies the functor C ↦ Fun q ( C ) . In particular, being a cartesian fibration, the projection (37)encodes the contravariance dependence of Fun q ( C ) in C . We will now show that Fun q ( C ) also depends covariantly in C via the formations of left Kan extensions. In particular, it will follow that the projec-tion (37) is also a cocartesian fibration. Since we constantly work with contravariant functors to spectralet us employ the following notation: given a functor 𝑔 ∶ D → E between ∞ -categories we denote by 𝑔 ! ∶ Fun( D op , S 𝑝 ) → Fun( E op , S 𝑝 ) the operation of left Kan extending contravariant functor to spectraalong 𝑔 op ∶ D op → E op .1.4.1. Lemma. i) If 𝑓 ∶ C → D is an exact functor between stable ∞ -categories then the associated left Kan extensionfunctor 𝑓 ! ∶ Fun( C op , S 𝑝 ) → Fun( D op , S 𝑝 ) sends exact functors to exact functors.ii) If 𝑓 ∶ C → D and 𝑔 ∶ A → B are exact functors between stable ∞ -categories then the associated leftKan extension functor ( 𝑓 × 𝑔 ) ! ∶ Fun( C op × A op , S 𝑝 ) → Fun( D op × B op , S 𝑝 ) sends bi-exact functors to bi-exact functors.iii) If 𝑓 ∶ C → D is an exact functor between stable ∞ -categories then the associated left Kan extensionfunctor 𝑓 ! ∶ Fun( C op , S 𝑝 ) → Fun( D op , S 𝑝 ) sends quadratic functors to quadratic functors.Proof. We first note that left Kan extension along any functor between pointed ∞ -categories preserve re-duced functors by the pointwise formula for left Kan extension. Let now 𝑓 ∶ C → D be an exact functorand R ∶ C op → S 𝑝 a reduced functor. Consider the following commutative diagram of stable ∞ -categories C D
Pro( C ) Pro( D ) 𝑓𝑖 𝑗̃𝑓 ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 27 Then the bottom arrow admits a left adjoint 𝑔 ∶ Pro( D ) ⟶ Pro( C ) , which corresponds to restrictionalong C → D under the identification of Pro(−) ≃ Ind((−) op ) op with (the opposite category of) rightexact functors to spaces. Now since the Yoneda embedding 𝑗 ∶ D → Pro( D ) is fully faithful we have that 𝑗 ∗ 𝑗 ! ∶ Fun( D op , S 𝑝 ) → Fun( D op , S 𝑝 ) is equivalent to the identity, and so 𝑓 ! R ≃ 𝑗 ∗ 𝑗 ! 𝑓 ! R ≃ 𝑗 ∗ ̃𝑓 ! 𝑖 ! R . Moreover, since 𝑔 is left adjoint to ̃𝑓 we have that the left Kan extension functor ̃𝑓 ! ∶ Fun ex (Pro( C ) op , S 𝑝 ) → Fun ex (Pro( D ) op , S 𝑝 ) is equivalent to restriction along 𝑔 op ∶ Pro( D ) op → Pro( C ) op , and so 𝑓 ! R ≃ 𝑗 ∗ 𝑔 ∗ 𝑖 ! R ≃ ( 𝑔𝑗 ) ∗ 𝑖 ! R . Applying [Lur17, Proposition 6.1.5.4] we now get that 𝑖 ! R is exact (resp. quadratic) if R is exact (resp.quadratic). Precomposition with the exact functor 𝑔𝑗 then preserve the properties of being exact or quadratic,and so 𝑓 ! R is exact (resp. quadratic) if R is exact (resp. quadratic). This proves i) and iii). To prove ii), wenow argue as follows. By the compatibility of left Kan extensions with composition of functors we mayreduce to the case where either 𝑓 or 𝑔 are the identity functor. By symmetry it will suffice to assume that itis 𝑓 which is the identity C → C . For any functor R ∶ C op × A op → S 𝑝 and every 𝑥 ∈ C we then have ((id × 𝑔 ) ! R ) | { 𝑥 }× B op ≃ 𝑔 ! ( R | { 𝑥 }× A ) , as can be seen by the pointwise formula for left Kan extension. We may then conclude that under theexponential equivalences Fun( C op × A op , S 𝑝 ) ≃ Fun( C op , Fun( A op , S 𝑝 )) and Fun( C op × B op , S 𝑝 ) ≃ Fun( C op , Fun( B op , S 𝑝 )) the left Kan extension functor (id × 𝑔 ) ! corresponds to post-composing with the left Kan extension functor 𝑔 ! ∶ Fun( A op , S 𝑝 ) → Fun( B op , S 𝑝 ) . Under the same equivalence the bi-exact functors correspond to thosefunctor C op → Fun( A op , S 𝑝 ) which are exact and which take values in Fun ex ( A op , S 𝑝 ) ⊆ Fun( A op , S 𝑝 ) .Since 𝑔 ! preserve exact functors by the first part of the lemma and post-composition with 𝑔 ! preserves exactfunctors since 𝑔 ! is colimit preserving (being a left adjoint), it now follows that (id × 𝑔 ) ! preserves bi-exactfunctors, as desired. (cid:3) Corollary.
The projection
Cat h∞ ⟶ Cat ex∞ is also a cocartesian fibration, with pushforward along 𝑓 ∶ C ⟶ D given by Ϙ ↦ 𝑓 ! Ϙ . Let us now discuss the compatibility of restriction and left Kan extensions with the decomposition ofthe ∞ -category of quadratic functors give by Corollary 1.3.12. We first observe that, given an exact functor 𝑓 ∶ C → D , the associated restriction functor 𝑓 ∗ ∶ Fun q ( D ) → Fun q ( C ) respects the square (34) in itsentirety: indeed, taking linear and bilinear parts is compatible with restriction by Remarks 1.1.6 and 1.1.23,and the bottom functor in (34) is also visibly compatible with restriction. We then get that if Ϙ is quadraticfunctor on D with bilinear part B , linear part L and structure map 𝛼 ∶ L → [B Δ ] tC , then 𝑓 ∗ Ϙ is the quadraticfunctor with bilinear part ( 𝑓 × 𝑓 ) ∗ B , linear part 𝑓 ∗ L and structure map 𝑓 ∗ 𝛼 ∶ 𝑓 ∗ L → 𝑓 ∗ [B Δ ] tC ≃ [(( 𝑓 × 𝑓 ) ∗ B) Δ ] tC . We now give a similar statement for left Kan extensions:1.4.3.
Proposition.
Let 𝑓 ∶ C ⟶ D be an exact functor between stable ∞ -categories and let Ϙ ∈ Fun q ( C ) be a quadratic functor on C . Then the natural transformations (38) ( 𝑓 × 𝑓 ) ! B Ϙ ⇒ B 𝑓 ! Ϙ 𝑓 ! L Ϙ ⇒ L 𝑓 ! Ϙ and 𝑓 ! (Ω ∞ Ϙ ) ⟶ Ω ∞ ( 𝑓 ! Ϙ ) are equivalences.Proof. Let 𝑖 ∶ C → Pro( C ) and 𝑗 ∶ D → Pro( D ) be the respective Yoneda embeddings. Arguing as in theproof of Lemma 1.4.1 using that ( 𝑗 × 𝑗 ) ∗ ( 𝑗 × 𝑗 ) ! is equivalent to the identity and that restriction commuteswith taking bilinear parts (Remark 1.1.6) we may identify the first map in (38) with the restriction along 𝑔𝑗 × 𝑔𝑗 ∶ D × D → Pro( C ) × Pro( C ) of(39) ( 𝑖 × 𝑖 ) ! ∶ B Ϙ ⇒ B 𝑖 ! Ϙ . We may hence assume without loss of generality that D = Pro( C ) and 𝑓 = 𝑖 . To prove the latter specialcase, we see that the component of the transformation (39) at a pair of pro-objects { 𝑥 𝛼 } 𝛼 ∈ I , { 𝑦 𝛽 } 𝛽 ∈ J in C identifies with the natural map colim ( 𝛼,𝛽 )∈ I op × J op f ib[ Ϙ ( 𝑥 𝛼 , 𝑦 𝛽 ) → Ϙ ( 𝑥 𝛼 ) ⊕ Ϙ ( 𝑦 𝛽 )] → f ib [ colim ( 𝛼,𝛽 )∈ I op × J op Ϙ ( 𝑥 𝛼 ⊕ 𝑦 𝛽 ) → colim 𝛼 ∈ I Ϙ ( 𝑥 𝛼 ) ⊕ colim 𝛽 ∈ J Ϙ ( 𝑦 𝛽 ) ] . Now since I and J are cofiltered the projections I op × J op → I op and I op × J op → J op are cofinal and hencewe can also rewrite the above map as colim ( 𝛼,𝛽 )∈ I op × J op f ib[ Ϙ ( 𝑥 𝛼 , 𝑦 𝛽 ) → Ϙ ( 𝑥 𝛼 ) ⊕ Ϙ ( 𝑦 𝛽 )] → f ib [ colim ( 𝛼,𝛽 )∈ I op × J op Ϙ ( 𝑥 𝛼 ⊕ 𝑦 𝛽 ) → colim ( 𝛼,𝛽 )∈ I op × J op Ϙ ( 𝑥 𝛼 ) ⊕ colim ( 𝛼,𝛽 )∈ I op × J op Ϙ ( 𝑦 𝛽 ) ] and so the desired result follows from the commutation of finite limits and filtered colimits in S 𝑝 . Thisalso implies that the second map in (38) is an equivalence since the formation of linear parts is obtained by L Ϙ ( 𝑥 ) ∶= cof[B Ϙ ( 𝑥, 𝑥 ) hC → Ϙ ( 𝑥 )] and left Kan extension commutes with colimits. Finally, the proof thatthe map 𝑓 ! (Ω ∞ Ϙ ) ⟶ Ω ∞ ( 𝑓 ! Ϙ ) is an equivalence is obtained via the same argument by reducing to the case of D = Pro( C ) and using thatthe formation of infinite loop spaces commutes with filtered colimits. (cid:3) Proposition 1.4.3 tells us that the formation of linear and bilinear parts is compatible with left Kanextensions. The situation is however slightly less simple then with restriction, since the bottom arrowin (34) does not commute with left Kan extensions. This is essentially due to the fact that the formation ofsymmetric hermitian structures B ↦ Ϙ sB = B( 𝑥, 𝑥 ) hC does not commute with left Kan extension. Instead,given Ϙ ∈ Fun q ( C ) we have a natural map 𝑓 ! Ϙ sB → Ϙ s( 𝑓 × 𝑓 ) ! B which is generally not an equivalence. This leads to the following description of the behavior of structuremaps under left Kan extensions:1.4.4. Corollary.
Let 𝑓 ∶ C → D be an exact functor. If Ϙ ∶ C op → S 𝑝 is quadratic functor on C withbilinear part B , linear part L and structure map 𝛼 ∶ L → [B Δ ] tC then 𝑓 ! Ϙ is the quadratic functor on D with bilinear part 𝑓 ! B , linear part 𝑓 ! L , and structure map the composite (40) 𝑓 ! L → 𝑓 ! [B Δ ] tC → [(( 𝑓 × 𝑓 ) ! B) Δ ] tC , which is the map induced on linear parts by the composite 𝑓 ! Ϙ → 𝑓 ! Ϙ sB → Ϙ s( 𝑓 × 𝑓 ) ! B . Remark.
In the situation of Corollary 1.4.4 we can also identify the second map in (40) with theBeck-Chevalley transformation on the lax commuting square on the right(41)
Fun s ( D ) Fun ex ( D op , S 𝑝 )Fun s ( C ) Fun ex ( C op , S 𝑝 ) (−) tC2Δ (−) tC2Δ Fun s ( C ) Fun ex ( C op , S 𝑝 )Fun s ( D ) Fun ex ( D op , S 𝑝 ) (−) tC2Δ (−) tC2Δ which is obtained from the commuting square on the left by replacing the vertical restriction functors ( 𝑓 × 𝑓 ) ∗ and 𝑓 ∗ by their left adjoints ( 𝑓 × 𝑓 ) ! and 𝑓 ! , respectively. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 29
2. P
OINCARÉ OBJECTS
In this section we will introduce and study another key element of the present paper, the notion of a
Poincaré object in a given Poincaré ∞ -category ( C , Ϙ ) . As reflected by Examples 1.2.12 and 1.2.13, wethink of a Poincaré structure on a given stable ∞ -category C as a way of encoding a particular notion ofhermitian form, e.g., quadratic, symmetric, or anti-symmetric forms on modules over rings. In the context ofa general Poincaré ∞ -category ( C , Ϙ ) and an object 𝑥 ∈ C , we will consequently call points in the underlyinginfinite loop space Ω ∞ Ϙ ( 𝑥 ) hermitian forms on 𝑥 . Such a form determines in particular a map 𝑥 → D 𝑥 from 𝑥 to its dual, and we will say that a form is Poincaré if this map is an equivalence. A Poincaré object is thenthe abstract analogous of a module equipped with (some flavour of) a hermitian form which is unimodular.We will begin in §2.1 by introducing the main definitions and establishing a few basic consequences.One of the simplest forms of Poincaré objects are the hyperbolic ones, which are the abstract analogue ofthe notion of hyperbolic quadratic forms. We will discuss these types of Poincaré objects in §2.2 and seehow their formation can be encoded as the action of a suitable Poincaré functor
Hyp( C ) → C from a certainPoincaré ∞ -category Hyp( C ) constructed from C . The Poincaré ∞ -category Hyp( C ) displays the interestingproperty that its Poincaré objects correspond to just objects in C , and we will study it in further depth in§7. We will also exploit this construction in order to prove that Poincaré objects with respect to symmetricPoincaré structures (Definition 1.2.11) correspond to C -fixed objects in C (see Proposition 2.2.11 below).In §2.3 we will study another important kind of Poincaré objects - the metabolic Poincaré objects. Thesecorrespond to metabolic forms in the classical sense, that is, forms which admit a Lagrangian. Similarly tothe hyperbolic case we will show how one can understand metabolic Poincaré objects via Poincaré objects ina certain Poincaré ∞ -category Met( C , Ϙ ) constructed from ( C , Ϙ ) . The notion of a metabolic Poincaré objectsis the main input in the definition of the L -groups of a given Poincaré ∞ -category (see Definition 2.3.11below). These are in fact the homotopy groups of the L -theory spectrum which was classically defined andstudied in the seminal work of Ranicki [Ran92], and transported to the context of Poincaré ∞ -categories byLurie [Lur11]. A key technique in studying L -group is Ranicki’s algebraic Thom construction , which wewill present in §2.3 in the setting of Poincaré ∞ -categories, and revisit in greater depth in §7.3.A key role in the present series of papers is played by the Grothendieck-Witt spectrum of a Poincaré ∞ -category, an invariant we will construct using the framework of cobordism categories in Paper [II]. Thezero’th homotopy group of the Grothendieck-Witt spectrum, also known as the Grothendieck-Witt group ,was classically defined in the context of rings as the group completion of the groupoid of unimodularforms (with respect to orthogonal sum). We will see how to define the Grothendieck-Witt group in theabstract setting of Poincaré ∞ -categories in §2.4, and extract some of its basic properties. In particular, theGrothendieck-Witt group GW ( C , Ϙ ) of a Poincaré ∞ -category ( C , Ϙ ) sits in an exact sequence K ( C ) C → GW ( C , Ϙ ) → L ( C , Ϙ ) → between the zero’th L -group of ( C , Ϙ ) and the C -orbits of the algebraic K -theory of C . This exact sequenceis in fact the tail of a long exact sequence issued from a fibre sequence of spectra K( C ) hC → GW( C , Ϙ ) → L( C , Ϙ ) that we will construct in Paper [II]. The existence of the above fibre sequence in this generality is a principalnovelty of our approach to hermitian K -theory, and yields a variety of consequences we will exploit in Paper[II], Paper [III] and Paper [IV].2.1. Hermitian and Poincaré objects.
In this section we will present the notions of hermitian and Poincaréobjects and extract some of their basic properties.2.1.1.
Definition.
Let ( C , Ϙ ) be a hermitian ∞ -category and 𝑥 ∈ C and an object. By a hermitian form on 𝑥 we will mean a point 𝑞 in the space Ω ∞ Ϙ ( 𝑥 ) . We will then refer to the pair ( 𝑥, 𝑞 ) as a hermitian object in C .Hermitian objects can be organized into an ∞ -category given by the total ∞ -category of the right fibration He( C , Ϙ ) ∶= ∫ 𝑥 ∈ C Ω ∞ Ϙ ( 𝑥 ) → C classified by the functor Ω ∞ Ϙ ∶ C op ⟶ S . We will refer to He( C , Ϙ ) as the ∞ -category of hermitian objects in ( C , Ϙ ) . We will denote by Fm( C , Ϙ ) ⊆ He( C , Ϙ ) the maximal subgroupoid of He( C , Ϙ ) , and refer to it asthe space of hermitian objects . Lemma.
The assignment ( C , Ϙ ) ↦ He( C , Ϙ ) canonically extends to a functor He ∶ Cat h∞ → Cat ∞ ,together with a natural transformation to the forgetful functor ( C , Ϙ ) ↦ C , whose component for a given ( C , Ϙ ) ∈ Cat h∞ is the defining right fibration He( C , Ϙ ) → C .Proof. The functor
Cat ex∞ ↪ Cat ∞ , together with the composed natural transformation Fun q (−) ⇒ Fun((−) op , S 𝑝 ) Ω ∞∗ ⟹ Fun((−) op , S ) , where Ω ∞∗ denotes post-composition with the infinite loop space functor Ω ∞ ∶ S 𝑝 → S , together inducesunder unstraightening a functor Cat h∞ → ∫ C ∈Cat ∞ Fun( C op , S ) . Invoking (the dual of) [GHN17, Corollary A.31], we may identify the Grothendieck construction on theright as ∫ C ∈Cat ∞ Fun( C op , S ) ≃ ∫ C ∈Cat ∞ RFib( C ) ≃ RFib where RFib( C ) denotes the ∞ -category of right fibrations over C and RFib ⊆ Ar(Cat ∞ ) is the full subcat-egory of the arrow category of Cat ∞ consisting of right fibrations. The resulting functor Cat h∞ → RFib → Ar(Cat ∞ ) then associates to a hermitian ∞ -category ( C , Ϙ ) the right fibration He( C , Ϙ ) → C , yielding thedesired functoriality. (cid:3) We will mostly be interested in hermitian forms which satisfy a unimodularity condition. To formulateit, we need to assume that ( C , Ϙ ) non-degenerate. In that case any hermitian object ( 𝑥, 𝑞 ) determines a map 𝑞 ♯ ∶ 𝑥 → D Ϙ ( 𝑥 ) as the image of 𝑞 under Ω ∞ Ϙ ( 𝑥 ) ⟶ Ω ∞ B Ϙ ( 𝑥, 𝑥 ) = Hom C ( 𝑥, D Ϙ ( 𝑥 )) . Definition.
We will say that a hermitian form 𝑞 on 𝑥 ∈ C is Poincaré if the associated map 𝑞 ♯ ∶ 𝑥 → D Ϙ ( 𝑥 ) is an equivalence. In this case we will also say that ( 𝑥, 𝑞 ) is a Poincaré object . We will denote by
Pn( C , Ϙ ) ⊆ Fm( C , Ϙ ) the full subgroupoid of Fm( C , Ϙ ) spanned by the Poincaré objects. We will refer to Pn( C , Ϙ ) ∈ S as the space of Poincaré objects in ( C , Ϙ ) .2.1.4. Remark.
Similarly to the ∞ -category He( C , Ϙ ) one could also form an ∞ -category of Poincaré objectsas a full subcategory of He( C , Ϙ ) . This construction is rather poorly behaved formally and will not play anyrole in this paper. Therefore we will only consider the space Pn( C , Ϙ ) of Poincaré objects here.2.1.5. Lemma. If ( 𝑓 , 𝜂 ) ∶ ( C , Ϙ ) → ( C ′ , Ϙ ′ ) is a non-degenerate hermitian functor between non-degeneratehermitian ∞ -categories then the induced functor 𝑓 ∗ ∶ Fm( C , Ϙ ) → Fm( C ′ , Ϙ ′ ) perserves Poincaré objects, that is, it maps the full subgroupoid Pn( C , Ϙ ) ⊆ Fm( C , Ϙ ) to the full subgroupoid Pn( C ′ , Ϙ ′ ) ⊆ Fm( C ′ , Ϙ ′ ) . In particular, the association ( C , Ϙ ) ↦ Pn( C , Ϙ ) thus extends to a functor Pn ∶ Cat p∞ → S . It is this functor Pn , that will play a pivotal role in the rest of the paper. Proof of Lemma 2.1.5.
By Remark 1.2.6 the natural transformation 𝜂 ∶ Ϙ → 𝑓 ∗ Ϙ ′ determines a commuta-tive diagram Ω ∞ Ϙ ( 𝑥 ) Ω ∞ Ϙ ′ ( 𝑓 ( 𝑥 ))Ω ∞ B Ϙ ( 𝑥, 𝑥 ) Ω ∞ B Ϙ ′ ( 𝑓 ( 𝑥 ) , 𝑓 ( 𝑥 ))Map C ( 𝑥, D Ϙ ( 𝑥 )) Map C ( 𝑓 ( 𝑥 ) , 𝑓 D Ϙ ( 𝑥 )) Map C ( 𝑓 ( 𝑥 ) , D Ϙ ′ 𝑓 ( 𝑥 )) Ω ∞ 𝜂 Ω ∞ 𝛽 𝜂 ≃ ≃ 𝑓 ( 𝜏 𝜂 ) ∗ In particular, if ( 𝑓 , 𝜂 ) is duality preserving then 𝜏 𝜂 is an equivalence and hence the top horizontal arrowsends Poincaré forms on 𝑥 to Poincaré forms on 𝑓 ( 𝑥 ) . (cid:3) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 31 Remark.
The map Ϙ ( 𝑥 ) → B Ϙ ( 𝑥, 𝑥 ) factors as Ϙ ( 𝑥 ) → B Ϙ ( 𝑥, 𝑥 ) hC → B Ϙ ( 𝑥, 𝑥 ) , see (8). It thenfollows that for any hermitian form 𝑞 on 𝑥 the corresponding map 𝑞 ♯ ∶ 𝑥 → D Ϙ ( 𝑥 ) is self-dual , that is, it isinvariant under the C -action on hom( 𝑥, D Ϙ 𝑥 ) ≃ B( 𝑥, 𝑥 ) . In particular, by Remark 1.2.3 there is a canonicalhomotopy rendering the diagram 𝑥 D Ϙ D op Ϙ ( 𝑥 )D Ϙ ( 𝑥 ) 𝑞 ♯ ev 𝑥 D Ϙ ( 𝑞 ♯ ) commutative.2.1.7. Remark.
Every Poincaré form 𝑞 on 𝑥 gives rise to a form ̂𝑞 on D Ϙ ( 𝑥 ) via the inverse of the inducedmap ( 𝑞 ♯ ) ∗ ∶ Ω ∞ Ϙ (D Ϙ ( 𝑥 )) → Ω ∞ Ϙ ( 𝑥 ) . By construction 𝑞 ♯ ∶ ( 𝑥, 𝑞 ) → (D Ϙ ( 𝑥 ) , ̂𝑞 ) is an equivalence in Pn( C , Ϙ ) and from Remark 2.1.6 we find ̂𝑞 ♯ ≃ D Ϙ ( 𝑞 ♯ ) −1 ∶ D Ϙ ( 𝑥 ) → D Ϙ D Ϙ ( 𝑥 ) . In particular, 𝑞 −1 ♯ ≃ ev −1 𝑥 ◦ ̂𝑞 .2.1.8. Example.
Let 𝑀 be a compact oriented topological 𝑛 -manifold with boundary 𝜕𝑀 ⊆ 𝑀 and fun-damental class [ 𝑀 ] ∈ 𝐻 𝑛 ( 𝑀, 𝜕𝑀 ) . Then the fundamental class together with the cup-product induces a (− 𝑛 ) -shifted hermitian form 𝑞 [ 𝑀 ] ∈ Ω ∞ Ϙ s[− 𝑛 ] ℤ ( 𝐶 ∗ ( 𝑀, 𝜕𝑀 )) = Map ℤ ( 𝐶 ∗ ( 𝑀, 𝜕𝑀 ) ⊗ 𝐶 ∗ ( 𝑀, 𝜕𝑀 ) , ℤ [− 𝑛 ]) hC sending ( 𝜑, 𝜓 ) to ( 𝜑 ∪ 𝜓 )([ 𝑀 ]) . We note that we are working with homological grading conventions, sothat, for example, the complex 𝐶 ∗ ( 𝑀, 𝜕𝑀 ) is concentrated in non-positive degrees with trivial homologyoutside the range [− 𝑛, . The associated map 𝑞 [ 𝑀 ] ♯ from 𝐶 ∗ ( 𝑀, 𝜕𝑀 ) to its dual can then be identified withthe canonical map 𝐶 ∗ ( 𝑀, 𝜕𝑀 ) → 𝐶 ∗ ( 𝑀 ) , which is an equivalence if and only if 𝐶 ∗ ( 𝜕𝑀 ) ≃ 0 , i.e., if and only if 𝜕𝑀 is empty. In particular, thehermitian form 𝑞 [ 𝑀 ] is Poincaré if and only if 𝑀 is closed.2.2. Hyperbolic and symmetric Poincaré objects.
Given a Poincaré ∞ -category ( C , Ϙ ) , the space ofPoincaré objects Pn( C , Ϙ ) is related to the underlying space of objects 𝜄 C in two different ways. First, onecan of course take a Poincaré object and forget its Poincaré form, yielding a forgetful map Pn( C , Ϙ ) → 𝜄 C .There is however also an interesting construction in the other direction, which takes a object 𝑥 ∈ C andassociates to it the object 𝑥 ⊕ D 𝑥 endowed with its hyperbolic Poincaré form, leading to a map 𝜄 C → Pn( C ) .Though these constructions seem different in nature, they are in fact closely related, and will both occupyour attention in this present section. A common feature they both share is equivariance with respect to the C -action on 𝜄 C induced by the duality. In the final part of this section we will show that when the Poincaréstructure is symmetric the resulting map Pn( C , Ϙ ) → 𝜄 C hC is an equivalence. We will further study therelationship between the hyperbolic and forgetful functors in §7.4 in the setting of C -categories.2.2.1. Definition.
Let ( C , Ϙ ) be a Poincaré ∞ -category with duality D . Given an object 𝑥 ∈ C we will denoteby hyp( 𝑥 ) ∈ Pn( C , Ϙ ) the Poincaré object whose underlying object is 𝑥 ⊕ D 𝑥 and whose Poincaré form isgiven by the the image of the identity under Map C ( 𝑥, 𝑥 ) (ev 𝑥 ) ∗ ←←←←←←←←←←←←←←←←←←←←←←←→ Map C ( 𝑥, DD( 𝑥 )) ≃ Ω ∞ B Ϙ ( 𝑥, D( 𝑥 )) ⟶ Ω ∞ Ϙ ( 𝑥 ⊕ D( 𝑥 )) . Unwinding the definitions one easily checks that this indeed defines a Poincaré object. We will refer to hyp( 𝑥 ) as the hyperbolic Poincaré object on 𝑥 .To understand systematically the role played by hyperbolic Poincaré objects in C it is most useful todescribe them as Poincaré objects in another Poincaré ∞ -category build from C .2.2.2. Definition.
Let C be a stable ∞ -category. We define its hyperbolic category Hyp( C ) to be the her-mitian ∞ -category whose underlying stable ∞ -category is C ⊕ C op , equipped with the hermitian structure Ϙ hyp ( 𝑥, 𝑦 ) = hom C ( 𝑥, 𝑦 ) . Unwinding the definitions, we see that the symmetric bilinear functor associated to the hyperbolic her-mitian structure is given by B hyp (( 𝑥, 𝑦 ) , ( 𝑥 ′ , 𝑦 ′ )) = hom C ( 𝑥, 𝑦 ′ ) ⊕ hom C ( 𝑥 ′ , 𝑦 ) , and its linear approximation is trivial. In particular, the bilinear functor B hyp is perfect with duality D hyp ( 𝑥, 𝑦 ) =( 𝑦, 𝑥 ) and consequently Hyp( C ) is always a Poincaré ∞ -category.2.2.3. Remark.
By construction, the quadratic functor Ϙ hyp is obtained by diagonally restricting the bilinearfunctor (( 𝑥, 𝑦 ) , ( 𝑥 ′ , 𝑦 ′ )) ↦ hom C ( 𝑥, 𝑦 ′ ) . It then follows that the canonical maps B hyp (( 𝑥, 𝑦 ) , ( 𝑥, 𝑦 )) → Ϙ hyp ( 𝑥, 𝑦 ) → B hyp (( 𝑥, 𝑦 ) , ( 𝑥, 𝑦 )) are given by the collapse and diagonal maps hom C ( 𝑥, 𝑦 ) ⊕ hom C ( 𝑥, 𝑦 ) → hom C ( 𝑥, 𝑦 ) → hom C ( 𝑥, 𝑦 ) ⊕ hom C ( 𝑥, 𝑦 ) , and Ϙ hyp coincides with both the quadratic and symmetric Poincaré structure associated to the symmetricbilinear functor B hyp .2.2.4. Remark.
The Poincaré ∞ -category Hyp( C ) is shift-invariant : for every 𝑛 ∈ ℤ the functor Σ 𝑛 ×id ∶ C × C op → C × C op refines to an equivalence ( C × C op , Ϙ hyp ) ≃ ( C × C op , Ϙ [ 𝑛 ]hyp ) , see Definition 1.2.16.For ( C , Ϙ ) a Poincaré ∞ -category with duality D , the associated hyperbolic category Hyp( C ) relates to C via Poincaré functors(42) Hyp( C ) hyp ←←←←←←←←←←←←←←→ ( C , Ϙ ) fgt ←←←←←←←←←←←←→ Hyp( C ) in both directions. Here the functor on the left in (42) is given by the exact functor ( 𝑥, 𝑦 ) ↦ 𝑥 ⊕ D 𝑦 ,promoted to a hermitian functor via the natural transformation hom C ( 𝑥, 𝑦 ) (ev 𝑦 ) ∗ ←←←←←←←←←←←←←←←←←←←←←←→ hom C ( 𝑥, DD 𝑦 ) ≃ B Ϙ ( 𝑥, D 𝑦 ) ⟶ Ϙ ( 𝑥 ⊕ D 𝑦 ) , while the second functor is given by the exact functor 𝑥 ↦ ( 𝑥, D 𝑥 ) , promoted to a hermitian functor via thenatural transformation Ϙ ( 𝑥 ) → B Ϙ ( 𝑥, 𝑥 ) → hom C ( 𝑥, D 𝑥 ) ≃ Ϙ hyp ( 𝑥, D 𝑥 ) . By definition, the ∞ -category He(Hyp( C )) sits in a right fibration He(Hyp( C )) → C ⊕ C op classified by the functor C × C op → S sending ( 𝑥, 𝑦 ) to the mapping space hom C ( 𝑥, 𝑦 ) . But this functor isalready known to classify the right fibration TwAr( C ) → C op ⊕ C where TwAr( C ) is the twisted arrow category of C (see, e.g., [Lur17, §5.2.1]), and we consequently obtainan equivalence He(Hyp( C )) ≃ TwAr( C ) over C ⊕ C op . In particular, hermitian objects in Hyp( C ) are simply given by arrows 𝛼 ∶ 𝑥 → 𝑦 in C , whilemorphisms between hermitian object correspond to diagrams in C of the form 𝑥 𝑥 ′ 𝑦 𝑦 ′ . 𝛼 𝛼 ′ Proposition.
For a stable ∞ -category C , the composite Pn(Hyp( C )) → 𝜄 C ⊕ 𝜄 C op → 𝜄 C is an equivalence of spaces. Here the first map is induced by the forgetful functor Cat p∞ → Cat ex∞ and the sec-ond is given by the projection onto the first factor. Under this equivalence, the natural map
Pn(Hyp( C )) → He(Hyp( C )) corresponds to the map 𝜄 C ≃ TwAr( 𝜄 C ) → TwAr( C ) . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 33 Proof.
An object [ 𝛼 ∶ 𝑥 → 𝑦 ] ∈ TwAr( C ) viewed as hermitian object ( 𝑥, 𝑦, 𝛼 ) in Hyp( C ) , has as associatedself dual map ( 𝑥, 𝑦 ) → D( 𝑥, 𝑦 ) = ( 𝑦, 𝑥 ) the map 𝛼 on both factors (viewed as either a map 𝑥 → 𝑦 in C or amap 𝑦 → 𝑥 in C op ). Consequently, the hermitian form ( 𝑥, 𝑦, 𝛼 ) is Poincaré if and only if 𝛼 is an equivalence.Together with the fact that right fibrations detect equivalences we obtain that Pn(Hyp( C )) ≃ TwAr( 𝜄 C ) ⊆ 𝜄 TwAr( C ) ≃ Fm(Hyp( C )) . We finish the proof by observing that the projection
TwAr( 𝜄 C ) → 𝜄 C is an equivalence since 𝜄 C is an ∞ -groupoid. (cid:3) Remark.
We will show in §7.4 that the association C ↦ Hyp( C ) organizes into a functor Cat ex∞ → Cat p∞ which is both left and right adjoint to the forgetful functor Cat p∞ → Cat ex∞ , with unit and counit givenby (42), see Corollary 7.2.20. Together with the corepresentability of Pn (Proposition 4.1.3) this will giveanother proof of Proposition 2.2.5.In light of Proposition 2.2.5 the hermitian functors (42) now induce a pair of maps(43) 𝜄 C → Pn( C , Ϙ ) → 𝜄 C Unwinding the definitions we see that the functor on the left sends an object 𝑥 to the associated hyperbolicPoincaré object hyp( 𝑥 ) , while, the functor on the right sends a Poincaré object ( 𝑥, 𝑞 ) to the underlyingobject 𝑥 . A key feature of both these maps is that they are C -equivariant with respect to the C -action on 𝜄 C induced by the duality of Ϙ and the trivial action on Pn( C , Ϙ ) . To make this idea precise will first constructthis action on the level of the Poincaré ∞ -category Hyp( C ) .2.2.7. Construction.
Let ( C , Ϙ ) be a Poincaré ∞ -category with associated duality D = D Ϙ . We construct a C -action on Hyp( C ) ∈ Cat p∞ as follows. To begin, consider the equivalence of stable ∞ -categories id × D op ∶ C × C ≃ ←←←←←←←→ C × C op . Transporting the flip C -action on C × C to C × C op we thus obtain a C -action on C × C op , given informally bythe formula ( 𝑥, 𝑦 ) ↦ (D 𝑦, D op 𝑥 ) . We wish to promote this action to C -action on the Poincaré ∞ -category Cat p∞ . Since every equivalence in Cat h∞ is a Poincaré functor may equivalently construct a C -action on Hyp( C ) as a hermitian ∞ -category. By the construction of Cat h∞ as the unstraightening of the functor C ↦ Fun q ( C ) , lifting the above C -action on C × C op to Hyp( C ) is equivalent to giving a C -fixed pointstructure on Ϙ hyp ∈ Fun q ( C × C op ) with respect to the induced C -action on Fun q ( C × C op ) . Since the relevant C -action was transported from the flip action on C × C via the equivalence (id , D op ) we may equivalentlyconstruct a C -fixed point structure on the quadratic functor (id × D op ) ∗ Ϙ hyp . The latter is readily discoveredto be [(id × D op ) ∗ Ϙ hyp ]( 𝑥, 𝑦 ) = hom C ( 𝑥, D 𝑦 ) = B Ϙ ( 𝑥, 𝑦 ) and so we need to construct a C -fixed point structure on B Ϙ , considered as an object of Fun q ( C × C ) . But B Ϙ lies in the full subcategory Fun b ( C ) ⊆ Fun q ( C × C ) , where it is equipped with a C -fixed structure byvirtue of Lemma 1.1.9.2.2.8. Remark.
The C -action on Hyp( C ) constructed in 2.2.7 induces a C -action on He(Hyp( C )) ≃TwAr( C ) . Unwinding the definitions, this actions sends an arrow [ 𝑥 → 𝑦 ] ∈ TwAr( C ) to the dual ar-row D Ϙ 𝑦 → D Ϙ 𝑥 . Similarly, this C -action determines an action on Pn(Hyp( C )) . Under the identification Hyp( C ) ≃ 𝜄 C of Proposition 2.2.5 this action can be written simply by 𝑥 ↦ D Ϙ 𝑥 . Here we point out thatsince 𝜄 C is an ∞ -groupoid it is canonically equivalent to its opposite via an equivalence which sends everyarrow to its inverse. Hence the contravariant equivalence D becomes a self-equivalence on the level of 𝜄 C .We may also state this as follows: the C -action (−) op ∶ Cat ∞ → Cat ∞ admits a canonical trivializationalong the full subcategory S ⊆ Cat ∞ (in fact, the space of self-equivalences of S is contractible by its uni-versal property [Lur09a, Theorem 5.1.5.6]), yielding an identification S hC ≃ Fun(BC , S ) . The duality D Ϙ then induces a duality on 𝜄 C and hence a C -action.2.2.9. Lemma.
Let ( C , Ϙ ) be a Poincaré ∞ -category with duality D = D Ϙ . Then the functors Hyp( C ) hyp ←←←←←←←←←←←←←←→ ( C , Ϙ ) fgt ←←←←←←←←←←←←→ Hyp( C ) both admit a distinguished refinement to C -equivariant maps with respect to the C -action on Hyp( C ) construction in 2.2.7 and the trivial action on ( C , Ϙ ) . Corollary.
The induced maps on Poincaré objects (which we denote by the same name) 𝜄 C hyp ←←←←←←←←←←←←←←→ Pn( C , Ϙ ) fgt ←←←←←←←←←←←←→ 𝜄 C are C -equivariant with respect to the duality induced action on 𝜄 C and the trivial action on Pn( C , Ϙ ) .Proof of Lemma 2.2.9. We first construct the C -equivariant structure on the underlying exact functors.For this, note that since the C -action on Hyp was constructed by transporting the flip action along theequivalence id × D op ∶ C × C → C × C op and D op D ≃ id it will suffice to to promote the resulting exactfunctors C × C → C → C × C to C -equivariant exact functors. Indeed, these are just the diagonal and fold map of C as an object inthe semi-additive ∞ -category Cat ex∞ , which are both canonically C -equivariant. To lift the resulting C -equivariant structure on (id , D op ) ∶ C → C × C op to a C -equivariant structure on the Poincaré functor f gt we need to promote the associated natural transformation Ϙ ⇒ (id , D op ) ∗ Ϙ hyp to a C -equivariant map in Fun q ( C ) . Transporting the problem again along the equivalence id × D op weneed to put a C -equivariant structure on the natural transformation Ϙ ⇒ Δ ∗ B where Δ ∶ C × C is the diagonal. Indeed, this is established in Lemma 1.1.10. By the same argument wesee that in order to obtain the desired C -equivariant structure on hyp it will suffice to put a C -equivariantstructure on the natural transformation B ⇒ ∇ ∗ Ϙ , where ∇ ∶ C ⊕ C → C is the collapse functor ( 𝑥, 𝑦 ) ↦ 𝑥 ⊕ 𝑦 . Using the adjunction between restriction andleft Kan extension we may instead put a C -equivariant structure on the adjoint transformation ∇ ! B ≃ Δ ∗ B ⇒ Ϙ . where we used that left Kan extension along ∇ op is obtained by restriction along its right adjoint Δ op ∶ C op → C op × C op . The desired C -equivariant structure was again established in Lemma 1.1.10. (cid:3) Let us now focus on the Poincaré functor f gt ∶ ( C , Ϙ ) → Hyp( C ) . Upon taking hermitian and Poincaréobjects (and using Proposition 2.2.5) this Poincaré functor induces a commutative diagram(44) Pn( C , Ϙ ) Fm( C , Ϙ ) He( C , Ϙ ) ( 𝑥, 𝑞 ) 𝜄 C 𝜄 TwAr( C ) TwAr( C ) [ 𝑞 ♯ ∶ 𝑥 → D Ϙ 𝑥 ]∋∋ in which the vertical maps inherit from f gt a C -equivariant structure with respect to the trivial action ontheir domains and the C -action induced by the C -action on Hyp( C ) on the target. Here the left squareconsists only of spaces and the horizontal maps are (up to equivalence) inclusions of components: for theupper left map these are the components of He( C , Ϙ ) consisting of Poincaré objects and for the lower left mapthese are the components of 𝜄 TwAr( C ) consisting of those arrows [ 𝑥 → 𝑦 ] which are equivalences. Sinceby definition a hermitian object ( 𝑥, }) ′ is Poincaré if and only if 𝑞 ♯ is an equivalence we see in particularthat the left square is cartesian. Now by the C -equivariance above the external rectangle in (44) induces acommutative square(45) Pn( C , Ϙ ) He( C , Ϙ ) ( 𝑥, 𝑞 )( 𝜄 C ) hC TwAr( C ) hC [ 𝑞 ♯ ∶ 𝑥 → D Ϙ 𝑥 ]∋∋ Proposition. If Ϙ = Ϙ sB is a symmetric Poincaré structure of some symmetric bilinear form B thenthe vertical maps in (45) are equivalences. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 35 Proof.
Consider the extended diagram(46)
Pn( C , Ϙ ) Fm( C , Ϙ ) He( C , Ϙ ) C TwAr( 𝜄 C ) hC 𝜄 TwAr( C ) hC TwAr( C ) hC ( C × C op ) hC TwAr( 𝜄 C ) 𝜄 TwAr( C ) TwAr( C ) C × C op in which the external rectangle of the left column is cartesian as observed above. In addition, since the fibresof the map TwAr( 𝜄 C ) → 𝜄 TwAr( C ) are (−1) -truncated then bottom left square is cartesian as well, and hencethe top left square is cartesian. Similarly, since the map from the homotopy fixed point is conservative, thebottom central square and therefore the top central square, are cartesian.Now the map C → C × C op is equivalent as a C -equivariant arrow to the diagonal inclusion C → C × C ,which exhibits the C -object C × C as coinduced from C . This implies in particular that the top right verticalmap in (46) is an equivalence. Thus to conclude it suffices to show that the top right square is cartesian.Now consider the right most column in (46). Since homotopy fixed points commute with fibre prod-ucts the fibres of the middle horizontal map over a fixed object in C is the homotopy fixed points of thecorresponding fibre of the bottom horizontal map in the same column. We hence obtain that the map onhorizontal fibres in the top right square can be identified with the induced map Ω ∞ Ϙ ( 𝑥 ) → Map C ( 𝑥, D 𝑥 ) hC , which is an equivalence by the assumption that Ϙ = Ϙ sB for some B . It then follows that the second and thirdvertical arrows in the top of (46) are equivalences. The left most vertical map in that row is consequentlyan equivalence as well since the top left square is cartesian. (cid:3) Metabolic objects and L -groups. In this section we will introduce the notion of a metabolic
Poincaréobject and use it to define the L -groups of a Poincaré ∞ -category. In the context of modules over rings thesewere first defined by Wall and Ranicki in their seminal work on surgery theory [Wal99], and were transportedto the setting of Poincaré ∞ -categories in [Lur11]. We will then develop an analogue of Ranicki’s algebraicThom construction [Ran80, Proposition 3.4] in this context. This construction will play an important rolein the framework of algebraic surgery which we will set up in Paper [II].2.3.1. Definition.
Let ( C , Ϙ ) be a Poincaré ∞ -category and ( 𝑥, 𝑞 ) a Poincaré object. By an isotropic object over 𝑥 we will mean a pair ( 𝑓 ∶ 𝑤 → 𝑥, 𝜂 ) where 𝑓 ∶ 𝑤 → 𝑥 is a map in C and 𝜂 ∶ 𝑓 ∗ 𝑞 ∼ 0 ∈ Ω ∞ Ϙ ( 𝑤 ) is anull-homotopy of the restriction of 𝑞 to 𝑤 . We will say that an isotropic object ( 𝑤 → 𝑥, 𝜂 ) is a Lagrangian if the null-homotopy of 𝑤 → 𝑥 ≃ D 𝑥 → D 𝑤 given by the image of 𝜂 in Ω ∞ B Ϙ ( 𝑤, 𝑤 ) = hom C ( 𝑤, D 𝑤 ) exhibits the sequence 𝑤 → 𝑥 → D 𝑤 as exact. We will say that ( 𝑥, 𝑞 ) is metabolic if it admits a Lagrangian.2.3.2. Example. If ( C , Ϙ ) is a Poincaré ∞ -category and 𝑥 ∈ C an object then the associated hyperbolicPoincaré object hyp( 𝑥 ) is metabolic with Lagrangian given by the component inclusion 𝑥 → 𝑥 ⊕ D 𝑥 .2.3.3. Example. In D p ( 𝔽 ) , let 𝑉 be a 2-dimensional 𝔽 -vector space with basis 𝑣, 𝑢 equipped with thesymmetric bilinear form 𝑏 ∶ 𝑉 ⊗ 𝔽 𝑉 → 𝔽 given by 𝑏 ( 𝑣, 𝑣 ) = 0 and 𝑏 ( 𝑣, 𝑢 ) = 𝑏 ( 𝑢, 𝑣 ) = 𝑏 ( 𝑢, 𝑢 ) = 1 . Thenthe Poincaré object ( 𝑉 , 𝑏 ) ∈ Pn( D p ( 𝔽 ) , Ϙ s 𝔽 ) is metabolic with Lagrangian 𝐿 = ⟨ 𝑣 ⟩ ↪ 𝑉 but 𝐿 is notisomorphic to hyp( 𝑈 ) for any 𝑈 ∈ D p ( 𝔽 ) . Indeed, since any object in D p ( 𝔽 ) breaks as a direct sum ofshifts of 𝔽 the only possible candidate is 𝑈 = 𝔽 , but ( 𝑉 , 𝑏 ) is not isomorphic to hyp( 𝔽 ) . In particular, notevery metabolic object is hyperbolic.2.3.4. Example.
Let 𝑀 be a closed oriented 𝑛 -manifold with fundamental class [ 𝑀 ] ∈ 𝐻 𝑛 ( 𝑀 ; ℤ ) , so thatwe have a symmetric Poincaré form 𝑞 [ 𝑀 ] ∈ Ω ∞ Ϙ 𝑠 [− 𝑛 ] ℤ ( 𝐶 ∗ ( 𝑀 )) as in Example 2.1.8. If 𝑊 is now an oriented ( 𝑛 + 1) -manifold with boundary 𝑀 then the relative fundamental class [ 𝑊 ] ∈ 𝐻 𝑛 +1 ( 𝑊 , 𝑀 ) can be usedto promote the map 𝐶 ∗ ( 𝑊 ) → 𝐶 ∗ ( 𝑀 ) to a Lagrangian of ( 𝐶 ∗ ( 𝑀 ) , 𝑞 [ 𝑀 ] ) . This can be considered as an algebraic reflection of the fact that 𝑊 exhibits 𝑀 as a boundary . In particular, if ( 𝐶 ∗ ( 𝑀 ) , 𝑞 [ 𝑀 ] ) is not metabolic then 𝑀 is not the boundary ofany oriented ( 𝑛 + 1) -manifold, that is, 𝑀 is not (oriented-ly) null-cobordant.As in the case of hyperbolic Poincaré objects, it would be desirable to have a description of metabolicPoincaré objects in terms of Poincaré objects in another Poincaré ∞ -category constructed from ( C , Ϙ ) .2.3.5. Definition.
For a Poincaré ∞ -category ( C , Ϙ ) , we define the associated metabolic category Met( C , Ϙ ) to be the hermitian ∞ -category with underlying ∞ -category Ar( C ) = Fun(Δ , C ) and hermitian structure Ϙ met ∶ Ar( C ) op = Ar( C op ) Ar( Ϙ ) ←←←←←←←←←←←←←←←←←←←←←→ Ar( S 𝑝 ) fib ←←←←←←←←←←←←→ S 𝑝 whose value on arrows is Ϙ met ([ 𝑤 → 𝑥 ]) = f ib( Ϙ ( 𝑥 ) → Ϙ ( 𝑤 )) .Unwinding the definitions we see that the underlying symmetric bilinear functor of Ϙ met is B met ( 𝑤 → 𝑥, 𝑤 ′ → 𝑥 ′ ) = f ib[B Ϙ ( 𝑥, 𝑥 ′ ) → B Ϙ ( 𝑤, 𝑤 ′ )] . From this formula we see that if B Ϙ is perfect with duality D then B met is perfect with duality D met ( 𝑤 → 𝑥 ) = ( f ib[D 𝑥 → D 𝑤 ]) → D 𝑥 ) . We note that by definition a hermitian form on [ 𝑓 ∶ 𝑤 → 𝑥 ] with respect to Ϙ met consists of a form 𝑞 ∈Ω ∞ Ϙ ( 𝑥 ) together with a null-homotopy 𝜂 of 𝑓 ∗ 𝑞 ∈ Ω ∞ Ϙ ( 𝑤 ) . Such a Ϙ met -form ( 𝑞, 𝜂 ) is Poincaré if and onlyif the associated self dual map encoded by the horizontal maps of the square(47) 𝑤 f ib[D 𝑥 → D 𝑤 ] 𝑥 D 𝑥 𝜂 ♯ 𝑞 ♯ is an equivalence. Here 𝑞 ♯ is the self dual map determined by 𝑞 and we denoted by 𝜂 ♯ the map correspondingto the null-homotopy of the composed map 𝑤 → 𝑥 → D 𝑥 → D 𝑤 determined by the image of 𝜂 in Ω ∞ B( 𝑤, 𝑤 ) = Map( 𝑤, D 𝑤 ) . We then see that (47) constitutes an equivalence between the vertical arrowsif and only if 𝑞 ♯ ∶ 𝑥 → D 𝑥 is an equivalence and the resulting sequence 𝑤 → 𝑥 ≃ D 𝑥 → D 𝑤 is exact, that is,if ( 𝑓 ∶ 𝑤 → 𝑥, 𝜂 ) is a Lagrangian. We may thus conclude that Poincaré objects in Met( C , Ϙ ) correspond tometabolic objects in ( C , Ϙ ) , or, more precisely, to Poincaré objects in C equipped with a specified Lagrangian.2.3.6. Definition.
For a Poincaré ∞ -category ( C , Ϙ ) we will denote by Pn 𝜕 ( C , Ϙ ) ∶= Pn(Met( C , Ϙ )) the space of Poincaré objects in Met( C , Ϙ ) , which we consider as above as the space of Poincaré objectsequipped with a specified Lagrangian.2.3.7. Lemma.
The maps (48) ( C , Ϙ [−1] ) 𝑖 ←←←←→ Met( C , Ϙ ) met ←←←←←←←←←←←←←←→ ( C , Ϙ ) given respectively by 𝑖 ( 𝑥 ) = [ 𝑥 → and met([ 𝑤 → 𝑥 ]) = 𝑥 extend to morphisms in Cat p∞ .Proof. For the first we observe that the map 𝑖 is fully faithful, that 𝑖 ∗ Ϙ met ≃ Ω Ϙ and that the image of 𝑖 isclosed under the duality in Met( C , Ϙ ) , so the result follows from Observation 1.2.19. For the second mapwe take the hermitian structure associated to the canoical map Ϙ met ([ 𝑤 → 𝑥 )) = f ib[ Ϙ ( 𝑥 ) → Ϙ ( 𝑤 )] → Ϙ ( 𝑥 ) . By the explicit description of the duality above we see that the resulting hermitian functor is Poincaré. (cid:3)
Unwinding the definitions we see that the map(49) Pn 𝜕 ( C , Ϙ ) → Pn( C , Ϙ ) induced by the right hand Poincaré functor in (48) corresponds to the forgetful map which takes a Poincaréobject equipped with a Lagrangian and forgets the Lagrangian. In particular, a Poincaré object in ( C , Ϙ ) ismetabolic if and only if it is in the image of (49) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 37 As observed earlier, every hyperbolic form is metabolic, but not every metabolic object is equivalent tothe associated hyperbolic form on its Lagrangian. This relation between metabolic and hyperbolic objects isbest expressed by relating the Poincaré ∞ -categories Met( C , Ϙ ) and Hyp( C ) via suitable Poincaré functors.2.3.8. Construction.
Let ( C , Ϙ ) be a Poincaré ∞ -category. We define Poincaré functors Hyp( C ) Met( C , Ϙ ) Met( C , Ϙ ) Hyp( C )( 𝑥, 𝑦 ) ( 𝑥 → 𝑥 ⊕ D 𝑦 ) ( 𝑤 → 𝑥 ) ( 𝑤, D cof[ 𝑤 → 𝑥 ]) can lag and Hyp( C ) Met( C , Ϙ ) Met( C , Ϙ ) Hyp( C )( 𝑥, 𝑦 ) (D 𝑦 → 𝑥 ⊕ D 𝑦 ) ( 𝑤 → 𝑥 ) (cof[ 𝑤 → 𝑥 ] , D 𝑤 ) dcan dlag via the indicated formulas on the level on the underlying exact functors, and with hermitian structures asfollows. For the two functors on the left hand side the hermitian structure is obtained via the identification hom C ( 𝑥, 𝑦 ) ≃ B Ϙ ( 𝑥, D 𝑦 ) ≃ f ib[ Ϙ ( 𝑥 ⊕ D 𝑦 ) → Ϙ ( 𝑥 ) ⊕ Ϙ (D 𝑦 )] , whose target visibly projects to both f ib[ Ϙ ( 𝑥 ⊕ D 𝑦 ) → Ϙ ( 𝑥 )] and f ib[ Ϙ ( 𝑥 ⊕ D 𝑦 ) → Ϙ (D 𝑦 )] . For the functorson the right hand side the hermitian structure is given by the natural transformation f ib[ Ϙ ( 𝑥 ) → Ϙ ( 𝑤 )] → f ib[B Ϙ ( 𝑥, 𝑤 ) → B Ϙ ( 𝑤, 𝑤 )] ≃ B Ϙ (cof[ 𝑤 → 𝑥 ] , 𝑤 ) where we recognize the target as naturally equivalent to both lag ∗ B hyp and dlag ∗ B hyp . The preservation ofthe duality by these hermitian functors is visible by the explicit descriptions of D met and D hyp above. Wealso note that the composites Hyp( C ) can ←←←←←←←←←←←←←→ Met( C , Ϙ ) lag ←←←←←←←←←←←←→ Hyp( C ) and Hyp( C ) dcan ←←←←←←←←←←←←←←←←←→ Met( C , Ϙ ) dlag ←←←←←←←←←←←←←←←←→ Hyp( C ) are naturally equivalent to the identity, and so exhibit Hyp( C ) as a retract of Met( C , Ϙ ) in Cat p∞ .2.3.9. Remark.
The Poincaré functors lag , dlag∶ Met( C ) → Hyp( C ) are closely related: they differ bypost-composition with the Poincaré involution Hyp( C ) ≃ ←←←←←←←→ Hyp( C ) of Construction 2.2.7. Similarly, thePoincaré functors can , dcan∶ Hyp( C ) → Met( C ) differ by pre-composition with this involution. It thenfollows from Corollary 2.2.10 that the composite Met( C , Ϙ ) → Hyp( C ) hyp ←←←←←←←←←←←←←←→ ( C , Ϙ ) is independent of whether the first functor is lag or dlag . The action of this composed functor on Pn(−) sends a Poincaré object ( 𝑥, 𝑞 ) equipped with a Lagrangian 𝑤 → 𝑥 to the associated hyperbolic object hyp( 𝑤 ) ≃ hyp(D 𝑤 ) . The difference between a metabolic Poincaré object and its hyperbolic counterpartplays a key role in the definition of the Grothendieck-Witt group, see §2.4 below. On the other hand, wealso observe that the composite Poincaré functor Hyp( C ) → Met( C , Ϙ ) met ←←←←←←←←←←←←←←→ ( C , Ϙ ) coincides with the functor hyp of (42), independently of whether the first functor is can or dcan . On thelevel of Poincaré objects we may interpret this as the observation that a hyperbolic Poincaré object hyp( 𝑤 ) can be considered as a metabolic object in two canonical ways: one via the Lagrangian 𝑤 → 𝑤 ⊕ D 𝑤 andone via the Lagrangian D 𝑤 → 𝑤 ⊕ D 𝑤 .A fundamental invariant of Poincaré ∞ -categories is their L -groups. To define them, we first observethat the set 𝜋 Pn( C , Ϙ ) of equivalence classes of Poincaré objects carries a natural commutative monoidstructure, with sum given by [ 𝑥, 𝑞 ] + [ 𝑥 ′ , 𝑞 ′ ] = [ 𝑥 ⊕ 𝑥 ′ , 𝑞 ⟂ 𝑞 ′ ] for [ 𝑥, 𝑞 ] , [ 𝑥 ′ , 𝑞 ′ ] ∈ 𝜋 Pn( C , Ϙ ) , where 𝑞 ⟂ 𝑞 ′ ∈ Ω ∞ Ϙ ( 𝑥 ⊕ 𝑥 ′ ) ≃ Ω ∞ Ϙ ( 𝑥 ) × Ω ∞ Ϙ ( 𝑥 ′ ) × Ω ∞ B Ϙ ( 𝑥, 𝑥 ′ ) corresponds to the tuple ( 𝑞, 𝑞 ′ , . Though this commutative monoid is generally not a group, every elementis invertible up to the class of metabolic objects. More precisely, we have the following:2.3.10. Lemma.
Let ( C , Ϙ ) be Poincaré ∞ -category. Then the cokernel of the map 𝜋 Pn 𝜕 ( C , Ϙ ) → 𝜋 Pn( C , Ϙ ) in the category of commutative monoids is a group. Explicitly, the inverse to [ 𝑥, 𝑞 ] is given by [ 𝑥, − 𝑞 ] .Proof. This follows from the fact that ( 𝑥 ⊕ 𝑥, 𝑞 ⟂ − 𝑞 ) is metabolic with Lagrangian given by the diagonalinclusion 𝑥 → 𝑥 ⊕ 𝑥 with the canonical null-homotopy 𝑞 + (− 𝑞 ) ∼ 0 . (cid:3) Definition.
Let ( C , Ϙ ) be a Poincaré ∞ -category. For 𝑛 ∈ ℤ we define the 𝑛 ’th L -group of ( C , Ϙ ) by L 𝑛 ( C , Ϙ ) ∶= coker[ 𝜋 Pn 𝜕 ( C , Ϙ [− 𝑛 ] ) → 𝜋 Pn( C , Ϙ [− 𝑛 ] )] which is an abelian group by Lemma 2.3.10.The terminology of L -groups goes back to Wall [Wal99], who defined the quadratic L -groups of a (notnecessarily commutative) ring with anti-involution. In the case of fields the zero’th quadratic L -group wasfirst defined by Witt [Wit37] and was later became to be known as the Witt group , playing an important rolein arithmetic geometry through its relation to Milnor’s K-theory and Galois cohomology as formulated inMilnor’s conjecture, proven by Veovodsky in a celebrated application of motivic homotopy theory. HigherWitt groups were later defined by Balmer [Bal05] in the setting of triangulated categories. These wereknown to coincide with Ranicki-Wall L -groups when is invertible, see [Bal05, §1.3]. For the preciserelation between classical L -groups and the ones defined above in the setting of Poincaré ∞ -categories, seeRemark 4.2.9 below.2.3.12. Remark.
The collection of L -groups are in fact the homotopy groups of a spectrum valued invariant L( C , Ϙ ) , known as the L -theory spectrum. A definition in the setting of Poincaré ∞ -categories was givenin [Lur11], but was defined much earlier in the setting of rings with anti-involution by Ranicki [Ran92],and plays a key role in surgery theory. We will give a precise definition of this invariant in Paper [II], proveits main properties and characterize it by a universal property. The interaction with the closely relatedGrothendieck-Witt spectrum is one of the principal themes of the present series of papers.2.3.13. Remark.
It follows from Lemma 2.3.10 that if ( 𝑓 , 𝜂 ) ∶ ( C , Ϙ ) → ( C ′ , Ϙ ′ ) is a Poincaré functor thenthe two induced abelian group homomorphisms ( 𝑓 , 𝜂 ) ∗ , ( 𝑓 , − 𝜂 ) ∗ ∶ L 𝑛 ( C , Ϙ ) → L 𝑛 ( C ′ , Ϙ ′ ) differ by a sign, where − 𝜂 is an additive inverse to 𝜂 in the E ∞ -group of natural transformations Ϙ → 𝑓 ∗ Ϙ ′ (well-defined up to homotopy).2.3.14. Remark.
We note that the Poincaré functor ( 𝑖, 𝜂 ) ∶ ( C , Ϙ [−1] ) → Met( C , Ϙ ) constructed in Lemma 2.3.7is fully-faithful and the natural transformation 𝜂 ∶ Ϙ ⇒ 𝑖 ∗ Ϙ met is an equivalence. It then follows that theinduced map 𝜋 Pn( C , Ϙ [−1] ) → 𝜋 Pn(Met( C , Ϙ )) = 𝜋 Pn 𝜕 ( C , Ϙ ) is injective. Since the essential image of 𝑖 coincides with the full subcategory spanned by those objects whose image under met ∶ Met( C , Ϙ ) → ( C , Ϙ ) is zero it follows that the sequence of monoids → 𝜋 Pn( C , Ϙ [−1] ) → 𝜋 Pn 𝜕 ( C , Ϙ ) → 𝜋 Pn( C , Ϙ ) is exact. We may consequently identify L 𝑛 ( C , Ϙ ) with the “middle homology” of the sequence of monoids 𝜋 Pn 𝜕 ( C , Ϙ [− 𝑛 ] ) → 𝜋 Pn 𝜕 ( C , Ϙ [− 𝑛 +1] ) → 𝜋 Pn 𝜕 ( C , Ϙ [− 𝑛 +2] ) . Given now a Poincaré ∞ -category ( C , Ϙ ) , the map Pn( C , Ϙ [−1] ) → Pn 𝜕 ( C , Ϙ ) induced by the the Poincaréfunctor of Lemma 2.3.7, sends a (−1) -shifted Poincaré object ( 𝑥, 𝑞 ) to the metabolic Poincaré object equipped with 𝑥 as its Lagrangian. In particular, we observe that the data of a Lagrangians of is equivalentto that of a Poincaré object with respect to the shifted Poincaré structure Ϙ [−1] . This idea fits in the moregeneral paradigm of the algebraic Thom isomorphism developed by Ranicki [Ran80, Proposition 3.4], underwhich Poincaré objects in ( C , Ϙ ) equipped with a Lagrangian can equivalently be encoded via a hermitianobject with respect to Ϙ [−1] . To construct this equivalence, it will be useful to introduce the followingconstruction: ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 39 Definition.
Let ( C , Ϙ ) be a Poincaré ∞ -category. We will denote by Ar( C , Ϙ ) the hermitian ∞ -category whose underlying stable ∞ -category is the arrow category Ar( C ) and whose quadratic functor Ϙ ar sits in a pullback diagram(50) Ϙ ar ([ 𝑓 ∶ 𝑧 → 𝑤 ]) Ϙ ( 𝑧 )B Ϙ ( 𝑧, 𝑤 ) B Ϙ ( 𝑧, 𝑧 ) 𝑓 ∗ where the vertical map is the canonical one from a quadratic functor to its diagonally restricted bilinear partwhile the bottom horizontal map is the natural transformation whose component at 𝑓 is given by restrictionalong 𝑓 .2.3.16. Remark.
For [ 𝑧 → 𝑤 ] ∈ Ar( C ) , the commutative square Ϙ ( 𝑤 ) Ϙ ( 𝑧 )B Ϙ ( 𝑧, 𝑤 ) B Ϙ ( 𝑧, 𝑧 ) 𝑓 ∗ determines a natural map Ϙ ( 𝑤 ) → Ϙ ar ( 𝑧 → 𝑤 ) . From Corollary 1.1.21 we then get that for an exact sequence 𝑧 → 𝑤 → 𝑥 the associated sequence Ϙ ( 𝑥 ) → Ϙ ( 𝑤 ) → Ϙ ar ( 𝑧 → 𝑤 ) is exact.Unwinding the definitions we see that the underlying symmetric bilinear functor of Ϙ ar sits in a fibresquare B ar ([ 𝑧 → 𝑤 ] , [ 𝑧 ′ → 𝑤 ′ ]) B Ϙ ( 𝑤, 𝑧 ′ )B Ϙ ( 𝑧, 𝑤 ′ ) B Ϙ ( 𝑧, 𝑧 ′ ) from which we we see that when B Ϙ is perfect with duality D then B ar is perfect with duality D ar ([ 𝑓 ∶ 𝑧 → 𝑤 ]) = [D 𝑓 ∶ D 𝑤 → D 𝑧 ] . In this case, identifying B Ϙ ( 𝑧, 𝑤 ) ≃ hom C ( 𝑤, D 𝑧 ) , we see that a hermitian form on an arrow [ 𝑓 ∶ 𝑧 → 𝑤 ] ∈ Ar( C ) consists of a triple ( 𝑞, 𝑔, 𝜂 ) where 𝑞 is hermitian form on 𝑧 with respect to Ϙ , 𝑔 ∶ 𝑤 → D 𝑧 isa map in C , and 𝜂 is a homotopy 𝑞 ♯ ∼ 𝑔 ◦ 𝑓 between the resulting two maps from 𝑧 to D 𝑧 . The self-dualmap [ 𝑧 → 𝑤 ] → [D 𝑤 → D 𝑧 ] associated to such a triple can then be expressed as the map between the twovertical arrows in the square 𝑧 D 𝑤𝑤 D 𝑧 D 𝑔𝑓 𝑞 ♯ D 𝑓𝑔 In particular, a triple ( 𝑞, 𝑔, 𝜂 ) constitutes a Poincaré form if and only if 𝑔 is an equivalence. We now notethat the map Ϙ ar ([ 𝑧 → 𝑤 ]) → Ϙ ( 𝑧 ) appearing in the defining square (50) promotes the domain projection Ar( C ) → C [ 𝑧 → 𝑤 ] ↦ 𝑧 to a hermitian functor and hence determines a map(51) He(Ar( C , Ϙ )) → He( C , Ϙ ) given on the level of tuples as above by ( 𝑧 → 𝑤, 𝑞, 𝑔, 𝜂 ) ↦ ( 𝑧, 𝑞 ) . We then have the following:2.3.17. Proposition.
The map (51) restricts to an equivalence
Pn(Ar( C , Ϙ )) → Fm( C , Ϙ ) ( 𝑧 → 𝑤, 𝑞, 𝑔, 𝜂 ) ↦ ( 𝑧, 𝑞 ) . An explicit inverse is given by the association ( 𝑧, 𝑞 ) ↦ ( 𝑞 ♯ ∶ 𝑧 → D 𝑧, 𝑞, id , id) . Concisely stated, Proposition 2.3.17 says that for a Poincaré ∞ -category ( C , Ϙ ) , hermitian objects in C can be described via Poincaré objects in its arrow category. Though a direct proof is perfectly possible atthe moment, we will postpone it to §7.3, where we will prove it this statement in a more general context inProposition 7.3.5. Meanwhile, let us connect the present conclusions to the above discussion of metabolicobjects.2.3.18. Notation.
For a stable ∞ -category C we will denote by Seq( C ) ⊆ Fun(Δ × Δ , C ) the full subcate-gory spanned by the exact squares of the form(52) 𝑧 𝑤 𝑥 In other words,
Seq( C ) is the ∞ -category of exact sequences [ 𝑧 → 𝑤 → 𝑥 ] in C , where we will often omitthe null-homotopy encoded by the commutative square (52) to simplify notation.We have two projections(53) Ar( C ) Seq( C ) Ar( C )[ 𝑧 → 𝑤 ] [ 𝑧 → 𝑤 → 𝑥 ] [ 𝑤 → 𝑥 ] Lemma.
The projections (53) are both equivalences. In addition, the restrictions of Ϙ ar and Ϙ [1]met to Seq( C ) via the left and right projection respectively, are naturally equivalent.Proof. The first claim follows from the fact that exact squares as in (52) are both left Kan extended from theirrestriction to Λ ⊆ Δ × Δ and right Kan extended from their restriction to Λ ⊆ Δ × Δ . The naturalhomotopy between Ϙ ar | Seq( C ) and Ϙ [1]met | Seq( C ) is then encoded by the -fold exact sequence of quadraticfunctors on Seq( C ) given by Ϙ met ([ 𝑤 → 𝑥 ]) → Ϙ ( 𝑥 ) → Ϙ ( 𝑤 ) → Ϙ ar ( 𝑧 → 𝑤 ) , where the last exact sequence is by Remark 2.3.16. (cid:3) Lemma 2.3.19 identifies the Poincaré ∞ -category Met( C , Ϙ ) with the Poincaré ∞ -category Ar( C , Ϙ ) upto a shift of the Poincaré structure. We note however that (Ar( C ) , Ϙ [1]ar ) ≃ Ar( C , Ϙ [1] ) , that is, the formationof arrow Poincaré ∞ -categories commutes with shifting the Poincaré structure. Using Lemma 2.3.19 andProposition 2.3.17 we then obtain a sequence of equivalences Pn(Met( C , Ϙ )) ≃ Pn(Ar( C , Ϙ [−1] )) ≃ Fm( C , Ϙ [−1] ) . Unwinding the definitions, this composed map sends a tuple ( 𝑤 → 𝑥, 𝑞, 𝜂 ) , consisting of a Poincaré object ( 𝑥, 𝑞 ) equipped with a Lagrangian ( 𝑤 → 𝑥, 𝜂 ) , to the object 𝑧 ∶= f ib( 𝑤 → 𝑥 ) , equipped with the shiftedhermitian structure encoded by the pair of null-homotopies of 𝑞 | 𝑧 (one restricted from 𝜂 and one inducedby the null-homotopy of the composed map 𝑧 → 𝑤 → 𝑥 ).2.3.20. Corollary (The algebraic Thom isomorphism) . The association [ 𝑤 → 𝑥 ] ↦ f ib( 𝑤 → 𝑥 ) underlinesa natural equivalence of spaces Pn(Met( C , Ϙ )) ≃ Fm( C , Ϙ [−1] ) between Poincaré objects in C equipped with a Lagrangian and (−1) -shifted hermitian objects in C . Remark.
Combining Remark 2.3.14 with the algebraic Thom isomorphism of Corollary 2.3.20 wemay identify the L -groups of ( C , Ϙ ) with the homology monoids of the chain complex of monoids of theform ... → 𝜋 Fm( C , Ϙ [− 𝑛 −1] ) → 𝜋 Fm( C , Ϙ [− 𝑛 ] ) → 𝜋 Fm( C , Ϙ [− 𝑛 +1] ) → ... where the map 𝜋 Fm( C , Ϙ [− 𝑖 ] ) → 𝜋 Fm( C , Ϙ [− 𝑖 +1] ) sends an (− 𝑖 ) -fold hermitian object ( 𝑥, 𝑞 ) to its Ranickiboundary cof[ 𝑥 → Ω 𝑖 D Ϙ ( 𝑥 )] , endowed with its associated (− 𝑖 + 1) -fold Poincaré form.We finish this section by framing the observation that the hyperbolic and arrow constructions naturallycommute with each other ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 41 Lemma.
For a stable ∞ -category C , the natural equivalence (54) Ar( C × C op ) ≃ Ar( C ) × Ar( C op ) ≃ Ar( C ) × Ar( C ) op in which the second equivalence sends ( 𝑓 ∶ 𝑧 → 𝑤, 𝑓 ′ ∶ 𝑧 ′ → 𝑤 ′ ) to ( 𝑓 ∶ 𝑧 → 𝑤, 𝑤 ′ ← 𝑧 ′ ∶ 𝑓 ′ ) , extendsto an equivalence of Poincaré ∞ -categories Ar(Hyp( C )) ≃ Hyp(Ar( C )) . Proof.
Transporting the Poincaré structure of
Ar(Hyp( C )) along the equivalence (54) yields the quadraticfunctor ( 𝑓 ∶ 𝑧 → 𝑤, 𝑓 ′ ∶ 𝑧 ′ → 𝑤 ′ ) ↦ Ϙ hyp ( 𝑧, 𝑤 ′ ) × B hyp (( 𝑧,𝑤 ′ ) , ( 𝑧,𝑤 ′ )) B hyp (( 𝑧, 𝑤 ′ ) , ( 𝑤, 𝑧 ′ )) ≃hom C ( 𝑧, 𝑤 ′ ) × hom C ( 𝑧,𝑤 ′ )×hom C ( 𝑧,𝑤 ′ ) [hom C ( 𝑧, 𝑧 ′ ) × hom C ( 𝑤, 𝑤 ′ )] ≃ hom C ( 𝑧, 𝑧 ′ ) × hom C ( 𝑧,𝑤 ′ ) hom C ( 𝑤, 𝑤 ′ )≃ hom Ar( C ) ( 𝑓 , 𝑓 ′ ) . We thus finish the proof by recognizing the last term as the quadratic functor of
Hyp(Ar( C )) . (cid:3) Combining Lemma 2.3.22 with Lemma 2.3.19 and Remark 2.2.4 we immediately conclude2.3.23.
Corollary.
For a stable ∞ -category C there is a natural equivalence of Poincaré ∞ -categories Met(Hyp( C )) ≃ Hyp(Ar( C )) , which, on the underlying stable ∞ -categories, is given by the equivalence Ar( C × C op ) ≃ Ar( C ) × Ar( C op ) ≃ Ar( C ) × Ar( C ) op , where the second equivalence is the product of the identity of Ar( C ) and the equivalence Ar( C op ) ≃ Ar( C ) op which sends an arrow 𝑥 ← 𝑤 ∶ 𝑔 in C op to the canonical arrow f ib( 𝑔 ) → 𝑤 . In particular, the Poincaréfunctor met ∶ Met Hyp( C ) → Hyp( C ) from Lemma 2.3.7 (applied to Hyp( C ) ) admits a section in Cat p∞ . The Grothendieck-Witt group.
In this section we define the Grothendieck-Witt group of a Poincaré ∞ -category. In the setting of ordinary rings, the Grothendieck-Witt group was classically defined as thegroup completion of the monoid of isomorphism classes of pairs ( 𝑃 , 𝑞 ) where 𝑃 is finite dimensionalprojective module and 𝑞 is a non-degenerate hermitian form of some flavour (symmetric, quadratic, anti-symmetric, etc.). It was later extended to more general contexts such as vectors bundles over algebraicvarieties [Kne77] and forms in abstract additive categories with duality [QSS79]. In doing so it was re-alized that the simple definition via group completion needs to be slightly modified to take into accountinformation coming from non-split short exact sequences. In particular, one had to quotient out the groupcompletion by the relation [ 𝑥, 𝑞 ] ∼ [hyp( 𝑤 )] identifying the class of a metabolic object with Lagrangian 𝑤 with that of the associated hyperbolic object. In fact, the latter relation already implies the group prop-erty for the resulting quotient (as we will see below in our context), and hence can be done on the level ofmonoids without explicitly group completing. On the other hand, in the case of modules over rings (or, moregenerally, in contexts in which all short exact sequences split) the relation [ 𝑥, 𝑞 ] ∼ [hyp( 𝑤 )] automaticallyholds in the group completion since every metabolic object is stably hyperbolic.In the present section we will give a definition of the Grothendieck-Witt group in the context of Poincaré ∞ -categories, and extract some of its basic properties. Using the work of the fourth and ninth authors [HS20],this definition can be compared with the classical one in the case of modules over rings using suitablePoincaré structures on D p ( 𝑅 ) , see Remark 4.2.17 below for the precise statement.For the following definition, recall the map hyp∶ 𝜄 C → Pn( C , Ϙ ) induced on Poincaré objects by thePoincaré functor hyp ∶ Hyp( C ) → ( C , Ϙ ) under the equivalence Pn(Hyp( C )) ≃ 𝜄 C of Proposition 2.2.5.Explicitly, this map sends an object 𝑥 ∈ C to the Poincaré object hyp( 𝑥 ) = 𝑥 ⊕ D 𝑥 equipped with itscanonical Poincaré form, see §2.2.2.4.1. Definition.
Let ( C , Ϙ ) be a Poincaré ∞ -category. We define GW ( C , Ϙ ) to be the quotient in thecategory of commutative monoids of 𝜋 Pn( C , Ϙ ) (with its commutative monoid structure given by directsums) by the relations(55) [ 𝑥, 𝑞 ] ∼ [hyp( 𝑤 )] for every Poincaré object ( 𝑥, 𝑞 ) with Lagrangian 𝑤 → 𝑥 . Remark.
By definition we may identify GW ( C , Ϙ ) with the coequilizer of the pair of maps 𝜋 Pn Met( C , Ϙ ) ⇉ 𝜋 Pn( C , Ϙ ) , where the first map is induced by the Poincaré functor met ∶ Met( C , Ϙ ) → ( C , Ϙ ) and the second is inducedby the composed Poincaré functor Met( C , Ϙ ) lag ←←←←←←←←←←←←→ Hyp( C ) hyp ←←←←←←←←←←←←←←→ C discussed in Remark 2.3.9.We quickly summarize a few properties which follows directly from the definition of GW .2.4.3. Lemma. i) For every exact sequence 𝑧 ′ → 𝑧 → 𝑧 ′′ in C the relation [hyp( 𝑧 )] ∼ [hyp( 𝑧 ′ )] + [hyp( 𝑧 ′′ )] holds in GW ( C , Ϙ ) .ii) For every Poincaré object [ 𝑥, 𝑞 ] ∈ Pn( C , Ϙ ) the relation [ 𝑥, 𝑞 ] + [ 𝑥, − 𝑞 ] + [hyp(Ω 𝑥 )] ∼ [hyp( 𝑥 )] + [hyp(Ω 𝑥 )] ∼ 0 holds in GW ( C , Ϙ ) .Proof. For i) note that if 𝑧 ′ → 𝑧 → 𝑧 ′′ is an exact sequence in C then 𝑤 ∶= 𝑧 ′ ⊕ D Ϙ 𝑧 ′′ is naturally aLagrangian in hyp( 𝑧 ) , and hyp( 𝑤 ) ≃ hyp( 𝑧 ′ ) ⊕ hyp( 𝑧 ′′ ) .To prove ii), the first identification is given by (55) applied to the metabolic object ( 𝑥 Δ ←←←←←←←←→ 𝑥 ⊕ 𝑥, 𝑞 ⊕ − 𝑞 ) and the second relation is given by i) applied to the exact sequence Ω 𝑥 → → 𝑥 . (cid:3) Corollary.
The commutative monoid GW ( C , Ϙ ) is always a group . We will refer to it as the Grothendieck-Witt group of ( C , Ϙ ) . Example.
For a stable ∞ -category C , the isomorphism(56) 𝜋 Pn(Hyp( C )) ≅ 𝜋 𝜄 ( C ) induced on components of the equivalence of Proposition 2.2.5, descends to a group isomorphism GW (Hyp( C )) ≅ K ( C ) . Indeed, the isomorphism (56) relates the class of 𝑧 ∈ C to the Poincaré class of ( 𝑧, 𝑧 ) ∈ C × C op equippedwith its canonical Poincaré form id 𝑧 ∈ hom C ( 𝑧, 𝑧 ) = Ϙ hyp ( 𝑧, 𝑧 ) . An isotropic object in (( 𝑧, 𝑧 ) , id 𝑧 ) is thengiven by a pair of maps 𝑧 ′ → 𝑧, 𝑧 → 𝑧 ′′ in C , such that the composite 𝑧 ′ → 𝑧 → 𝑧 ′′ , which corresponds tothe pullback of the Poincaré form id 𝑧 to Ϙ hyp ( 𝑧 ′ , 𝑧 ′′ ) = hom C ( 𝑧 ′ , 𝑧 ′′ ) , vanishes. Such an isotropic object is aLagrangian precisely when the resulting sequences 𝑧 ′ → 𝑧 → 𝑧 ′′ is exact. Moreover the hyperbolic objecton ( 𝑧 ′ , 𝑧 ′′ ) is given by the object ( 𝑧 ′ ⊕𝑧 ′′ , 𝑧 ′ ⊕𝑧 ′′ ) ∈ C × C op . It then follows that under the isomorphism (56),the defining relations of the Grothendieck-Witt group can be written as [ 𝑧 ] = [ 𝑧 ′ ] + [ 𝑧 ′′ ] for every exactsequence 𝑧 ′ → 𝑧 → 𝑧 ′′ , which are exactly the relations defining the quotient 𝜋 𝜄 ( C ) ↠ K ( C ) .Recall from Definition 2.3.11 that zero’th L -group L ( C , Ϙ ) is defined as the cokernel of the monoidhomomorphism 𝜋 Pn 𝜕 ( C , Ϙ ) → 𝜋 Pn( C , Ϙ ) . In particular, the quotient map 𝜋 Pn( C , Ϙ ) → L ( C , Ϙ ) sendsthe class of any metabolic object (and in particular any hyperbolic object) to zero, and hence factors througha group homomorphism GW ( C , Ϙ ) → L ( C , Ϙ ) , which is necessarily surjective, as we can identify L ( C , Ϙ ) with the quotient group of GW ( C , Ϙ ) by thesubgroup spanned by the classes of metabolic objects. To obtain more information about this kernel, wenote that the map 𝜋 Pn 𝜕 ( C , Ϙ ) → 𝜋 ( 𝜄 C ) [ 𝑤 → 𝑥 ] ↦ 𝑤 induced by the Poincaré functor lag ∶ Met( C , Ϙ ) → Hyp( C ) of Construction 2.3.8, is surjective: any object 𝑧 ∈ C is a Lagrangian in the Poincaré object hyp( 𝑧 ) . In addition, the canonical map 𝜋 ( 𝜄 C ) → K ( C ) to the zero’th algebraic K -theory group of C is surjective as well: we may identify K ( C ) with the quotient of 𝜋 ( 𝜄 C ) in the category of commutative monoids by the relations [ 𝑧 ] ∼ [ 𝑧 ′ ] + [ 𝑧 ′′ ] for every exact sequence ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 43 𝑧 ′ → 𝑧 → 𝑧 ′′ . By Lemma 2.4.3i) the homomorphism hyp ∶ 𝜋 ( 𝜄 C ) → 𝜋 Pn( C , Ϙ ) then descends to ahomomorphism of abelian groups(57) [hyp] ∶ K ( C ) → GW ( C , Ϙ ) , which by Example 2.4.5 we may also identify with the map induced on GW by the Poincaré functor hyp ∶ Hyp( C ) → C . Comparing the relevant universal properties we then see that GW ( C , Ϙ ) sits in apushout square of commutative monoids of the form(58) 𝜋 Pn 𝜕 ( C , Ϙ ) 𝜋 Pn( C , Ϙ )K ( C ) GW ( C , Ϙ ) [met][lag] [hyp] where [lag] denotes the composed map 𝜋 Pn 𝜕 ( C , Ϙ ) → 𝜋 ( 𝜄 C ) → K ( C ) . It then follows that L ( C , Ϙ ) canequivalently by obtained as the cokernel of the map of abelian groups [hyp] ∶ K ( C ) → GW ( C , Ϙ ) . On theother hand, by Corollary 2.2.10 the map [hyp] is C -equivariant with respect to the induced C -action on K and the trivial C -action on GW ( C , Ϙ ) . We then obtain an induced exact sequence of abelian groups(59) K ( C ) C → GW ( C , Ϙ ) → L ( C , Ϙ ) → . In Paper [II] we will show that this sequence comes from an exact sequence of spectra K( C ) hC → GW( C , Ϙ ) → L( C , Ϙ ) , the Tate exact sequence , encoding the fundamental relationship between these three invariants. In par-ticular, this allows one to extend (59) to a long exact sequence involving the higher L -groups and higherGrothendieck-Witt groups, yielding a powerful tool for computing the latter. In Paper [III] we will exploitthese ideas for computing the higher Grothendieck-Witt groups of the integers.3. P OINCARÉ STRUCTURES ON MODULES CATEGORIES
In this section we will discuss hermitian and Poincaré structures on ∞ -categories of modules over ringspectra. In particular, we fix a base E ∞ -ring spectrum 𝑘 and consider an E -algebra 𝐴 in the symmetricmonoidal ∞ -category Mod 𝑘 of 𝑘 -module spectra. We will denote Alg E ∶= Alg E (Mod 𝑘 ) the ∞ -categoryof E -algebra objects in Mod 𝑘 , which we will simply refer to as E -algebras . Given an E -algebra 𝐴 ∈Alg E , we will denote by Mod 𝐴 the ∞ -category of left 𝐴 -module objects in Mod 𝑘 , which we will refer toas 𝐴 -modules . Our goal is to describe and study hermitian and Poincaré structures on the full subcategories Mod f 𝐴 ⊆ Mod 𝜔𝐴 ⊆ Mod 𝐴 of finitely presented and compact, respectively, 𝐴 -modules in Mod 𝑘 , as theseconstitute a large majority of the examples which arise in practice (see also §4 for more specific examples).We will begin in §3.1 by introducing the notion of a module with involution, and show how it can beused to model bilinear functors on module ∞ -categories. We will then refine this notion §3.2 to a modulewith genuine involution, that which will allow us to encode not only bilinear functors but also hermitianand Poincaré structures. Then, in §3.3 we will discuss the basic operations of restriction and induction ofmodules with genuine involution along maps of ring spectra.We point out that the ∞ -categories Mod 𝜔𝐴 and Mod f 𝐴 are independent of 𝑘 , and the reader who wishesto avoid this additional layer of structure is invited to take 𝑘 = 𝕊 . On the other hand, the reader who prefersto reason in terms of chain-complexes instead of spectra is invited to set 𝑘 = ℤ . The latter case (and moregenerally, that of a complex-oriented 𝑘 ) also leads to better periodicity properties, which we will discuss§3.4.3.1. Ring spectra and involutions.
In this section we will define the notion of a module with involution over an E -algebra 𝐴 and show how it can be used to construct bilinear functors on Mod f 𝐴 and Mod 𝜔𝐴 . Wewill identify the modules with involution which lead to perfect bilinear functors as those which are invertible in a suitable sense. An important class of invertible modules with involution arise from E -algebras withanti-involutions, a case for which we will present a convenient recognition criterion, see Proposition 3.1.11below.To begin, we note that since ⊗ 𝑘 is a symmetric monoidal structure on Mod 𝑘 the monoidal product ⊗ 𝑘 ∶ Mod 𝑘 × Mod 𝑘 → Mod 𝑘 itself refines to a monoidal functor. This monoidal functor then sends the algebra object ( 𝐴, 𝐴 ) in Mod 𝑘 × Mod 𝑘 to the algebra object 𝐴 ⊗ 𝑘 𝐴 in Mod 𝑘 , and thus refines to a functor(60) Mod 𝐴 × Mod 𝐴 → Mod 𝐴⊗ 𝑘 𝐴 ( 𝑋, 𝑌 ) ↦ 𝑋 ⊗ 𝑘 𝑌 .
In addition, since ⊗ 𝑘 is a symmetric monoidal structure the functor ⊗ 𝑘 ∶ Mod 𝐴 × Mod 𝐴 → Mod 𝐴 is C -equivariant with respect the flip action on the domain and trivial action on the target, and consequently (60)inherits a C -equivariant structure with respect to the flip action on the domain and the C -action on thetarget induced by the flip action on 𝐴 ⊗ 𝑘 𝐴 . Given an ( 𝐴 ⊗ 𝑘 𝐴 ) -module 𝑀 , let us denote by B 𝑀 ∶ Mod 𝜔𝐴 × Mod 𝜔𝐴 → S 𝑝 ( 𝑋, 𝑌 ) ↦ hom 𝐴⊗ 𝑘 𝐴 ( 𝑋 ⊗ 𝑘 𝑌 , 𝑀 ) the resulting bilinear form. The association 𝑀 ↦ B 𝑀 then assembles to form a functor(61) B (−) ∶ Mod 𝐴⊗ 𝑘 𝐴 → Fun b (Mod 𝜔𝐴 ) which by the above inherits a C -equivariant structure with respect to the flip-induced C -actions on bothsides.3.1.1. Definition.
Let 𝐴 be an E -algebra. By a module with involution over 𝐴 we will mean an object 𝑀 of the ∞ -category (Mod 𝐴⊗ 𝑘 𝐴 ) hC , where as above C acts on Mod 𝐴⊗ 𝑘 𝐴 via its flip action on 𝐴 ⊗ 𝑘 𝐴 .Concretely, a module with involution over 𝐴 consists of a spectrum 𝑀 with a C -action and an ( 𝐴⊗ 𝑘 𝐴 ) -module structure, such that the involution is linear over the ring map 𝐴 ⊗ 𝑘 𝐴 → 𝐴 ⊗ 𝑘 𝐴 which switchesthe two factors. Since (61) is C -equivariant the bilinear functor B 𝑀 ( 𝑋, 𝑌 ) = hom 𝐴⊗ 𝑘 𝐴 ( 𝑋 ⊗ 𝑘 𝑌 , 𝑀 ) . on Mod 𝜔𝐴 associated to 𝑀 consequently inherits the structure of a symmetric bilinear functor.We may consider an ( 𝐴 ⊗ 𝑘 𝐴 ) -module 𝑀 as a 𝑘 -module spectrum equipped with two commuting actionsof 𝐴 . The first 𝐴 -action then promotes 𝑀 to an object of Mod 𝐴 , while the second action refines to an actionof 𝐴 on 𝑀 via 𝐴 -module maps. In particular, the second 𝐴 -action can be encoded via a map(62) 𝐴 → hom 𝐴 ( 𝑀, 𝑀 ) . If 𝑀 is a module with involution over 𝐴 then the involution determines an equivalence between the twodifferent 𝐴 -module structures. In particular, in this case it does not matter which action is considered firstand which is considered second.3.1.2. Definition.
We will say that a module with involution 𝑀 over 𝐴 is invertible if it is compact as an 𝐴 -module (with respect to either the first or the second 𝐴 -action) and the map (62) is an equivalence.3.1.3. Proposition.
Let 𝐴 be an E -algebra and 𝑀 a module with involution over 𝐴 . Then the symmetricbilinear functor B 𝑀 ∈ Fun s (Mod 𝜔𝐴 ) is non-degenerate if and only if 𝑀 belongs to Mod 𝜔𝐴 , where 𝑀 isconsidered as an 𝐴 -module via its first 𝐴 -action. Similarly, the restriction of B 𝑀 to Mod f 𝐴 is non-degenerateif and only if 𝑀 belongs to Mod f 𝐴 . In both cases the associated duality is then given by D 𝑀 ( 𝑋 ) ∶= hom 𝐴 ( 𝑋, 𝑀 ) , where hom 𝐴 ( 𝑋, 𝑀 ) is considered as an 𝐴 -module via the residual second 𝐴 -action. In addition, in bothcases the B 𝑀 is perfect if and only if 𝑀 is in addition invertible. Remark.
It follows from Proposition 3.1.3 that 𝑀 is invertible if and only if it is compact as an 𝐴 -module and the functor 𝑋 ↦ hom 𝐴 ( 𝑋, 𝑀 ) from Mod 𝜔𝐴 to itself is an equivalence. Now for any E -algebra 𝐴 the functor 𝑋 ↦ hom 𝐴 ( 𝑋, 𝐴 ) determines an equivalence Mod 𝜔𝐴 ≃ ←←←←←←←→ (Mod 𝜔𝐴 op ) op . In addition, for 𝑀 compact the natural map hom 𝐴 ( 𝑋, 𝐴 ) ⊗ 𝐴 𝑀 → hom 𝐴 ( 𝑋, 𝑀 ) is an equivalence. Itthen follows that for a module with involution 𝑀 which is compact as an 𝐴 -module the condition of beinginvertible is equivalent to the condition that (−) ⊗ 𝐴 𝑀 ∶ Mod 𝜔𝐴 op → Mod 𝜔𝐴 is an equivalence of ∞ -categories. Equivalently, such an 𝑀 is invertible if there exists an ( 𝐴 op × 𝐴 op ) -module 𝑁 such that 𝑁 ⊗ 𝐴 𝑀 ≃ 𝐴 as 𝐴 -bimodules and 𝑀 ⊗ 𝐴 op 𝑁 ≃ 𝐴 op as 𝐴 op -bimodules. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 45 Proof of Proposition 3.1.3.
We begin with the first claim. For a fixed compact 𝐴 -module 𝑌 , the functor 𝑋 ↦ hom 𝐴⊗ 𝑘 𝐴 ( 𝑋 ⊗ 𝑘 𝑌 , 𝑀 ) from Mod op 𝐴 to Mod 𝐴 is represented by the 𝐴 -module hom 𝐴 ( 𝑌 , 𝑀 ) , where thelatter is considered as an 𝐴 -module via its second 𝐴 -action. In particular, if 𝑀 is compact as an 𝐴 -modulethen hom 𝐴 ( 𝑌 , 𝑀 ) is compact and hence also represents the functor B 𝑀 (− , 𝑌 ) on Mod 𝜔𝐴 , in which case B 𝑀 is non-degenerate with duality as stated. In the other direction, if B 𝑀 is non-degenerate with duality D thenthe canonically associated bilinear form 𝑌 ⊗ 𝑘 D( 𝑌 ) → 𝑀 determines a map 𝜃 ∶ D( 𝑌 ) → hom 𝐴 ( 𝑌 , 𝑀 ) ,which by construction induces an equivalence on mapping spaces from every compact 𝐴 -module 𝑋 . Sincecompact 𝐴 -modules generate all 𝐴 -modules under colimits the map 𝜃 must be an equivalence, and so hom 𝐴 ( 𝑌 , 𝑀 ) is compact for every compact 𝑌 . Since Mod 𝜔𝐴 is generated under finite colimits and retractsby 𝐴 this condition is equivalent to hom 𝐴 ( 𝐴, 𝑀 ) ≃ 𝑀 being compact.The same argument holds if we replace everywhere compact by finitely presented, since finitely presented 𝐴 -modules also generate all modules by colimits, and Mod f 𝐴 itself is generated under finite colimits by 𝐴 .Now assume that 𝑀 is compact so that B 𝑀 is non-degenerate with duality D 𝑀 as above. We wish toshow that the evaluation map 𝑋 → D 𝑀 D 𝑀 ( 𝑋 ) = hom 𝐴 (hom 𝐴 ( 𝑋, 𝑀 ) , 𝑀 )) is an equivalence if and only if 𝑀 is invertible. Since Mod 𝜔𝐴 is generated under finite colimits and retractsby 𝐴 it will suffice to check the component of the evaluation map at 𝑋 = 𝐴 , in which case it becomes themap 𝐴 → hom 𝐴 (hom 𝐴 ( 𝐴, 𝑀 ) , 𝑀 )) = hom 𝐴 ( 𝑀, 𝑀 ) which by definition is an equivalence if and only if 𝑀 is invertible. The same argument works for the caseof Mod f 𝐴 . (cid:3) Definition.
Let 𝑀 be a module with involution over an E -algebra 𝐴 . We will denote by Ϙ q 𝑀 , Ϙ s 𝑀 ∶ (Mod 𝜔𝐴 ) op → S 𝑝 the quadratic and symmetric hermitian structures associated to the symmetric bilinear form B 𝑀 ∈ Fun s (Mod 𝜔𝐴 ) as in Example 1.1.17 and Definition 1.2.11. These are given explicitly by the formulas Ϙ q 𝑀 ( 𝑋 ) = hom 𝐴⊗ 𝑘 𝐴 ( 𝑋 ⊗ 𝑘 𝑋, 𝑀 ) hC and Ϙ s 𝑀 ( 𝑋 ) = hom 𝐴⊗ 𝑘 𝐴 ( 𝑋 ⊗ 𝑘 𝑋, 𝑀 ) hC . We now consider some examples. An E -algebra with anti-involution is an object of Alg hC E , where theaction of C on the ∞ -category of E -algebras is given by sending an algebra 𝐴 to its opposite 𝐴 op .3.1.6. Example.
Let 𝐴 be an E -algebra with anti-involution ̄ ∙ ∶ 𝐴 op → 𝐴 . Then then 𝐴 can be naturallyconsidered as an invertible module with involution over itself. Indeed, by using that the forgetful functor Alg E → S 𝑝 is equivariant with respect to the trivial action on the target, we obtain a functor Alg hC E → Fun(BC , S 𝑝 ) , which allows us to view 𝐴 as a spectrum with C -action. In addition, by construction thisaction switches between the canonical left and right actions of 𝐴 on itself. More precisely, viewing 𝐴 as an ( 𝐴 ⊗ 𝑘 𝐴 op ) -module, this C -action is linear over the C -action on 𝐴 ⊗ 𝑘 𝐴 op which flips the two componentsand applies the anti-involution ̄ ∙ . Equivalently, the anti-involution determines an equivalence of E -algebras 𝐴 ⊗ 𝑘 𝐴 op ≃ 𝐴 ⊗ 𝑘 𝐴 intertwining the above C -action with the flip action on 𝐴 ⊗ 𝑘 𝐴 , and we may henceview 𝐴 as an object of Mod hC 𝐴⊗ 𝑘 𝐴 . Informally, the 𝐴 ⊗ 𝑘 𝐴 -action on 𝐴 is given by ( 𝑎 ⊗ 𝑏 ) ⋅ 𝑥 = 𝑎 ⋅ 𝑥 ⋅ ̄𝑏 .We may then recover the anti-involution ̄ ∙ as the induced map of E -algebras 𝐴 → hom 𝐴 ( 𝐴, 𝐴 ) = 𝐴 op . Thelatter is therefore an equivalence, and so 𝐴 is invertible.3.1.7. Example.
The restriction of the C -action on Alg E to E ∞ -algebras is canonically trivialized, so thatwe obtain a functor Fun(BC , Alg E ∞ ) → Alg hC E . In particular, E ∞ -algebras with C -actions give rise to an E -algebra with involution. For example, any E ∞ -algebra equipped with the trivial C -action, determines an E -algebra with involution. More generally,when 𝐴 is an E ∞ -algebra, any 𝐴 -module 𝑀 with the trivial C -action canonically defines a module withinvolution over 𝐴 , with 𝐴 ⊗ 𝑘 𝐴 acting via the multiplication map 𝐴 ⊗ 𝑘 𝐴 → 𝐴 .3.1.8. Example.
An important source of E -algebras with anti-involution arises from the group algebraconstruction. We will study this example and its relation to visible L -theory in further detail in §4.3. Examples.
Let 𝑅 be an ordinary associative ring. Using the Eilenberg-MacLane embedding H ∶ A 𝑏 ↪ S 𝑝 we may associate to 𝑅 an E -ring spectrum H 𝑅 , and by [Lur17, Theorem 7.1.2.1] we have a naturalequivalence Mod 𝜔 H 𝑅 ≃ D p ( 𝑅 ) between the ∞ -category of compact H 𝑅 -module spectra and the perfect derived ∞ -category of 𝑅 , definedas in Example 1.2.12. An anti-involution on H 𝑅 is then the same as an anti-involution on 𝑅 , that is, anisomorphism (∙) ∶ 𝑅 ≅ ←←←←←←←→ 𝑅 op such that 𝑟 = 𝑟 for 𝑟 ∈ 𝑅 . We will study this case in further detail in §4.2.Ordinary rings which carry anti-involutions are then fairly common, see Examples 4.2.3.3.1.10. Example.
As in Example 3.1.9, suppose that 𝑅 is an ordinary associative ring. Recall that a Wallanti-structure [Wal70] on 𝑅 consists of an anti-automorphism (∙) ∶ 𝑅 → 𝑅 op and a unit 𝜖 ∈ 𝑅 ∗ such that 𝑟 = 𝜖 −1 𝑟𝜖 and 𝜖 = 𝜖 −1 . The most common type of these are the central Wall anti-structures, namely, thosein which 𝜖 is in the center and (−) is an anti-involution. Given a Wall anti-structure we can consider 𝑅 asan ( 𝑅 ⊗ 𝑅 ) -module via the action ( 𝑎 ⊗ 𝑏 )( 𝑐 ) = 𝑎𝑏𝑐 , and endow it with an involution given by 𝑥 ↦ 𝜖𝑥 .Applying the Eilenberg-MacLane functor this results in an invertible module with involution over the E -ring spectrum H 𝑅 , whose underline H 𝑅 -module is H 𝑅 , but which is generally not the one associated to anyanti-involution on H 𝑅 .The following lemma gives a recognition criterion for modules with involution over 𝐴 which comefrom anti-involutions of 𝐴 . Essentially, it reflects the idea that the datum of an anti-involution on 𝐴 isequivalent to that of a perfect duality D ∶ Mod 𝜔𝐴 → (Mod 𝜔𝐴 ) op together with a symmetric Poincaré form 𝑢 ∈ hom 𝐴 ( 𝐴, D( 𝐴 )) hC on the 𝐴 -module 𝐴 :3.1.11. Proposition.
Let 𝐴 be an E -algebra and 𝑀 a module with involution over 𝐴 . Then 𝑀 comes froman anti-involution on 𝐴 as in Example (3.1.6) if and only if there exists a C -equivariant map of spectra 𝑢 ∶ 𝕊 → 𝑀 (where C -acts trivially on 𝕊 ) such that the induced 𝐴 -module map 𝐴 → 𝑀 (using, say thefirst 𝐴 -action on 𝑀 ), is an equivalence. In this case, the anti-involution on 𝐴 can be recovered via the map 𝐴 → hom 𝐴 ( 𝑀, 𝑀 ) ≃ hom 𝐴 ( 𝐴, 𝐴 ) = 𝐴 op associated to the second 𝐴 -action on 𝑀 . The proof of Proposition 3.1.11 will require some preparation. Given a stable ∞ -category C , the forma-tion of mapping spectra allows one to consider C as an ∞ -category enriched in spectra, see [GH15, Example7.4.14]. This enrichment is functorial in C , that is, it can be organized into a functor(63) Cat ex∞ → Cat S 𝑝 C ↦ C S 𝑝 where the latter is the ∞ -category of S 𝑝 -enriched ∞ -categories, see [BGT13, Proposition 4.10] (and [Hau15,Theorem 1.1] for the comparison of the model categorical and ∞ -categorical approaches to spectrally en-riched categories). The functor (63) is in fact fully-faithful and exhibits Cat ex∞ as an accessible localisationof
Cat S 𝑝 by the collection of triangulated equivalences , see [BGT13, Theorem 4.22].3.1.12. Remark.
Every spectrally enriched ∞ -category has an “underlying ∞ -category” obtained by apply-ing the functor Ω ∞ = Map( 𝕊 , −) ∶ S 𝑝 → S to all mapping spectra. In particular, the underlying ∞ -categoryof the spectrally enriched category C S 𝑝 is just C itself (or rather, its image in Cat ∞ ). More formally, thefunctor (63) constitutes a lift of the inclusion Cat ex∞ ↪ Cat ∞ along the underlying ∞ -category functor Cat S 𝑝 → Cat ∞ .Let Cat ex∞ , ∗ ∶= Cat ex∞ × Cat ∞ (Cat ∞ ) Δ ∕ denote the ∞ -category of stable ∞ -categories C equipped with adistinguished object 𝑥 ∈ C , and similarly, let Cat S 𝑝, ∗ = Cat S 𝑝 × Cat ∞ (Cat ∞ ) Δ ∕ denote the ∞ -category ofspectrally enriched categories equipped with a distinguished object. Then we may consider the composite(64) Cat ex∞ , ∗ → Cat S 𝑝, ∗ → Alg E where the first functor is induced from (63) and the second is the functor of [GH15, Theorem 6.3.2(iii)],which can be described on objects as sending a pointed spectrally enriched category ( D , 𝑥 ) to the en-domorphism spectrum Map D ( 𝑥, 𝑥 ) . As shown in loc. cit. this functor has a fully-faithful left adjoint B ∶ Alg E → Cat S 𝑝, ∗ which sends a ring spectrum 𝐴 to the pointed spectrally enriched category (B 𝐴, 𝑥 ) containing a single object 𝑥 whose endomorphism ring is 𝐴 . The essential image of B is then given by thosepointed spectrally enriched categories ( D , 𝑥 ) for which 𝑥 is the only object up to equivalence. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 47 Proposition.
Let ( C , Ϙ ) be a Poincaré ∞ -category and ( 𝑥, 𝑞 ) a Poincaré object in C . Then the image hom C ( 𝑥, 𝑥 ) ∈ Alg E of ( C , 𝑥 ) under (64) inherits a canonical anti-involution, that it, it lifts to an object of Alg C E .Proof. We first note that the spectral enrichment functor (63) commutes with taking opposites. Indeed,since (63) is fully-faithful and its essential image, spanned by the pre-triangulated spectrally enriched cat-egories, is closed under opposites, the op action on Cat S 𝑝, ∗ induces a C -action on Cat ex∞ . This action thencoincides with the action induced by the inclusion of
Cat ex∞ in Cat ∞ by Remark 3.1.12 and the fact that theunderlying ∞ -category functor Cat S 𝑝, ∗ → Cat ∞ visibly commutes with opposites since it is induced by afunctor S 𝑝 → S on the level of enriching ∞ -categories.Now since Δ ≃ (Δ ) op via an essentially unique isomorphism the op action on Cat ex∞ induces a C -actionon Cat ex∞ , ∗ , which can be described on objects by the formula ( C , 𝑥 ) ↦ ( C op , 𝑥 ) . Similarly, the op actionon Cat S 𝑝 induces a C -action on (Cat ex∞ ) S 𝑝, ∗ . In addition, the endomorphism functor (Cat ex∞ ) S 𝑝, ∗ → Alg E in (64) is C -equivariant with respect to the op actions on both sides, as can be seen by the fact that itadmits a fully-faithful right adjoint B ∶ Alg E → Cat S 𝑝, ∗ which is itself compatible with taking oppositesessentially by construction. Now consider the diagram ( C , 𝑥 ) Cat ex∞ , ∗ Alg E hom C ( 𝑥, 𝑥 ) C Cat ex∞ ∈ ∋∈ in which both arrows are C -equivariant with respect to the op action. Now the perfect duality D Ϙ promotes C to a C -fixed object of Cat ex∞ (see (27)). The fibre of
Cat ex∞ , ∗ → Cat ex∞ over C can be identified with 𝜄 C ,with its C -action induced by D Ϙ . To finish the proof it will suffice to show that 𝑥 ∈ 𝜄 C refines to a C -fixedpoint. Indeed, this now follows from Corollary 2.2.10. (cid:3) Proof of Proposition 3.1.11.
The only if direction is clear, since any anti-involution on 𝐴 preserves the unitmap 𝑢 ∶ 𝕊 → 𝐴 . To prove the other direction, suppose that 𝑀 is a module with involution over 𝐴 and we aregiven a C -equivariant map 𝑢 ∶ 𝕊 → 𝑀 with respect to the trivial C -action on 𝕊 , such that the induced 𝐴 -module map 𝐴 → 𝑀 is an equivalence. Since 𝑢 is C -invariant it follows that this condition holds for boththe first and second 𝐴 -actions. On the other hand, it also holds for the 𝐴 op -action on 𝐴 that the analogouslydefined map is an equivalence. The corresponding statement also holds for the hom 𝐴 ( 𝑀, 𝑀 ) -action on 𝑀 .We hence obtain a commutative diagram(65) 𝐴 hom 𝐴 ( 𝑀, 𝑀 ) 𝐴 op 𝑀 𝑀 ≃ ≃ ≃ ≃ in which both vertical maps (induced by the base point 𝑢 ∶ 𝕊 → 𝑀 ) are equivalences, and the bottomhorizontal map is the involution on 𝑀 . It then follows that the top horizontal map 𝐴 → hom 𝐴 ( 𝑀, 𝑀 ) isan equivalence as well. In particular, 𝑀 is invertible and hence the induced bilinear functor on Mod 𝜔𝐴 isperfect by Lemma 3.1.3, with associated duality 𝑋 ↦ hom 𝐴 ( 𝑋, 𝑀 ) on Mod 𝜔𝐴 . The C -equivariant map 𝑢 ∶ 𝕊 → 𝑀 then determines a form 𝑞 𝑢 ∈ Ϙ s 𝑀 ( 𝐴 ) = hom 𝐴 ( 𝐴 ⊗ 𝑘 𝐴, 𝑀 ) hC which is Poincaré by the condition that the induced map 𝐴 → 𝑀 = D( 𝐴 ) is an equivalence. We maythen identify the top horizontal equivalence in (65) as the underlying equivalence of the anti-involution on hom 𝐴 ( 𝐴, 𝐴 ) = hom 𝐴 (D( 𝐴 ) , D( 𝐴 )) op induced by the Poincaré form 𝑞 𝑢 by Proposition 3.1.13. (cid:3) Modules with genuine involution.
Our goal in this section is to refine the definition of a module withinvolution studied in §3.1 above in order to obtain a notion capable of encoding not just bilinear but alsoquadratic functors on
Mod 𝜔𝐴 and Mod f 𝐴 . To begin, recall that for a spectrum 𝑋 there is a canonical map 𝑋 → ( 𝑋 ⊗ 𝕊 𝑋 ) tC , known as the Tate diagonal , which enjoys a variety of favorable formal properties, see [NS18]. If 𝑋 is now a 𝑘 -module then we can consider the composed map(66) 𝑋 → ( 𝑋 ⊗ 𝕊 𝑋 ) tC → ( 𝑋 ⊗ 𝑘 𝑋 ) tC , where the second map is induced by the lax monoidal structure of the forgetful functor Mod 𝑘 → S 𝑝 . For 𝑋 = 𝑘 the map (66) gives the composed map Fr ∶ 𝑘 → ( 𝑘 ⊗ 𝕊 𝑘 ) tC → 𝑘 tC which is known as the Tate Frobenius map . For a 𝑘 -module spectrum 𝑋 we may then consider the spectrum ( 𝑋 ⊗ 𝑘 𝑋 ) tC , which is naturally a 𝑘 tC -module, as a 𝑘 -module spectrum, by restricting structure along theTate Frobenius. With this 𝑘 -module structure the map (66) becomes 𝑘 -linear, and we will henceforth referto it as the 𝑘 -linear Tate diagonal .3.2.1. Warning.
Since 𝑘 is the unit of Mod 𝑘 the flip C -action on 𝑘 ⊗ 𝑘 𝑘 ≃ 𝑘 is trivial. However, the TateFrobenius 𝑘 → 𝑘 tC is generally not equivalent to the composed map 𝑘 → 𝑘 hC → 𝑘 tC . In particular, ifwe were to endow ( 𝑋 ⊗ 𝑘 𝑋 ) tC with the 𝑘 -module structure restricted from its 𝑘 tC -module structure along 𝑘 → 𝑘 hC → 𝑘 tC (which would be the 𝑘 -module structure we would obtain by applying to 𝑋 ⊗ 𝑘 𝑋 theTate construction internally in Mod 𝑘 ) then (66) would generally not be 𝑘 -linear.3.2.2. Definition.
Let 𝐴 be a E -algebra. A module with genuine involution over 𝐴 is a triple ( 𝑀, 𝑁, 𝛼 ) which consists of- a module with involution 𝑀 over 𝐴 in the sense of Definition 3.1.1,- an 𝐴 -module 𝑁 , and- an 𝐴 -linear map 𝑓 ∶ 𝑁 → 𝑀 tC .Here we view 𝑀 tC , which is canonically an ( 𝐴 ⊗ 𝑘 𝐴 ) tC -module, as an 𝐴 -module through the 𝑘 -linearTate diagonal 𝐴 → ( 𝐴 ⊗ 𝑘 𝐴 ) tC .3.2.3. Remark.
When 𝑘 = 𝕊 the data of a module with genuine involution over 𝐴 can equivalently bedescribed as a module over the Hill-Hopkins-Ravenel norm of 𝐴 : this is a genuine C -spectrum whoseunderlying C -spectrum is given by 𝐴 ⊗ 𝑘 𝐴 with the flip C -action, whose geometric fixed points are givenby 𝐴 , and whose reference map 𝐴 → ( 𝐴 ⊗ 𝑘 𝐴 ) tC is the Tate-diagonal considered in [NS18]. This is analgebra object with respect to the symmetric monoidal structure on genuine C -spectra, and the data of amodule over this algebra object consists exactly to an ( 𝐴 ⊗ 𝑘 𝐴 ) -module with C -action (which is 𝑀 in ourcase) and a module over the geometric fixed points 𝐴 (which is 𝑁 in our case), such that 𝑁 is the geometricfixed points of a C -genuine refinement of 𝑀 (this is the datum of the map 𝑓 ∶ 𝑁 → 𝑀 tC in our case).3.2.4. Lemma.
For 𝑀 ∈ Mod hC 𝐴⊗ 𝑘 𝐴 and 𝑋 ∈ Mod 𝜔𝐴 there is an equivalence (67) hom 𝐴⊗ 𝑘 𝐴 ( 𝑋 ⊗ 𝑘 𝑋, 𝑀 ) tC ≃ hom 𝐴 ( 𝑋, 𝑀 tC ) natural in 𝑀 and 𝑋 .Proof. Consider the functor 𝐹 ∶ (Mod 𝜔𝐴 ) op → S 𝑝 given by 𝑋 ↦ hom 𝐴⊗ 𝑘 𝐴 ( 𝑋 ⊗ 𝑘 𝑋, 𝑀 ) tC . This functoris exact, thus by Morita theory it is of the form hom 𝐴 ( 𝑋, 𝑁 ) for some 𝐴 -module 𝑁 . Setting 𝑋 = 𝐴 wefind that 𝑁 = 𝑀 tC as a spectrum. Furthermore the right action of Ω ∞ 𝐴 on 𝐴 translates under the functor 𝐹 to the diagonal action of Ω ∞ 𝐴 on 𝑀 tC , i.e. the action through the composite Ω ∞ 𝐴 Δ ←←←←←←←←→ (Ω ∞ 𝐴 × Ω ∞ 𝐴 ) hC can ←←←←←←←←←←←←←→ Ω ∞ ( 𝐴 ⊗ 𝐴 ) tC underlying the Tate diagonal. Applying the same observation to shifts of 𝐴 and using the exactness thenshows the claim. (cid:3) Construction.
Let ( 𝑀, 𝑁, 𝛼 ) be an 𝐴 -module with genuine involution. We define a quadratic functor Ϙ 𝛼𝑀 on perfect 𝐴 -modules by the pullback(68) Ϙ 𝛼𝑀 ( 𝑋 ) hom 𝐴 ( 𝑋, 𝑁 ) Ϙ s 𝑀 ( 𝑋 ) hom 𝐴 ( 𝑋, 𝑀 tC ) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 49 where the lower horizontal map is given by the composite Ϙ s 𝑀 ( 𝑋 ) = hom 𝐴⊗ 𝑘 𝐴 ( 𝑋 ⊗ 𝑘 𝑋, 𝑀 ) hC → hom 𝐴⊗ 𝑘 𝐴 ( 𝑋 ⊗ 𝑘 𝑋, 𝑀 ) tC ≃ hom 𝐴 ( 𝑋, 𝑀 tC ) , where the last equivalence is via Lemma 3.2.4.By construction the underlying bilinear part of Ϙ 𝛼𝑀 is B 𝑀 , and hence the condition that Ϙ 𝛼𝑀 is Poincarédepends only on 𝑀 , via the criterion of Proposition 3.1.3. In addition, by Lemma 1.1.10 the norm map Ϙ q 𝑀 → Ϙ s 𝑀 factors canonically as Ϙ q 𝑀 ⟶ Ϙ 𝛼𝑀 ⟶ Ϙ s 𝑀 . If 𝑀 tC = 0 , for example if 𝐴 is an 𝕊 [ ] -algebra, then Ϙ 𝛼 ( 𝑋 ) ≃ Ϙ s 𝑀 ( 𝑋 ) ⊕ hom 𝐴 ( 𝑋, 𝑁 ) .3.2.6. Example.
Let 𝑀 be a module with involution over an E -algebra 𝐴 . The modules with genuineinvolution ( 𝑀, , → 𝑀 tC ) and ( 𝑀, 𝑀 tC , 𝑀 tC = 𝑀 tC ) give rise respectively to the quadratic andsymmetric functors Ϙ q 𝑀 and Ϙ s 𝑀 of Definition 3.1.5.3.2.7. Example.
Let 𝑀 be a module with involution over an E -algebra 𝐴 and assume that 𝐴 is connective(so that the truncation of an 𝐴 -module admits a canonical 𝐴 -module structure). Then there is for every 𝑚 ∈ ℤ a module with genuine involution given by ( 𝑀, 𝜏 ≥ 𝑚 𝑀 tC , 𝜏 ≥ 𝑚 𝑀 tC → 𝑀 tC ) with the reference map being the 𝑚 -connective cover. Applying Construction 3.2.5 these give rise to qua-dratic functors Ϙ ≥ 𝑚𝑀 that sit between the quadratic and the symmetric one, i.e., there are maps(69) Ϙ q 𝑀 → ⋯ → Ϙ ≥ 𝑀 → Ϙ ≥ 𝑀 → Ϙ ≥ −1 𝑀 → ⋯ → Ϙ s 𝑀 . If 𝑀 tC vanishes, e.g. if 2 is invertible in 𝐴 , then all of these maps are equivalences. The limit and colimitof these diagrams of Poincaré structures are given by lim[ ⋯ → Ϙ ≥ 𝑚𝑀 → Ϙ ≥ 𝑚 −1 𝑀 → ⋯ ] ≃ Ϙ q 𝑀 and colim[ ⋯ → Ϙ ≥ 𝑚𝑀 → Ϙ ≥ 𝑚 −1 𝑀 → ⋯ ] ≃ Ϙ s 𝑀 . Indeed, inspecting the defining pullback squares (68) for Ϙ ≥ 𝑚𝑀 and using that pullbacks and mapping spectraout of compact 𝐴 -modules respect both limits and colimits this follows from the fact that lim 𝜏 ≥ 𝑚 𝑀 tC = 0 while the induced map colim 𝜏 ≥ 𝑚 𝑀 tC → 𝑀 tC is an equivalence. We may hence consider the tower (69).as interpolating between the quadratic and symmetric Poincaré structures on Mod 𝜔𝐴 . We will study thisconstruction in further detail in §4.2 in the case where 𝐴 is (the Eilenberg-MacLane spectrum of) an ordinaryring.3.2.8. Example.
Let 𝐴 be a E -algebra with anti-involution. In Example 3.1.6 we have seen that 𝐴 can beconsider as a module with involution over itself. In order to promote 𝐴 to a module with genuine involutionwe need an 𝐴 -module 𝑁 and a map 𝑁 → 𝐴 tC of 𝐴 -modules. Such a triple ( 𝐴, 𝑁, 𝛼 ) is called an E -algebra with genuine involution . Any such E -algebra with genuine involution has an underlying genuine C -spectrum, the module 𝑁 taking the role of the geometric fixed points.3.2.9. Example.
Let 𝐴 be an orthogonal ring spectrum with anti-involution in the sense of [DMPR17]. Thisgives rise to a genuine C -spectrum whose underlying spectrum with C -action is the underlying spectrumof 𝐴 , whose geometric fixed points 𝐴 𝜑 C is canonically an 𝐴 -module, and where the map 𝛼 ∶ 𝐴 𝜑 C → 𝐴 tC is 𝐴 -linear. We therefore obtain a ring spectrum with genuine involution ( 𝐴, 𝐴 𝜑 C , 𝛼 ) .3.2.10. Example.
Consider the sphere spectrum 𝕊 with the trivial C -action. We may then view 𝕊 as anassociative ring spectrum with anti-involution, and refine it to a ring spectrum with genuine anti-involutionusing as reference map the composite 𝕊 → 𝕊 hC → 𝕊 tC , which also agrees in this case with the Tatediagonal. The Poincaré structure associated to this genuine anti-involution on 𝕊 is the universal Poincaréstructure Ϙ u of Example 1.2.15. Example.
Let 𝐴 be an E ∞ - 𝑘 -algebra equipped with a C -action. We may then view 𝐴 as an as-sociative 𝑘 -algebra with anti-involution, and refine it to a 𝑘 -algebra with genuine anti-involution using asreference map the composite t ∶ 𝐴 → ( 𝐴 ⊗ 𝑘 𝐴 ) tC → 𝐴 tC , where the first map is the 𝑘 -linear Tate diagonal and the second map is induced by the C -equivariantcommutative 𝑘 -algebra map 𝐴 ⊗ 𝑘 𝐴 → 𝐴 whose restriction to the first component is the identity 𝐴 → 𝐴 and restriction to the second component is given by the generator of the C -action. This yields a Poincaréstructure on Mod 𝜔𝐴 , which we denote by Ϙ t 𝑘 , and refer to as the Tate Poincaré structure associated to thegiven C -action. The universal Poincaré structure Ϙ u on S 𝑝 f then corresponds to the case where 𝐴 = 𝑘 = 𝕊 and the C -action is trivial.3.2.12. Example.
Given an E -algebra 𝐴 we may form the E -algebra with anti-involution 𝐴 ⊕ 𝐴 op wherethe involution filps the two factors. We may then refine this involution to a genuine involution by takingthe zero map → ( 𝐴 ⊕ 𝐴 op ) tC = 0 , which at the same time is also an equivalence. The resulting Poicaréstructure Ϙ 𝐴⊕𝐴 op on Mod 𝜔𝐴⊕𝐴 op is then both quadratic and symmetric (see Example 3.2.6). We now notethat the projections 𝐴 ⊕ 𝐴 op → 𝐴 and 𝐴 ⊕ 𝐴 op → 𝐴 op induce an equivalence Mod 𝜔𝐴⊕𝐴 op → Mod 𝜔𝐴 × Mod 𝜔𝐴 op ≃ Mod 𝜔𝐴 ×(Mod 𝜔𝐴 ) op , under which the Poincaré structure in question corresponds to the hyperbolic structure of Definition 2.2.2,and so (Mod 𝜔𝐴⊕𝐴 op , Ϙ 𝐴⊕𝐴 op ) ≃ Hyp(Mod 𝜔𝐴 ) . We shall henceforth focus on the case where 𝑘 = 𝕊 is the sphere spectrum. We will show that in this case,modules with genuine involution do not only provide a convenient way of producing hermitian structureson Mod 𝜔𝐴 and Mod f 𝐴 , but these two notions become in fact equivalent. To formulate this more precisely,weproceed to organize modules with genuine involution over 𝐴 into an ∞ -category Mod N 𝐴 , defined as thepullback(70) Mod N 𝐴 Ar(Mod 𝐴 )Mod hC 𝐴⊗ 𝑘 𝐴 Mod 𝐴 t(−) tC2 of the arrow category Ar(Mod 𝐴 ) and the ∞ -category (Mod 𝐴⊗ 𝑘 𝐴 ) hC of modules with involution. The rightvertical map is the projection onto the target, and the bottom horizontal map sends a module with involution 𝑀 to the Tate construction 𝑀 tC , considered as an 𝐴 -module via the Tate diagonal map 𝐴 → ( 𝐴 ⊗ 𝑘 𝐴 ) 𝑡𝐶 .We would like to relate Mod N 𝐴 and Fun q (Mod 𝜔𝐴 ) by constructing a commutative diagram(71) Mod N 𝐴 Ar(Mod 𝐴 )Fun q (Mod 𝜔𝐴 ) Ar(Fun ex ((Mod 𝜔𝐴 ) op , S 𝑝 ))Mod hC 𝐴⊗ 𝑘 𝐴 Mod 𝐴 Fun s (Mod 𝜔𝐴 ) Fun ex ((Mod 𝜔𝐴 ) op , S 𝑝 ) where the front square is the pullback square of Corollary 1.3.12 which exhibits the analogous decomposi-tion of the ∞ -category of quadratic functors Fun q (Mod 𝜔𝐴 ) into linear and bilinear parts. We then define theright face of (71) to be the square induced by the Yoneda map Mod 𝐴 → Fun ex ((Mod 𝜔𝐴 ) op , S 𝑝 ) 𝑁 ↦ hom 𝐴 (− , 𝑁 ) , ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 51 and the bottom arrow in the left face is the functor Mod hC 𝐴⊗ 𝑘 𝐴 → Fun s (Mod 𝜔𝐴 ) 𝑀 ↦ hom 𝐴⊗ 𝑘 𝐴 ((−) ⊗ 𝑘 (−) , 𝑀 ) introduced above. The commuting homotopy in the bottom square of (71) is then provided by Lemma 3.2.4.The cube is then uniquely determined by the fact that its front face is a cartesian square.3.2.13. Theorem. If 𝑘 = 𝕊 then the cube (71) , considered as a natural transformation from its back face toits front face, is an equivalence. In particular, the resulting arrow Mod N 𝐴 → Fun q (Mod 𝜔𝐴 ) is an equivalence of ∞ -categories, whose action on objects is given by ( 𝑀, 𝑁, 𝛼 ) ↦ Ϙ 𝛼𝑀 . Remark.
For a general 𝑘 one can still define the pullback of ∞ -categories Mod 𝑘 N 𝐴 ∶= Mod hC 𝐴⊗ 𝑘 𝐴 × Mod 𝐴 Ar(Mod 𝐴 ) and construct a functor(72) Mod 𝑘 N 𝐴 → Fun q (Mod 𝜔𝐴 ) which in general will not be an equivalence. Instead, we may identify Mod 𝑘 N 𝐴 with the ∞ -category ofquadratic functors on Mod 𝜔𝐴 equipped with a certain 𝑘 -bilinear compatibility on their bilinear parts. Thefunctor (72) then corresponds to forgetting the 𝑘 -bilinear compatibility data. Proof of Theorem 3.2.13.
Since the back and front faces are both cartesian squares it will suffice to showthat its right and back faces determine equivalences from their back edge to their front edge. For this, it willsuffice to show that the functors
Mod 𝐴 → Fun ex ((Mod 𝜔𝐴 ) op , S 𝑝 ) and Mod hC 𝐴⊗ 𝕊 𝐴 → Fun s (Mod 𝜔𝐴 ) are equivalences. For the former, we note that since S 𝑝 is stable we have that post-composition with Ω ∞ ∶ S 𝑝 → S induces an equivalence Fun ex ((Mod 𝜔𝐴 ) op , S 𝑝 ) ≃ Fun rex ((Mod 𝜔𝐴 ) op , S ) ≃ Ind(Mod 𝜔𝐴 ) and consequently the claim follows from the fact that Mod 𝐴 is generated by Mod 𝜔𝐴 under colimits. For thesecond map, by its construction it will suffice to show that the functor(73) Mod 𝐴⊗ 𝕊 𝐴 → Fun b (Mod 𝜔𝐴 ) 𝑀 ↦ hom 𝐴⊗ 𝕊 𝐴 ((−) ⊗ 𝕊 (−) , 𝑀 ) is an equivalence of ∞ -categories. For this, recall that by [Lur17, Theorem 4.8.5.16] and [Lur17, Remark4.8.5.19] the association 𝐴 ↦ Mod 𝐴 refines to a symmetric monoidal functor Θ S 𝑝 ∶ Alg E = Alg E ( S 𝑝 ) → LMod S 𝑝 (Pr 𝐿 ) from E -ring spectra to the ∞ -category S 𝑝 -module presentable ∞ -categories, and the latter can be identifiedwith the full subcategory Pr 𝐿 spanned by the stable presentable ∞ -categories. In particular, the bilinearfunctor Mod 𝐴 × Mod 𝐴 → Mod 𝐴⊗ 𝕊 𝐴 ( 𝑋, 𝑌 ) ↦ 𝑋 ⊗ 𝕊 𝑌 induces an equivalence(74) Mod 𝐴 ⊗ S 𝑝 Mod 𝐴 ≃ ←←←←←←←→ Mod 𝐴⊗ 𝕊 𝐴 , where ⊗ S 𝑝 denotes the tensor product of stable presentable ∞ -categories. Since Mod 𝐴 and Mod 𝐴⊗ 𝕊 𝐴 arecompactly generated by Mod 𝜔𝐴 and Mod 𝜔𝐴⊗ 𝕊 𝐴 respectively and the bilinear functor ( 𝑋, 𝑌 ) ↦ 𝑋 ⊗ 𝕊 𝑌 mapsa pair of compact 𝐴 -modules to a compact ( 𝐴 ⊗ 𝕊 𝐴 ) -module we see that the equivalence (74) is inducedon Ind-categories by the functor(75) Mod 𝜔𝐴 ⊗ Mod 𝜔𝐴 → Mod 𝜔𝐴⊗ 𝕊 𝐴 , induced by the same bilinear functor ( 𝑋, 𝑌 ) ↦ 𝑋 ⊗ 𝕊 𝑌 , where ⊗ is the tensor product of stable ∞ -categories, and we use the fact that Ind(−) is symmetric monoidal. It then follows that restriction along (75)induces an equivalence
Fun ex ((Mod 𝜔𝐴⊗ 𝕊 𝐴 ) op , S 𝑝 ) ≃ ←←←←←←←→ Fun ex ((Mod 𝜔𝐴 ⊗ Mod 𝜔𝐴 ) op , S 𝑝 ) ≃ Fun ex ((Mod 𝜔𝐴 ) op ⊗ (Mod 𝜔𝐴 ) op , S 𝑝 ) ≃ Fun b (Mod 𝜔𝐴 ) . Since the Yoneda map
Mod 𝐴⊗ 𝕊 𝐴 → Fun ex ((Mod 𝜔𝐴⊗ 𝕊 𝐴 ) op , S 𝑝 ) is an equivalence by the argument above wemay now conclude that (73) is an equivalence, and so the proof is complete. (cid:3) Remark.
The inclusion
Mod f 𝐴 ⊆ Mod 𝜔𝐴 exhibits the latter as an idempotent completion of the for-mer. Since S 𝑝 is idempotent complete, it follows that restriction induces equivalences Fun ex ((Mod 𝜔𝐴 ) op , S 𝑝 ) ≃Fun ex ((Mod f 𝐴 ) op , S 𝑝 ) and Fun b (Mod 𝜔𝐴 ) ≃ Fun b (Mod f 𝐴 , S 𝑝 ) . The proof of Theorem 3.2.13 then shows thatthe the association ( 𝑀, 𝑁, 𝛼 ) ↦ ( Ϙ 𝛼𝑀 ) Mod f 𝐴 induces an equivalence Mod N 𝐴 ≃ Fun q (Mod f 𝐴 ) . Alternatively,we will also show in Paper [II] (cf. [II].1.3.4) that restriction along dense fully-faithful inclusions inducesan equivalence on Fun q (−) , and so in particular the restriction map Fun q (Mod 𝜔𝐴 ) → Fun q (Mod f 𝐴 ) , so thatTheorem 3.2.13 implies its analogue for Mod f 𝐴 .3.2.16. Remark.
It follows from Theorem 3.2.13 and Proposition 3.1.3 that the association ( 𝑀, 𝑁, 𝛼 ) ↦ Ϙ 𝛼𝑀 determines a 1-1 correspondence between equivalence classes of objects ( 𝑀, 𝑁, 𝛼 ) ∈ Mod N 𝐴 for which 𝑀 is invertible and equivalence classes of quadratic functors Fun q (Mod 𝜔𝐴 ) that are Poincaré. Similarly,using Remark 3.2.15 we get that the association ( 𝑀, 𝑁, 𝛼 ) ↦ ( Ϙ 𝛼𝑀 ) | Mod f 𝐴 determines a 1-1 correspondencebetween those ( 𝑀, 𝑁, 𝛼 ) for which 𝑀 is finitely presented and invertible and the Poincaré structures in Fun q (Mod f 𝐴 ) . We note that under these correspondences the maps of Poincaré structures, which are bydefinition duality preserving, correspond to those maps ( 𝑀, 𝑁, 𝛼 ) → ( 𝑀 ′ , 𝑁 ′ , 𝛼 ′ ) in Mod N 𝐴 for which themap 𝑀 → 𝑀 ′ is an equivalence.3.3. Restriction and induction.
In the present section we assume 𝑘 = 𝕊 and consider the functorial de-pendence of Mod N 𝐴 in 𝐴 , and the compatibility of this functoriality with the one for hermitian structuresalong the equivalence of Theorem 3.2.13. Let 𝜙 ∶ 𝐴 → 𝐵 be a map E -algebras and let(76) 𝑝 𝜙 ∶ Mod 𝜔𝐴 → Mod 𝜔𝐵 be the induction functor sending 𝑋 to 𝐵 ⊗ 𝐴 𝑋 . Then the restriction functor 𝑝 ∗ 𝜙 ∶ Fun q (Mod 𝜔𝐵 ) → Fun q (Mod 𝜔𝐴 ) corresponds, under the equivalence of Theorem 3.2.13, to a functor(77) 𝜙 ∗ ∶ Mod N 𝐵 → Mod N 𝐴 , which we consider as the restriction of structure operation for modules with genuine involution. As ex-plained in §1.3, restriction of quadratic functors commutes with taking linear and bilinear parts and withthe formation of symmetric Poincaré structures, that is, it acts compatibly on the entire pullback square(78) Fun q ( C ) Ar(Fun ex ( C op , S 𝑝 ))Fun s ( C ) Fun ex ( C op , S 𝑝 ) . 𝜏 B t
Under the equivalence of Theorem 3.2.13 we obtain the same for the restriction functor (77), that is, itextends to the entire defining squares (70) for 𝐴 and 𝐵 . On the other hand, the Yoneda equivalences Mod 𝐴 ≃Fun ex ((Mod 𝜔𝐴 ) op , S 𝑝 ) and Mod 𝐵 ≃ Fun ex ((Mod 𝜔𝐵 ) op , S 𝑝 ) fit into a commutative square(79) Mod 𝐵 Mod 𝐴 Fun ex ((Mod 𝜔𝐵 ) op , S 𝑝 ) Fun ex ((Mod 𝜔𝐴 ) op , S 𝑝 ) ≃ 𝜙 ∗ ≃ 𝑝 ∗ 𝜙 in which the top horizontal arrow is the forgetful functor from 𝐵 -modules to 𝐴 -modules and the bottomhorizontal functor is restriction along 𝑝 𝜙 . Indeed, the commutativity is given by the adjunction equiv-alence hom 𝐵 ( 𝑝 𝜙 𝑋, 𝑀 ) ≃ hom 𝐴 ( 𝑋, 𝜙 ∗ 𝑀 ) . Similarly, the equivalences Mod 𝐴⊗ 𝕊 𝐴 ≃ Fun b (Mod 𝜔𝐴 ) and ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 53 Mod 𝐵⊗ 𝕊 𝐵 ≃ Fun b (Mod 𝜔𝐵 ) fit into a commutative square(80) Mod 𝐵⊗ 𝕊 𝐵 Mod 𝐴⊗ 𝕊 𝐴 Fun b (Mod 𝜔𝐵 ) Fun b (Mod 𝜔𝐴 ) ≃ ( 𝜙⊗𝜙 ) ∗ ≃( 𝑝 𝜙 × 𝑝 𝜙 ) ∗ We may thus conclude that for ( 𝑀, 𝑁, 𝛼 ) ∈ Mod N 𝐵 the restriction functor (77) is obtain by simply re-stricting the ( 𝐵 ⊗ 𝕊 𝐵 ) -module structure on 𝑀 to 𝐴 ⊗ 𝕊 𝐴 , the 𝐵 -module structure on 𝑁 to 𝐴 , and viewing 𝛼 as a map of 𝐴 -modules by forgetting its compatibility with the 𝐵 -module structures on its domain andcodomain.We now proceed to discuss how the operation of left Kan extension is mirrored along the equivalenceof Theorem 3.2.13. Recall that by Lemma 1.4.1 the operation of left Kan extensions Fun((Mod 𝜔𝐴 ) op , S 𝑝 ) → Fun((Mod 𝜔𝐵 ) op , S 𝑝 ) preserves quadratic functors, and the resulting functor(81) ( 𝑝 𝜙 ) ! ∶ Fun q (Mod 𝜔𝐴 ) → Fun q (Mod 𝜔𝐵 ) is compatible with taking linear and bilinear parts, that is B ( 𝑝 𝜙 ) ! Ϙ ≃ ( 𝑝 𝜙 × 𝑝 𝜙 ) ! B Ϙ and L ( 𝑝 𝜙 ) ! Ϙ ≃ ( 𝑝 𝜙 ) ! L Ϙ . By Lemma 1.4.1 and Corollary 1.4.4 we may then conclude the following:3.3.1.
Corollary.
Under the equivalence of Theorem 3.2.13, the left Kan extension functor (81) correspondsto the functor 𝜙 ! ∶ Mod N 𝐴 → Mod N 𝐵 sending a module with genuine involution ( 𝑀, 𝑁, 𝛼 ) ∈ Mod N 𝐴 to the module with genuine involution 𝜙 ! ( 𝑀, 𝑁, 𝛼 ) = ( 𝑝 𝜙⊗𝜙 𝑀, 𝑝 𝜙 𝑁, 𝜙 ! 𝛼 ) = (( 𝐵 ⊗ 𝕊 𝐵 ) ⊗ 𝐴⊗ 𝕊 𝐴 𝑀, 𝐵 ⊗ 𝐴 𝐵, 𝜙 ! 𝛼 ) ∈ Mod N 𝐵 where 𝜙 ! 𝛼 is given by the composite 𝐵 ⊗ 𝐴 𝑁 → 𝐵 ⊗ 𝐴 𝑀 tC → (( 𝐵 ⊗ 𝕊 𝐵 ) ⊗ 𝐴⊗ 𝕊 𝐴 𝑀 ) tC of 𝐵 ⊗ 𝐴 𝛼 and the Beck-Chevalley transformation on the lax commuting square on the right (82) (Mod 𝐵⊗ 𝕊 𝐵 ) hC Mod 𝐵 (Mod 𝐴⊗ 𝕊 𝐴 ) hC Mod 𝐴 (−) tC2 (−) tC2 (Mod 𝐴⊗ 𝕊 𝐴 ) hC Mod 𝐴 (Mod 𝐵⊗ 𝕊 𝐵 ) hC Mod 𝐵 (−) tC2 (−) tC2 which is obtained from the commuting square on the left after replacing the vertical forgetful functors bytheir left adjoints 𝐵 ⊗ 𝐴 (−) and ( 𝐵 ⊗ 𝕊 𝐵 ) ⊗ 𝐴⊗ 𝕊 𝐴 (−) , respectively. We now wish to apply the above discussion in order to obtain explicit data which permits to refine 𝑝 𝜙 ∶ Mod 𝜔𝐴 → Mod 𝜔𝐵 to a hermitian functor with respect to a pair of hermitian structures coming frommodules with genuine involution ( 𝑀 𝐴 , 𝑁 𝐴 , 𝛼 ) ∈ Mod N 𝐴 and ( 𝑀 𝐵 , 𝑁 𝐵 , 𝛽 ) ∈ Mod N 𝐵 . In terms of quadraticfunctors, this data is by a natural transformation 𝜂 ∶ Ϙ 𝛼𝑀 𝐴 ⇒ 𝑝 ∗ 𝜙 Ϙ 𝛽𝑀 𝐵 . Under the equivalences above, the natural transformation 𝜂 corresponds to a map of ( 𝑀 𝐴 , 𝑁 𝐴 , 𝛼 ) → 𝜙 ∗ ( 𝑀 𝐵 , 𝑁 𝐵 , 𝛽 ) in Mod N 𝐴 , or equivalently by adjunction, to a map 𝜙 ! ( 𝑀 𝐴 , 𝑁 𝐴 , 𝛼 ) → ( 𝑀 𝐵 , 𝑁 𝐵 , 𝛽 ) . Let us summarize thesituation in explicit terms as follows:3.3.2. Corollary.
Keeping the notation above, the data of a hermitian functor (Mod 𝜔𝐴 , Ϙ 𝛼𝑀 𝐴 ) → (Mod 𝜔𝐵 , Ϙ 𝛽𝑀 𝐵 ) covering the induction functor 𝑝 𝜙 can be encoded by a triple ( 𝛿, 𝛾, 𝜎 ) where 𝛿 ∶ 𝑀 𝐴 → 𝑀 𝐵 and 𝛾 ∶ 𝑁 𝐴 → 𝑁 𝐵 in Mod hC 𝐴⊗ 𝕊 𝐴 and Mod 𝐴 respectively, and 𝜎 is a commutation homotopy in the square 𝑁 𝐴 𝑁 𝐵 𝑀 tC 𝐴 𝑀 tC 𝐵𝛾𝛼 𝛽𝛿 tC2
Equivalently, we may provide the adjoints 𝛿 ∶ ( 𝐵 ⊗ 𝕊 𝐵 ) ⊗ 𝐴⊗ 𝕊 𝐴 𝑀 𝐴 → 𝑀 𝐵 and 𝛾 ∶ 𝐵 ⊗ 𝑁 𝐴 → 𝑁 𝐵 in Mod hC 𝐵⊗𝐵 and
Mod 𝐵 respectively, together with a commutative square of the form 𝐵 ⊗ 𝐴 𝑁 𝐴 𝑁 𝐵 𝐵 ⊗ 𝐴 𝑀 tC 𝐴 ( ( 𝐵 ⊗ 𝕊 𝐵 ) ⊗ 𝐴⊗ 𝕊 𝐴 𝑀 𝐴 ) tC 𝑀 tC 𝐵𝛾 id ⊗ 𝐴 𝛼 𝛽𝛿 tC2 where the left lower horizontal map is the Beck-Chevalley map (82) . It can also be identifies with thecomposition of the Tate diagonal and the lax monoidal structure of (−) tC . Lemma.
In the situation of Corollary 3.3.2, the hermitian functor ( 𝑝 𝜙 , 𝜂 ) ∶ (Mod 𝜔𝐴 , Ϙ 𝛼𝑀 𝐴 ) → (Mod 𝜔𝐵 , Ϙ 𝛽𝑀 𝐵 ) associated to a map ( 𝛿, 𝛾, 𝜎 ) ∶ ( 𝑀 𝐴 , 𝑁 𝐴 , 𝛼 ) → 𝜙 ∗ ( 𝑀 𝐵 , 𝑁 𝐵 , 𝛽 ) is Poincaré if and only if the composed map 𝐵 ⊗ 𝐴 𝑀 𝐴 → ( 𝐵 ⊗ 𝕊 𝐵 ) ⊗ 𝐴⊗ 𝕊 𝐴 𝑀 𝐴 𝛿 ←←←←←←→ 𝑀 𝐵 is an equivalence, where the first map is induced by the left unit 𝐵 → 𝐵 ⊗ 𝕊 𝐵 .Proof. By definition, the hermitian functor ( 𝑝 𝜑 , 𝜂 ) is Poincaré if and only if the induced map 𝐵 ⊗ 𝐴 D 𝑀 𝐴 ( 𝑋 ) = 𝐵 ⊗ 𝐴 hom 𝐴 ( 𝑋, 𝑀 𝐴 ) ⟶ hom 𝐵 ( 𝐵 ⊗ 𝐴 𝑋, 𝑀 𝐵 ) = D 𝑀 𝐵 ( 𝐵 ⊗ 𝕊 𝐴 ) is an equivalence of 𝐵 -modules for every perfect 𝐴 -module 𝑋 . Since Mod 𝜔𝐴 is generated under finite col-imits and retracts by 𝐴 this map is an equivalence for every 𝑋 ∈ Mod 𝜔𝐴 if and only if it is an equivalencefor 𝑋 = 𝐴 . But this is exactly the statement that the induced map 𝐵 ⊗ 𝐴 𝑀 𝐴 = 𝐵 ⊗ 𝐴 hom 𝐴 ( 𝐴, 𝑀 𝐴 ) → hom 𝐵 ( 𝐵, 𝑀 𝐵 ) = 𝑀 𝐵 is an equivalence, as desired. (cid:3) Definition.
In the situation of Lemma 3.3.3, when the condition that the induced map
𝐵 ⊗ 𝐴 𝑀 𝐴 → 𝑀 𝐵 is an equivalence holds, we say that the morphism ( 𝛿, 𝛾, 𝜎 ) is 𝜙 -invertible . In particular, Lemma 3.3.3says that the hermitian functor induced by ( 𝛿, 𝛾, 𝜎 ) is Poincaré if and only if ( 𝛿, 𝛾, 𝜎 ) is 𝜙 -invertible .3.3.5. Example.
Suppose that ( 𝜙, 𝜏 ) ∶ ( 𝐴, 𝑁 𝐴 , 𝛼 ) → ( 𝐵, 𝑁 𝐵 , 𝛽 ) is a map of E -algebras with genuine anti-involution (Example 3.2.8), so that 𝜙 ∶ 𝐴 → 𝐵 is a map of rings with anti-involution and 𝜏 ∶ 𝑁 𝐴 → 𝜙 ∗ 𝑁 𝐵 isa map of 𝐴 -modules. Then both ( 𝐴, 𝑁 𝐴 , 𝛼 ) and ( 𝐵, 𝑁 𝐵 , 𝛽 ) are invertible as modules with genuine involutionover 𝐴 and 𝐵 respectively and 𝜏 is 𝜙 -invertible. In particular, in this situation we always obtain an inducedPoincaré functor ( 𝑝 𝜙 , 𝜏 ) ∶ (Mod 𝜔𝐴 , Ϙ 𝛼𝐴 ) → (Mod 𝜔𝐵 , Ϙ 𝛽𝐵 ) .3.3.6. Example.
Suppose that 𝜙 ∶ 𝐴 → 𝐵 is a map of E -algebras. A module with involution 𝑀 𝐴 ∈(Mod 𝐴⊗ 𝕊 𝐴 ) hC then determines a symmetric bilinear functor with associated quadratic hermitian structure Ϙ q 𝑀 𝐴 on 𝐴 encoded by the module with genuine involution ( 𝑀 𝐴 , , → 𝑀 tC 𝐴 ) . The left Kan extensionof Ϙ q 𝑀 to Mod 𝜔𝐵 is then encoded by the module with genuine involution ( 𝑀 𝐵 , , → 𝑀 tC 𝐵 ) for 𝑀 𝐵 ∶= 𝑀 ⊗ 𝐴⊗ 𝕊 𝐴 ( 𝐵 ⊗ 𝕊 𝐵 ) , and so ( 𝑝 𝜙 ) ! Ϙ q 𝑀 ≃ Ϙ q 𝑀 𝐵 . On the other hand, the associated symmetric hermitian structure Ϙ s 𝑀 𝐴 is encoded by the module with gen-uine involution ( 𝑀 𝐴 , 𝑀 tC 𝐴 , id ∶ 𝑀 tC 𝐴 → 𝑀 tC 𝐴 ) , and hence its left Kan extension to Mod 𝜔𝐵 is encoded by ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 55 the module with genuine involution ( 𝑀 𝐵 , 𝐵 ⊗ 𝐴 𝑀 tC 𝐴 , 𝐵 ⊗ 𝐴 𝑀 tC 𝐴 → 𝑀 tC 𝐵 ) , which is generally not thesymmetric hermitian structure Ϙ s 𝑀 𝐵 , unless 𝐵 is perfect as an 𝐴 op -module (indeed, for a fixed 𝑀 the under-lying map of spectra 𝐵 ⊗ 𝐴 𝑀 tC → ( 𝐵 ⊗ 𝐴 𝑀 ) tC can be considered as a natural transformation betweentwo exact functors in the argument 𝐵 ∈ Mod 𝜔𝐴 op which is an equivalence on 𝐵 = 𝐴 op and hence on anyperfect 𝐵 ). As a counter-example consider 𝕊 → 𝕊 [ ] : by Lin’s theorem [Lin80] one has that 𝕊 tC ≃ 𝕊 ∧2 so 𝕊 [ ] ⊗ 𝕊 𝕊 tC ≃ H ℚ whereas ( 𝕊 [ ] ⊗ 𝕊 𝕊 [ ]) tC ≃ 0 since is invertible on 𝕊 [ ] ⊗ 𝕊 𝕊 [ ] ≃ 𝕊 [ ] .3.4. Shifts and periodicity.
In this final section we will discuss periodicity phenomena for Poincaré struc-tures on module ∞ -categories. In the case of ordinary rings, this is the basis for the classical -fold period-icity phenomenon in L -theory. In Paper [II] we will also show how it leads to a generalization of Ranickiperiodicity and the fundamental theorem for Grothendieck-Witt theory.Recall that for a quadratic functor Ϙ on a stable ∞ -category C , and an integer 𝑛 ∈ ℤ , we denoted by Ϙ [ 𝑛 ] the quadratic functor on C given by Ϙ [ 𝑛 ] ( 𝑥 ) = Σ 𝑛 Ϙ ( 𝑥 ) (see Definition 1.2.16). Since the formation oflinear and bilinear parts is exact the shifted quadratic functor Ϙ [ 𝑛 ] has bilinear part Σ 𝑛 B Ϙ , linear part Σ 𝑛 L Ϙ ,and structure map Σ 𝑛 L Ϙ ( 𝑥 ) → Σ 𝑛 B( 𝑥, 𝑥 ) tC induced by the structure map of Ϙ on 𝑛 -fold suspensions. If C is now of the form Mod 𝜔𝐴 for some E -algebra 𝐴 and Ϙ = Ϙ 𝛼𝑀 for some module with genuine involution ( 𝑀, 𝑁, 𝛼 ) over 𝐴 , then Ϙ [ 𝑛 ] = Σ 𝑛 Ϙ 𝛼𝑀 is the quadratic functor associated to module with genuine involution (Σ 𝑛 𝑀, Σ 𝑛 𝑁, Σ 𝑛 𝛼 ) , and we write(83) ( Ϙ 𝛼𝑀 ) [ 𝑛 ] ≃ Ϙ Σ 𝑛 𝛼 Σ 𝑛 𝑀 . A second natural operation we can perform on the quadratic functor Ϙ 𝛼𝑀 is to pre-compose it with Σ 𝑛 .In this case the hermitian ∞ -category (Mod 𝜔𝐴 , Ϙ 𝛼𝑀 ◦ Σ 𝑛 ) is canonically equivalent to (Mod 𝜔𝐴 , Ϙ 𝛼𝑀 ) via thefunctor Σ 𝑛 ∶ Mod 𝜔𝐴 → Mod 𝜔𝐴 . However, if we fix the identification of Mod 𝜔𝐴 as compact 𝐴 -modules, thereparametrized quadratic functor Ϙ 𝛼𝑀 ◦ Σ 𝑛 can be identified with the quadratic functor associated to anothermodule with genuine involution. To see this, let us introduce some notation. Given a finite dimensionalreal representation 𝑉 for the group C , let us denote by 𝑆 𝑉 the associated one-point compactification of 𝑉 , which is a sphere of dimension dim( 𝑉 ) equipped with a based C -action. Similarly, we will denote by 𝕊 𝑉 = Σ ∞ ( 𝑆 𝑉 ) the associated suspension spectrum with C -action. Given a spectrum 𝑋 with a C -action,we will denote by Σ 𝑉 𝑋 ∶= 𝕊 𝑉 ⊗ 𝑋 the smash product of 𝕊 𝑉 and 𝑋 equipped with its diagonal C -action.Similarly, the ∞ -category Mod hC 𝐴⊗𝐴 of modules with involution over 𝐴 is naturally tensored over spectrawith C -action, and given a module with involution 𝑀 over 𝐴 we will denote by Σ 𝑉 𝑀 = 𝕊 𝑉 ⊗ 𝑀 theassociated tensor of 𝑀 by 𝕊 𝑉 . We will denote by 𝜎 the -dimensional sign representation of C and 𝜌 the -dimensional regular representation of C . In particular, 𝜌 decomposes of the direct sum of a trivialrepresentation and a sign representation, which we write as 𝜌 = 1 + 𝜎 . More generally, we will denote by 𝑎 + 𝑏𝜌 + 𝑐𝜎 the direct sum of 𝑎 copies of the -dimensional trivial representation, 𝑏 copies of 𝜌 , and 𝑐 copiesof 𝜎 .We will require the following lemma:3.4.1. Lemma.
Let 𝑋 be a spectrum with a C -action. Then the map 𝑋 tC → (Σ 𝜎 𝑋 ) tC , induced by the C -equivariant map 𝑆 → 𝑆 𝜎 , is an equivalence. In particular, the Tate construction isinvariant under tensoring with the sign representation sphere spectrum 𝕊 𝜎 .Proof. The cofibre of the map 𝑆 → 𝑆 𝜎 is given by the pointed 𝐶 -space Σ( 𝐶 ) + . Thus the claim followssince 𝐶 ⊗ 𝑋 is a free 𝐶 -spectrum (by a shearing equivalence) and thus its Tate construction vanishes. (cid:3) Proposition.
Let ( 𝑀, 𝑁, 𝛼 ) be a module with genuine involution. Then for 𝑛 ∈ ℤ there are equiva-lences of quadratic functors ( Ϙ 𝛼𝑀 ) [ 𝑛 + 𝑚 ] ◦ Σ 𝑛 ≃ Ϙ Σ 𝑚 𝛼 Σ 𝑚 − 𝑛𝜎 𝑀 . where the right hand side is the quadratic functor associated to the module with genuine involution (Σ 𝑚 − 𝑛𝜎 𝑀, Σ 𝑚 𝑁, Σ 𝑚 𝛼 ) , defined using the identification (Σ 𝑚 − 𝑛𝜎 𝑀 ) tC ≃ Σ 𝑚 𝑀 tC issued from Lemma 3.4.1. In particular, thefunctor Ω 𝑛 ∶ Mod 𝜔𝐴 → Mod 𝜔𝐴 refines to an equivalence (Mod 𝜔𝐴 , ( Ϙ 𝛼𝑀 ) [2 𝑛 ] ) ≃ ←←←←←←←→ (Mod 𝜔𝐴 , Σ 𝑛 Ϙ 𝛼𝑀 ◦ Σ 𝑛 ) ≃ (Mod 𝜔𝐴 , Ϙ Σ 𝑛 𝛼 Σ 𝑛 (1− 𝜎 ) 𝑀 ) . of hermitian ∞ -categories (or Poincaré when 𝑀 is invertible). The same statement holds for Mod f 𝐴 inplace of Mod 𝜔𝐴 .Proof. In light of (83) it will suffice to prove for the case 𝑚 = 0 . We now observe that the operations Ϙ ↦ (B Ϙ ◦ Δ) hC , Ϙ ↦ (B Ϙ ◦ Δ) hC both commute with both pre-composition and post-composition with Σ 𝑛 .Consequently, the same holds for the functors Ϙ ↦ (B Ϙ ◦ Δ) tC and Ϙ ↦ L Ϙ . Since exact functors preservesuspensions we may conclude that the functor Fun q (Mod 𝜔𝐴 ) → Ar(Fun ex ((Mod 𝜔𝐴 ) op , S 𝑝 )) Ϙ ↦ [L Ϙ ⇒ (B Ϙ ◦ Δ) tC ] is invariant under replacing Ϙ with Ϙ [ 𝑛 ] ◦ Σ 𝑛 . In particular, the image of Ϙ 𝛼 [ 𝑛 ] 𝑀 ◦ Σ 𝑛 in Ar(Fun ex ((Mod 𝜔𝐴 ) op , S 𝑝 )) is naturally equivalent to that of Ϙ 𝛼𝑀 . On the other hand, the bilinear form of Ϙ 𝛼 [ 𝑛 ] 𝑀 ◦ Σ 𝑛 is equivalent to B Σ − 𝑛𝜎 𝑀 ; indeed, by adjunction we have an equivalence Σ 𝑛 hom 𝐴⊗𝐴 (Σ 𝑛 𝑋 ⊗ Σ 𝑛 𝑌 , 𝑀 ) ≃ Σ 𝑛 hom 𝐴⊗𝐴 (Σ 𝑛𝜌 ⊗ 𝑋 ⊗ 𝑌 , 𝑀 ) ≃Σ 𝑛 hom 𝐴⊗𝐴 ( 𝑋 ⊗ 𝑌 , Σ − 𝑛𝜌 𝑀 ) ≃ hom 𝐴⊗𝐴 ( 𝑋 ⊗ 𝑌 , Σ 𝑛 (1− 𝜌 ) 𝑀 ) ≃ hom 𝐴⊗𝐴 ( 𝑋 ⊗ 𝑌 , Σ − 𝑛𝜎 𝑀 ) . To finish the proof it will have suffice to construct a commuting homotopy in the resulting diagram ofequivalences [Σ − 𝑛𝜎 hom 𝐴⊗𝐴 ( 𝑋 ⊗ 𝑋, 𝑀 )] tC hom 𝐴⊗𝐴 ( 𝑋 ⊗ 𝑋, Σ − 𝑛𝜎 𝑀 ) tC hom 𝐴⊗𝐴 ( 𝑋 ⊗ 𝑋, (Σ − 𝑛𝜎 𝑀 ) tC )hom 𝐴⊗𝐴 ( 𝑋 ⊗ 𝑋, 𝑀 ) tC hom 𝐴⊗𝐴 ( 𝑋 ⊗ 𝑋, 𝑀 ) tC hom 𝐴⊗𝐴 ( 𝑋 ⊗ 𝑋, 𝑀 tC ) ≃ ≃ ≃≃ ≃≃ where the vertical maps are induced by the map 𝕊 − 𝑛𝜎 → 𝕊 and the horizontal right facing arrows are issuedfrom Lemma 3.2.4. Indeed the equivalence Lemma 3.2.4 is by construction natural in 𝑀 and the square onthe left commutes even before the Tate construction since for a spectrum 𝑍 and ( 𝐴 ⊗ 𝐴 ) -modules 𝐵, 𝐶 theequivalence
𝑍 ⊗ hom
𝐴⊗𝐴 ( 𝐵, 𝐶 ) ≃ hom
𝐴⊗𝐴 ( 𝐵, 𝑍 ⊗ 𝐶 ) is natural in 𝑍 , 𝐴 and 𝐵 . (cid:3) Remark.
We will show in §7.4 (see Corollary 7.4.17) that any quadratic functor Ϙ ∶ C op → S 𝑝 canon-ically refines to the genuine fixed points of a functor ̃ Ϙ ∶ C op → S 𝑝 gC to genuine C -spectra. In that contextone can make sense of tensoring a given quadratic functor with 𝕊 𝜎 , equipped with its genuine C -structurein which the geometric fixed points are 𝕊 . We may consider this operation as “twisting by a sign”. A ver-sion of the above calculations also holds in this generality. We note however that in this context there are (atleast) two non-equivalent sign actions on the sphere spectrum, corresponding to 𝕊 𝜎 and 𝕊 𝜎 −1 respectively(see Warning 3.4.8 below). Twisting a quadratic functor by the former sign then corresponds to the opera-tion Ϙ ↦ Ϙ [2] ◦ Σ while twisting with respect to the latter sign corresponds to the operation Ϙ ↦ Ϙ [−2] ◦ Ω .3.4.4. Construction (Symmetry and sign actions) . Let 𝑘 be an E ∞ -ring spectrum. By a symmetry on 𝑘 wewill mean a refinement of the unit map 𝕊 → 𝑘 to a C -equivariant map 𝕊 → Σ 𝜎 𝑘 (where C acts triviallyon 𝕊 ). In particular, a symmetry determines a trivialization of the C -action on Σ 𝜎 𝑘 . For an E ∞ -ringequipped with a symmetry we will denote Σ 𝜎 𝑘 simply by − 𝑘 , and refer to it as the sign action on 𝑘 . Wenote that (− 𝑘 ) ⊗ 𝑘 (− 𝑘 ) ≃ 𝑘 equivariantly thanks to the symmetry structure, and hence − 𝑘 indeed behaveslike a sign action. In particular, − 𝑘 is also equivariantly equivalent to 𝕊 𝜎 −1 and more generally to 𝕊 𝑛 − 𝑛𝜎 for every odd 𝑛 ∈ ℤ . If 𝑀 is any 𝑘 -module with C -action then we will denote by − 𝑀 = 𝑀 ⊗ 𝑘 (− 𝑘 ) thesame 𝑘 -module with the C -action twisted by the sign − 𝑘 .3.4.5. Example.
The commutative ring 𝑘 = ℤ , considered as an E ∞ -ring spectrum, admits a unique sym-metry. Indeed, since Map ℤ ( ℤ , ℤ ) = ℤ is discrete and the action of the generator of C on 𝑆 𝜎 is homotopicto the identity, it follows that the induced action on Σ 𝜎 ℤ is trivial and ℤ admits a unique fixed pointstructure. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 57 Example.
Generalizing Example 3.4.5 we claim that any complex orientation on 𝑘 determines asymmetry on it. This statement is essentially equivalent to exhibiting a symmetry on the universal case MU . To show that such a universal symmetry exists, it will suffice to show that 𝜋 (MU) lifts to 𝜋 (Σ 𝜎 MU) hC . Now the homotopy fixed points spectral sequence degenerates at the 𝐸 page since thispage is concentrated in even degrees: this follows from the fact that the C -action on homotopy groups 𝜋 ∗ (Σ 𝜎 MU) is trivial since the generator of C acts through a map 𝑆 𝜎 → 𝑆 𝜎 which is homotopic to theidentity. The homotopy fixed points spectral sequence thus strongly converges and we can pick universalsymmetry as desired.3.4.7. Example.
The sphere spectrum 𝑘 = 𝕊 does not admit a symmetry. Indeed, otherwise by Lemma 3.4.1we would get an equivalence Σ ( 𝕊 tC ) ≃ (Σ 𝕊 ) tC ≃ (Σ 𝜎 𝕊 ) tC ≃ 𝕊 tC , and so the Tate spectrum for the trivial action would be -periodic, whereas by Lin’s theorem 𝕊 tC ≃ 𝕊 isconnective and non-trivial. More generally, this argument shows that the -periodicity of 𝑘 tC is a necessarycondition the existence of a symmetry. In that respect we point out that for complex orientable ring spectrathese Tate constructions are known to be -periodic.3.4.8. Warning.
It follows from Example 3.4.7 that the two C -spectra 𝕊 𝜎 −1 and 𝕊 𝜎 are not C -equivariantlyequivalent. In particular, there are at least two equally natural candidates for the “sign action” on the spherespectrum. Using a similar argument to that of Example 3.4.7 one can actually show that no two of 𝕊 𝑛 − 𝑛𝜎 for 𝑛 ∈ ℤ are equivalent. Consequently, in the absence of a symmetry in the sense of Construction 3.4.4,one cannot talk about the sign action, or about “twisting by a sign”, without further specifications.3.4.9. Corollary (Periodicity for quadratic and symmetric structures) . Let 𝑘 be an E ∞ -ring spectrum equippedwith a symmetry (e.g, any ordinary commutative ring or any complex oriented E ∞ -ring spectrum). Let 𝐴 be an E -algebra over 𝑘 and let 𝑀 be an invertible module with involution over 𝐴 . Then the loop functor Ω refines to give equivalences of Poincaré ∞ -categories ( Mod 𝜔𝐴 , ( Ϙ s 𝑀 ) [2] ) ≃ ←←←←←←←→ ( Mod 𝜔𝐴 , Ϙ s− 𝑀 ) and ( Mod 𝜔𝐴 , ( Ϙ q 𝑀 ) [2] ) ≃ ←←←←←←←→ ( Mod 𝜔𝐴 , Ϙ q− 𝑀 ) , and the double loop functor Ω refines to give equivalences of Poincaré ∞ -categories ( Mod 𝜔𝐴 , ( Ϙ s 𝑀 ) [4] ) ≃ ←←←←←←←→ ( Mod 𝜔𝐴 , Ϙ s 𝑀 ) and ( Mod 𝜔𝐴 , ( Ϙ q 𝑀 ) [4] ) ≃ ←←←←←←←→ ( Mod 𝜔𝐴 , Ϙ q 𝑀 ) . To formulate a version of the above periodicity for Poincaré structures which are not symmetric or qua-dratic note that if 𝑘 is an E ∞ -ring spectrum equipped with a symmetry and 𝑀 is a 𝑘 -module equipped witha C -action then 𝑀 tC ≃ ←←←←←←←→ (Σ 𝜎 𝑀 ) tC ≃ (Σ 𝜎 𝑘 ⊗ 𝑘 𝑀 ) tC ≃ ←←←←←←←→ (Σ 𝑘 ⊗ 𝑘 𝑀 ) tC ≃ (Σ 𝑀 ) tC ≃ Σ 𝑀 tC , and so 𝑀 tC is an -periodic spectrum. In this case, for every module with genuine involution ( 𝑀, 𝑁, 𝛼 ) wemay consider the associated module with genuine involution ( 𝑀, Σ 𝑁, Σ 𝛼 ) , where Σ 𝛼 is the composedmap Σ 𝑁 → Σ 𝑀 ≃ 𝑀. Corollary ( -fold periodicity for genuine structures) . Let 𝑘 be an E ∞ -ring spectrum equipped witha symmetry (e.g, any ordinary commutative ring or any complex oriented E ∞ -ring spectrum). Let 𝐴 be an E -algebra over 𝑘 and let ( 𝑀, 𝑁, 𝛼 ∶ 𝑁 → 𝑀 tC ) be an invertible module with involution over 𝐴 . Thenthe double loop functor Ω refines to give an equivalence of Poincaré ∞ -categories ( Mod 𝜔𝐴 , ( Ϙ 𝛼𝑀 ) [4] ) ≃ ←←←←←←←→ ( Mod 𝜔𝐴 , Ϙ Σ 𝛼𝑀 ) . Corollary (Periodicity for truncated structures) . Let 𝑘 be an E ∞ -ring spectrum equipped with asymmetry (e.g, any ordinary commutative ring or any complex oriented E ∞ -ring spectrum). Let 𝐴 be an E -algebra over 𝑘 and let 𝑀 be an invertible module with involution over 𝐴 . Then for every 𝑚 ∈ ℤ thefunctors Ω and Ω respectively refine to give equivalences of Poincaré ∞ -categories ( Mod 𝜔𝐴 , ( Ϙ ≥ 𝑚𝑀 ) [2] ) ≃ ←←←←←←←→ ( Mod 𝜔𝐴 , Ϙ ≥ 𝑚 +1− 𝑀 ) and ( Mod 𝜔𝐴 , ( Ϙ ≥ 𝑚𝑀 ) [4] ) ≃ ←←←←←←←→ ( Mod 𝜔𝐴 , Ϙ ≥ 𝑚 +2 𝑀 ) . Proof.
The second equivalence is obtained by composing two equivalences of the first type. To prove thefirst one, Proposition 3.4.2 tells us that the quadratic functor ( Ϙ ≥ 𝑚𝑀 ) [2] ◦ Σ is naturally equivalent to the oneassociated to the module with genuine involution (− 𝑀, Σ( 𝜏 ≥ 𝑚 𝑀 tC ) , 𝛼 ∶ Σ( 𝜏 ≥ 𝑚 𝑀 tC ) → Σ 𝑀 tC ≃ (− 𝑀 ) tC ) . It will hence suffice to show that the reference map 𝛼 is an ( 𝑚 + 1) -connected cover. Indeed, it is thesuspension of an 𝑚 -connected cover. (cid:3) The following -periodicity for L -groups immediately follows (see also Remark 4.2.9 for the relationwith classical L -group periodicity):3.4.12. Corollary ( L -group skew periodicity) . Let 𝑘 be an E ∞ -ring spectrum equipped with a symmetry(e.g, any ordinary commutative ring or any complex oriented E ∞ -ring spectrum). Let 𝐴 be an E -algebraover 𝑘 and let 𝑀 be an invertible module with involution over 𝐴 . Then for 𝑑 ∈ ℤ we have canonicalisomorphisms L 𝑑 +2 ( Mod 𝜔𝐴 , Ϙ s 𝑀 ) ≅ L 𝑑 ( Mod 𝜔𝐴 , Ϙ s− 𝑀 ) and L 𝑑 +2 ( Mod 𝜔𝐴 , Ϙ q 𝑀 ) ≅ L 𝑑 ( Mod 𝜔𝐴 , Ϙ q− 𝑀 ) , and for 𝑑, 𝑚 ∈ ℤ we have canonical isomorphisms L 𝑑 +2 ( Mod 𝜔𝐴 , Ϙ ≥ 𝑚𝑀 ) ≅ L 𝑑 ( Mod 𝜔𝐴 , Ϙ ≥ 𝑚 +1− 𝑀 ) . Corollary ( L -group -fold periodicity) . Let 𝑘 be an E ∞ -ring spectrum equipped with a symmetry(e.g, any ordinary commutative ring or any complex oriented E ∞ -ring spectrum). Let 𝐴 be an E -algebraover 𝑘 and let 𝑀 be an invertible module with involution over 𝐴 . Then for 𝑑 ∈ ℤ we have canonicalisomorphisms L 𝑑 +4 ( Mod 𝜔𝐴 , Ϙ s 𝑀 ) ≅ L 𝑑 ( Mod 𝜔𝐴 , Ϙ s 𝑀 ) and L 𝑑 +4 ( Mod 𝜔𝐴 , Ϙ q 𝑀 ) ≅ L 𝑑 ( Mod 𝜔𝐴 , Ϙ q 𝑀 ) , and for 𝑑, 𝑚 ∈ ℤ we have canonical isomorphisms L 𝑑 +4 ( Mod 𝜔𝐴 , Ϙ ≥ 𝑚𝑀 ) ≅ L 𝑑 ( Mod 𝜔𝐴 , Ϙ ≥ 𝑚 +2 𝑀 ) .
4. E
XAMPLES
In this section we will discuss several examples of interest of Poincaré ∞ -categories in further detail.We will begin in §4.1 with the important example of the universal Poincaré ∞ -category ( S 𝑝 f , Ϙ u ) , which ischaracterized by the property that Poincaré functors out of it pick out Poincaré objects in the codomain. In§4.2 we will consider perfect derived ∞ -categories of ordinary rings and show how to translate the clas-sical language of forms on projective modules into that of the present paper via the process of deriving quadratic functors. These examples form the main link between the present work and classical hermitian K -theory, and will be the main focus of applications in Paper [III]. In §4.3 we will turn to a specific familyof ring spectra with anti-involution, the group ring spectra , which carries a special interest due to its rela-tion with surgery theory. In particular, we will explain how to construct modules with genuion involutionover such ring spectra whose associated L -theory captures visible L -theory as studied by [Wei92], [Ran92]and [WW14]. Finally, in §4.4 we will discuss the closely related case of parameterized spectra , whichserves as a base-point-free variant of group rings, and show how to construct Poincaré structures producingthe parameterized spectra variant of visible L -theory as studied in [WW14].4.1. The universal Poincaré category.
In this section we will discuss the Poincaré ∞ -category ( S 𝑝 f , Ϙ u ) ofExample 1.2.15, which we call the universal Poincaré ∞ -category . This term is motivated by the followingmapping property which we will prove below: the Poincaré ∞ -category ( S 𝑝 f , Ϙ u ) corepresents the functor Pn ∶ Cat p∞ → S which assigns to a Poincaré ∞ -category its space of Poincaré forms. To exhibit this,consider the map(84) 𝕊 → Ϙ u ( 𝕊 ) given by the identity 𝕊 → D 𝕊 = 𝕊 on the linear part and by the unit map 𝕊 → (D 𝕊 ⊗ D 𝕊 ) hC = 𝕊 hC on thebilinear part. These two maps canonically lead to the ‘same’ map to (D 𝕊 ⊗ D 𝕊 ) tC = 𝕊 tC since the Tate ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 59 diagonal of 𝕊 agrees with the composite 𝕊 → 𝕊 hC → 𝕊 tC by construction. In particular, the composite 𝕊 → Ϙ u ( 𝕊 ) → B Ϙ u ( 𝕊 , 𝕊 ) = D( 𝕊 ⊗ 𝕊 ) ≃ 𝕊 is the identity. We then note that the map (84) corresponds to ahermitian form 𝑞 u ∈ Ω ∞ Ϙ u ( 𝕊 ) such that 𝑞 u ♯ ∶ 𝕊 → D 𝕊 = 𝕊 is the identity, and in particular 𝑞 u is Poincaré.We will refer to it as the universal Poincaré form .4.1.1. Lemma.
For every quadratic functor Ϙ ∶ ( S 𝑝 f ) op → S 𝑝 , the map hom( Ϙ u , Ϙ ) → Ϙ ( 𝕊 ) , (85) induced by the universal form (84) , is an equivalence of spectra.Proof. Since the collection of Ϙ ∈ Fun q ( S 𝑝 f for which (85) is an equivalence is closed under limits it willsuffice by Proposition 1.3.11 to prove the claim whenever Ϙ is either exact or of the form Ϙ ( 𝑥 ) = B( 𝑥, 𝑥 ) hC for some symmetric bilinear functor B . In the former case, the claim follows since the linear part of Ϙ u is D by construction and the composite 𝕊 → Ϙ u ( 𝕊 ) → D 𝕊 exhibits D as (stably) represented by 𝕊 . On theother hand, if Ϙ is of the form B( 𝑥, 𝑥 ) hC for some symmetric bilinear functor B then the result follows fromLemma 1.1.7 since the image of the universal form (84) in B Ϙ u ( 𝕊 , 𝕊 ) = D( 𝕊 ⊗ 𝕊 ) = hom( 𝕊 ⊗ 𝕊 , 𝕊 ) = hom( 𝕊 , 𝕊 ) corresponds to the identity 𝕊 → 𝕊 by construction, and thus exhibits the underlying bilinear part B Ϙ u ∈ Fun b ( S 𝑝 f ) ≃ Fun ex (( S 𝑝 f ⊗ S 𝑝 f ) op , S 𝑝 f ) ≃ Fun ex (( S 𝑝 f ) op , S 𝑝 ) as stably represented by 𝕊 as well. (cid:3) Lemma.
The sphere spectrum 𝕊 ∈ S 𝑝 f exhibits S 𝑝 f as corepresenting the core groupoid functor Cr ∶ Cat ex∞ → S C ↦ 𝜄 C . Proof.
Let S fin∗ be the ∞ -category of finite pointed spaces. Then the inclusion S fin∗ → S 𝑝 f exhibits S 𝑝 f asthe Spanier-Whitehead stabilisation of S fin∗ , and in particular, for every stable ∞ -category D the restrictionfunctor Fun ex ( S 𝑝 f , D ) → Fun rex ( S fin∗ , D ) is an equivalence, where Fun rex (− , −) ⊆ Fun(− , −) denotes the full subcategory spanned by the right exact(i.e., finite colimits preserving) functors, see, e.g., [Lur16, Proposition C.1.1.7]. On the other hand, for D stable we may identify right exact functors S fin∗ → D with those which are reduced and excisive, and hencewith spectrum objects in D . We thus obtain that for D stable the evaluation functor Fun ex ( S 𝑝 f , D ) ≃ Fun rex ( S fin∗ , D ) ≃ S 𝑝 ( D ) → D is an equivalence, and hence induces an equivalence Map( S 𝑝 f , D ) ≃ Cr D on the level of core groupoids. (cid:3) We now come to the main result of this section:4.1.3.
Proposition (Universality of the universal Poincaré ∞ -category) . The universal Poincaré object ( 𝕊 , 𝑞 u ) exhibits ( S 𝑝 f , Ϙ u ) as corepresenting the functor Pn . Similarly, when considered as a hermitian ∞ -category,the underlying universal hermitian object exhibits ( S 𝑝 f , Ϙ u ) as corepresenting the functor Fm .Proof. We begin with the second claim. We need to show that for every ( C , Ϙ ) ∈ Cat h∞ the map Map
Cat h∞ (( S 𝑝 f , Ϙ u ) , ( C , Ϙ )) → Fm( C , Ϙ ) sending ( 𝑓 , 𝜂 ) ∶ ( S 𝑝 f , Ϙ u ) → ( C , Ϙ ) to ( 𝑓 ( 𝕊 ) , 𝜂 𝕊 𝑞 ) is an equivalence. Consider the commutative square(86) Map
Cat h∞ (( S 𝑝 f , Ϙ u ) , ( C , Ϙ )) Fm( C , Ϙ )Map Cat ex∞ ( S 𝑝 f , C ) Cr( C ) ≃ furnished by Lemma 2.1.2, where the bottom horizontal map is the one induced by the object 𝕊 ∈ 𝜄 S 𝑝 f ,which is an equivalence by Lemma 4.1.2. It will hence suffice to show that for every exact functor 𝑓 ∶ S 𝑝 f → C the top horizontal map in (86) induces an equivalence on vertical fibres. Now by the construction of Cat h∞ as a cartesian fibration we have that the fibre of the left vertical map over 𝑓 ∶ S 𝑝 f → C is given by the spaceof natural transformations Nat( Ϙ u , 𝑓 ∗ Ϙ ) . On the other hand, the fibres of the right vertical map over 𝑓 ( 𝕊 ) isthe space Ω ∞ Ϙ ( 𝑓 ( 𝕊 ) of hermitian forms on 𝑓 ( 𝕊 ) ∈ C by construction. Lemma 4.1.1 then implies that theinduced map Nat( Ϙ u , 𝑓 ∗ Ϙ ) → 𝑓 ∗ Ϙ ( 𝕊 ) = Ϙ ( 𝑓 ( 𝕊 )) on vertical fibres is an equivalence. This shows that ( 𝕊 , 𝑞 u ) exhibits ( S 𝑝 f , Ϙ u ) as representing the functor Fm in Cat h∞ . To obtain the analogous claim for Poincaré forms we observe that the corresponding map Nat(B Ϙ u , ( 𝑓 × 𝑓 ) ∗ B Ϙ ) → ( 𝑓 × 𝑓 ) ∗ B Ϙ ( 𝕊 , 𝕊 ) = B Ϙ ( 𝑓 ( 𝕊 ) , 𝑓 ( 𝕊 )) , induced by the image of the universal Poincaré form in B Ϙ ( 𝕊 , 𝕊 ) , identifies under Lemma 1.2.4 with themap(87) Nat( 𝑓 D , D Ϙ 𝑓 op ) → Map( 𝑓 ( 𝕊 ) , D Ϙ 𝑓 ( 𝕊 )) which sends a natural transformation 𝜏 ∶ 𝑓 D ⇒ D Ϙ 𝑓 op to the composite 𝑓 ( 𝕊 ) 𝑓 ∗ 𝑞 u ♯ ←←←←←←←←←←←←←←←←←←→ 𝑓 (D 𝕊 ) 𝜏 𝕊 ←←←←←←←←←←→ D Ϙ 𝑓 ( 𝕊 ) . Since 𝑞 u ♯ is an equivalence this map sends natural equivalences 𝑓 D ≃ ←←←←←←←→ D Ϙ 𝑓 op to equivalences 𝑓 ( 𝕊 ) ≃ ←←←←←←←→ D Ϙ 𝑓 ( 𝕊 ) . To finish the proof it will hence suffice to show that (87) also detects equivalences. Indeed,since 𝑞 u ♯ is an equivalence, this follows from the fact that 𝕊 generates S 𝑝 f under finite colimits and so anatural transformation between two exact functors on S 𝑝 f is an equivalence if and only if it evaluates to anequivalence on 𝕊 . (cid:3) Ordinary rings and derived structures.
In this section we consider the case of ordinary rings andexplain how classical inputs for Grothendieck-Witt- and L -theory can be encoded as Poincaré structures onthe ∞ -category of perfect chain complexes. Our main result (see Proposition 4.2.12 below) is that suchPoincaré structures are essentially uniquely determined by their values on projective modules. This leadsto the formation of non-abelian derived versions of classical notions of hermitian forms, constituting themain link through which the point of view taken in this paper series interacts with its classical algebraiccounterpart.Let 𝑅 be an ordinary associative ring. Recall the Eilenberg-MacLane inclusion H ∶ A 𝑏 = S 𝑝 ♡ ↪ S 𝑝 of abelian groups as the heart of the canonical 𝑡 -structure on spectra. Since this 𝑡 -structure is compatiblewith tensor products H is naturally lax symmetric monoidal and consequently H 𝑅 is naturally an E -ringspectrum, and furthermore an E -algebra over the E ∞ -ring spectrum H ℤ . We then have natural equivalences[Lur17, Theorem 7.1.2.1](88) Mod H ℤ ≃ D ( ℤ ) and Mod 𝜔 H 𝑅 ≃ D p ( 𝑅 ) between the ∞ -categories of H ℤ -module spectra and compact H 𝑅 -modules in H ℤ -module spectra, as con-sidered in §3, and the derived and perfect derived ∞ -categories of ℤ and 𝑅 , respectively, see Example 1.2.12and Example 3.1.9. Similarly, we have an equivalence Mod fH 𝑅 ≃ D f ( 𝑅 ) between the ∞ -category of finitelypresented 𝑅 -modules in Mod H ℤ and the finitely presented derived category D f ( 𝑅 ) , obtained from the cat-egory of bounded complexes of finitely generated free 𝑅 -modules by inverting quasi-isomorphisms.4.2.1. Notation.
In what follows we will need to consider both ordinary tensor products over ℤ and tensorproduct of H ℤ -module spectra over H ℤ , that which corresponds, under the equivalence (88), to the derived tensor product ⊗ L ℤ of complexes over ℤ . We will consequently write ⊗ ℤ for the former and ⊗ H ℤ for thelatter. In particular, for two ordinary 𝑅 -modules 𝑀 and 𝑁 one has a canonical map H 𝑀 ⊗ H ℤ H 𝑁 → H( 𝑀 ⊗ ℤ 𝑁 ) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 61 which is generally not an isomorphism, though it does exhibit its codomain as the ’th truncation of itsdomain, and in particular determines an isomorphism 𝜋 (H 𝑀 ⊗ H ℤ H 𝑁 ) ≅ 𝑀 ⊗ ℤ 𝑁. As described in §3, we may construct bilinear functors on
Mod 𝜔 H 𝑅 from modules with involution over H 𝑅 (Definition 3.1.1), and hermitian structures on Mod 𝜔 H 𝑅 from modules with genuine involution over H 𝑅 (Definition 3.2.2). In this context, it is natural to focus attention on modules with involution whicharise by taking Eilenberg-MacLane spectra of ordinary modules. More precisely, the latter would consistsimply of an ( 𝑅 ⊗ ℤ 𝑅 ) -module 𝑀 in the ordinary sense together with an involution of abelian groups 𝜎 ∶ 𝑀 ≃ ←←←←←←←→ 𝑀 which is linear over the flip isomorphism 𝑅 ⊗ ℤ 𝑅 ≃ ←←←←←←←→ 𝑅 ⊗ ℤ 𝑅 , that is, which satisfies 𝜎 (( 𝑎 ⊗ 𝑏 ) 𝑥 ) = ( 𝑏 ⊗ 𝑎 ) 𝜎 ( 𝑥 ) . We will generally write such modules with involution as pairs ( 𝑀, 𝜎 ) . We willthen denote by B 𝑀 the bilinear functor on D p ( 𝑅 ) corresponding to the bilinear functor B H 𝑀 on Mod H 𝑅 viathe equivalence (88). The bilinear functor B 𝑀 is then perfect if and only if H 𝑀 is invertible , that is, H 𝑀 isperfect as an H 𝑅 -module and the map H 𝑅 → End H 𝑅 (H 𝑀 ) is an equivalence (see Definition 3.1.2). We willsay that 𝑀 is invertible over 𝑅 if H 𝑀 is invertible over H 𝑅 and in addition 𝑀 is projective as an 𝑅 -module.This insures, for example, that the duality D 𝑀 ∶ D p ( 𝑅 ) op ≃ ←←←←←←←→ D p ( 𝑅 ) associated to the perfect bilinear functor B 𝑀 preserves the (ordinary) full subcategory Proj( 𝑅 ) ⊆ D p ( 𝑅 ) of finitely generated projective 𝑅 -modules,and determines in particular a duality of ordinary categories D 𝑀 ∶ Proj( 𝑅 ) op ≃ ←←←←←←←→ Proj( 𝑅 ) D 𝑀 ( 𝑋 ) = Hom 𝑅 ( 𝑋, 𝑀 ) , where Hom 𝑅 ( 𝑋, 𝑀 ) is given the 𝑅 -module structure using the second 𝑅 -action. This is also consistentwith the classical terminology concerning invertible modules, see, e.g., Example 4.2.2 just below.4.2.2. Example. If 𝑅 is commutative then any 𝑅 -module 𝑀 can be considered as an ( 𝑅 ⊗ ℤ 𝑅 ) -modulevia the multiplication homomorphism 𝑅 ⊗ ℤ 𝑅 → 𝑅 . In particular, the two 𝑅 -actions coincide, and wemay endow 𝑀 with the trivial involution. For a projective 𝑀 this results in an invertible module withinvolution if and only if 𝑀 is invertible as an object in the symmetric monoidal category Proj( 𝑅 ) . Fromthe perspective of algebraic geometry, such modules correspond to line bundles over spec( 𝑅 ) .4.2.3. Examples.
Suppose that 𝑅 is equipped with an anti-involution , that is, an abelian group involution ∙ ∶ 𝑅 ≃ ←←←←←←←→ 𝑅 which satisfies 𝑎𝑏 = 𝑏𝑎 . In this case 𝑅 can be considered as a module with involution over itselfvia the ( 𝑅 ⊗ ℤ 𝑅 ) -action ( 𝑎 ⊗ 𝑏 ) 𝑥 = 𝑎𝑥𝑏 and the involution 𝜎 = ∙ . Some examples of interest of such ringsinclude:i) Any commutative ring with an automorphism of order gives rise to a ring with anti-involution. Forexample, the field ℂ of complex numbers can be considered as a ring with anti-involution via complexconjugation.ii) The group ring ℤ [ 𝐺 ] associated to a discrete group 𝐺 carries a natural anti-involution given on additivegenerators by 𝑔 ↦ 𝑔 −1 . This example is a recurring one in geometric applications of L -theory (seealso §4.3). More generally one can consider an orientation character 𝜒 ∶ 𝐺 → 𝐶 = {±1} and definethe 𝜒 -twisted anti-involution by setting 𝑔 ↦ 𝜒 ( 𝑔 ) 𝑔 −1 .iii) For a commutative ring 𝑘 the 𝑘 -algebra of 𝑛 × 𝑛 -matrices Mat 𝑛 ( 𝑘 ) admits an anti-involution 𝐴 ↦ 𝐴 𝑡 given by sending a matrix to its transpose. This more generally works for 𝑘 a ring with anti-involution.iv) For a commutative ring 𝑘 and 𝑎, 𝑏 ∈ 𝑘 , the quaternion 𝑘 -algebra Q 𝑘 ( 𝑎, 𝑏 ) is the algebra generated over 𝑘 by elements 𝑖, 𝑗 under the relation 𝑖 = 𝑎, 𝑗 = 𝑏, 𝑖𝑗 = − 𝑖𝑗 . It admits an anti-involution sending 𝑖 to − 𝑖 and 𝑗 to − 𝑗 .Another common source of invertible modules with involution in the discrete case is the following.Recall that an anti-structure in the sense of Wall [Wal70] on a ring 𝑅 consists of a ring isomorphism ∙ ∶ 𝑅 op → 𝑅, together with a unit 𝜖 ∈ 𝑅 ∗ , such that 𝜖 = 𝜖 −1 and 𝑟 = 𝜖 −1 𝑟𝜖 . In particular, if 𝜖 belongs to the center of 𝑅 then ∙ is an anti-involution, and this is arguably the most common case studied in the literature. Specifically,one often considers the case where 𝜖 = ±1 , which, for example, in the case of the integers ℤ , are also theonly possibilities. Given a Wall anti-structure (∙ , 𝜖 ) , we may consider 𝑅 as an ( 𝑅 ⊗ ℤ 𝑅 ) -module with action given by ( 𝑎 ⊗ 𝑏 ) ⋅ 𝑟 = 𝑎𝑟𝑏 . The map 𝑎 ↦ 𝜖𝑎 is then an involution on 𝑅 which is linear over the flip action on 𝑅 ⊗ ℤ 𝑅 , and so we obtain the structure of a module with involution. This module with involution is always invertible : the induced map 𝑅 → Hom 𝑅 ( 𝑅, 𝑅 ) ≅ 𝑅 op identifies with 𝑟 ↦ 𝑟 .4.2.4. Remark.
A Wall anti-structure captures the most general form of an invertible module with involutionover 𝑅 whose underlying 𝑅 -module is 𝑅 . Indeed, giving the 𝑅 -module 𝑅 a second commuting 𝑅 -actionis equivalent to providing a ring homomorphism (∙) ∶ 𝑅 → Hom 𝑅 ( 𝑅, 𝑅 ) = 𝑅 op , which is furthermore anisomorphism if the desired ( 𝑅 ⊗ ℤ 𝑅 ) -module is to be invertible. The ( 𝑅 ⊗ ℤ 𝑅 ) -action can then be writtenin terms of (∙) by ( 𝑎 ⊗ 𝑏 )( 𝑐 ) = 𝑎𝑏𝑐 . If 𝜎 ∶ 𝑅 → 𝑅 is now an abelian group isomorphism which switchesthe two 𝑅 -action then 𝜎 is completely determined by the value 𝜖 ∶ 𝜎 (1) ∈ 𝑅 , in terms of which 𝜎 can bewritten as 𝜎 ( 𝑟 ) = 𝜎 ( 𝑟 ⋅
1) = 𝜎 (1) 𝑟 = 𝜖𝑟. Since 𝜎 and (−) are both isomorphisms of abelian groups so is the map 𝑟 ↦ 𝜖 ⋅ 𝑟 , and hence 𝜖 must be aunit. In addition, since 𝜎 switches the two 𝑅 -actions we also have 𝑟𝜖 = 𝑟𝜎 (1) = 𝜎 (1 ⋅ 𝑟 ) = 𝜖𝑟 and hence 𝑟 = 𝜖 −1 𝑟𝜖 . Finally, the condition that 𝜎 is an involution implies that 𝜎𝜎 (1) = 𝜖𝜖 and hence 𝜖 = 𝜖 −1 . In particular, the pair (∙ , 𝜖 ) is a Wall anti-structure on 𝑅 and the module with involutionwe obtain is the one associated to that structure.4.2.5. Remark. If (∙ , 𝜖 ) is a Wall anti-structure on a ring 𝑅 and 𝑢 ∈ 𝑅 ∗ is a unit then we can obtain a newWall anti-structure by replacing (∙) with 𝑢 −1 (−) 𝑢 and 𝜖 with 𝜖 ( 𝑢 ) −1 𝑢 . One then says that the two Wall anti-structures (∙ , 𝜖 ) and ( 𝑢 −1 (−) 𝑢, 𝜖 ( 𝑢 ) −1 𝑢 ) are conjugated . In this case the associated modules with involutionover 𝑅 are isomorphic via the map 𝑅 → 𝑅 sending 𝑥 to 𝑥𝑢 . In fact, any isomorphism between the moduleswith involution associated to two Wall anti-structures is of this form, and so two Wall anti-structures areconjugated if and only if their associated modules with involution are isomorphic as such.The construction below, which was shared with the authors by Uriya First, gives an example of a Wallanti-structure which is not conjugated to any central Wall anti-structure:4.2.6. Example.
Let 𝐾 be a field which admits an automorphism 𝜎 ∶ 𝐾 → 𝐾 of order . Define 𝑅 = 𝐾 [ 𝑥, 𝑥 −1 ; 𝜎 ] to be the twisted Laurent polynomial ring generated over 𝐾 by an invertible generator 𝑥 which satisfies the relation 𝑥 −1 𝛼𝑥 = 𝜎 ( 𝛼 ) for 𝛼 ∈ 𝐾 . We may then extend 𝜎 to an anti-automorphism on 𝑅 defined on monomials by 𝛼𝑥 𝑖 = 𝑥 − 𝑖 𝜎 ( 𝛼 ) = 𝜎 𝑖 ( 𝛼 ) 𝑥 − 𝑖 . Then 𝛼𝑥 𝑖 = 𝜎 ( 𝛼 ) 𝑥 𝑖 = 𝑥 −1 ( 𝛼𝑥 𝑖 ) 𝑥 and 𝑥𝑥 = 1 , so that we obtain a Wall anti-structure (∙ , 𝜖 ) with 𝜖 = 𝑥 . ThisWall anti-structure is not conjugated to any central Wall anti-structure: indeed, the units of 𝑅 are exactly themonomials 𝛼𝑥 𝑖 , and if we conjugate the above Wall anti-structure by 𝛼𝑥 𝑖 then the new 𝜖 will be of the form 𝛽𝑥 𝑖 +1 for a suitable 𝛽 ∈ 𝐾 , and as such cannot be in the center, since it does not commute with 𝐾 ⊆ 𝑅 .Let us now also give a non-commutative example of an invertible module with involution whose under-lying module is not the ring itself:4.2.7.
Example.
Let 𝐵 = Q ℚ (−5 , −13) be the quaternion algebra over ℚ and let 𝐴 ⊆ 𝐵 be the subringgenerated over ℤ by , 𝑖, 𝑗 and 𝛽 = 𝑖 + 𝑗 + 𝑖𝑗 . Then 𝐴 is a maximal order in the quaternion algebra 𝐵 ,that is, it is finitely generated as a ℤ -module and is not contained in any other subring with this property.Invertible bimodules over maximal orders are relatively well understood, and can all be realized as invertibletwo-sided ideals in 𝐴 . In particular, the two-sided ideal 𝐼 ⊆ 𝐴 which is generated by , 𝑖 − 1 , 𝑗 − 1 is aninvertible ideal of index in 𝐴 (the quotient 𝐴 ∕ 𝐼 ≅ 𝔽 [ 𝛽 ]∕( 𝛽 − 𝛽 + 1) is a finite field of order ). One canthen verify that this ideal is not principal by checking that 𝐴 contains no elements of norm , and so this 𝐴 -bimodule is not isomorphic to 𝐴 as a left 𝐴 -module. At the same time, the involution on 𝐵 sending 𝑖 to − 𝑖 and 𝑗 to − 𝑗 restricts to an involution on 𝐴 , through which we can consider 𝐼 as an 𝐴 ⊗ 𝐴 -module, and
ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 63 since this involution preserves 𝐼 it endows it with the structure of an involution which is linear over the flipmap 𝐴 ⊗ 𝐴 → 𝐴 ⊗ 𝐴 . Then 𝐼 gives an invertible module with involution over 𝐴 which is not isomorphicto 𝐴 as a left 𝐴 -module.Let us now fix a (projective) invertible module with involution 𝑀 over 𝑅 . As in Definition 3.1.5 wewrite Ϙ s 𝑀 and Ϙ q 𝑀 for the symmetric and quadratic Poincaré structures on D p ( 𝑅 ) associated to the bilinearfunctor B 𝑀 . Similarly, as in Example 3.2.7, for an integer 𝑚 ∈ ℤ , we consider the associated module withgenuine involution (H 𝑀, 𝜏 ≥ 𝑚 H 𝑀 tC , 𝑡 𝑚 ∶ 𝜏 ≥ 𝑚 H 𝑀 tC → H 𝑀 tC ) over H 𝑅 , where 𝑡 𝑚 ∶ 𝜏 ≥ 𝑚 H 𝑀 tC → H 𝑀 tC is the 𝑚 -connective cover of H 𝑀 tC . We will denote theassociated Poincaré structure on D p ( 𝑅 ) ≃ Mod 𝜔 H 𝑅 as in Construction 3.2.5 by Ϙ ≥ 𝑚𝑀 ∶ D p ( 𝑅 ) op → S 𝑝. Warning.
The H ℤ -module structure on H 𝑀 tC above is not the one obtained by performing the Tateconstruction in Mod H ℤ ≃ D ( ℤ ) , but rather via the Tate Frobenius of H ℤ , see Warning 3.2.1. For example,if 𝑅 is an 𝔽 -algebra and 𝑀 = 𝑅 with trivial involution then H 𝑀 tC = ⊕ 𝑛 ∈ ℤ Σ 𝑛 H 𝑀 as a spectrum, but theaction of ℤ on 𝜋 (H 𝑀 tC ) = 𝑅 is not the canonical action of ℤ on 𝑅 but rather the action 𝑎 ( 𝑥 ) = 𝑎 𝑥 .4.2.9. Remark.
We will study the Grothendieck-Witt and L -groups of the Poincaré ∞ -categories ( D p ( 𝑅 ) , Ϙ ≥ 𝑚𝑀 ) for −∞ ≤ 𝑚 ≤ ∞ in depth in Paper [III]. In particular, we will show in Paper [III] that the L -groups of ( D p ( 𝑅 ) , Ϙ q 𝑀 ) are naturally isomorphic to the quadratic L -groups of Ranicki-Wall [Wal99]. These are -periodic, cf. Corollary 3.4.13. The symmetric L -groups were first introduced by Ranicki [Ran80] using 𝑛 -dimensional Poincaré complexes to define the 𝑛 ’th L -group. These are not -periodic in general, andwe will show in Paper [III] that they are naturally isomorphic to the L -groups of the Poincaré ∞ -category ( D p ( 𝑅 ) , Ϙ ≥ 𝑀 ) . A periodic variant of the symmetric L -groups was later introduced by Ranicki in [Ran92] us-ing Poincaré complexes of arbitrary length. Those are naturally isomorphic to the L -groups of the Poincaré ∞ -category ( D p ( 𝑅 ) , Ϙ s 𝑀 ) , which are indeed -periodic by Corollary 3.4.13.4.2.10. Remark.
By construction the linear part of the Poincaré structure Ϙ ≥ 𝑚𝑀 is given by the formula 𝑋 ↦ hom H 𝑅 ( 𝑋, 𝜏 ≥ 𝑚 H 𝑀 tC ) , where in the last term we have identified 𝑋 with the corresponding H 𝑅 -modulespectrum via the equivalence (88). In particular, Ϙ ≥ 𝑚𝑀 ( 𝑋 ) sits in an exact sequence of spectra Ϙ q 𝑀 ( 𝑋 ) → Ϙ ≥ 𝑚𝑀 ( 𝑋 ) → hom H 𝑅 ( 𝑋, 𝜏 ≥ 𝑚 H 𝑀 tC ) , which can be considered as a way of measuring the gap between Ϙ ≥ 𝑚𝑀 and Ϙ q 𝑀 . On the other hand, from thefibre sequence 𝜏 ≥ 𝑚 H 𝑀 tC → H 𝑀 tC → 𝜏 ≤ 𝑚 −1 H 𝑀 tC we see that Ϙ ≥ 𝑚𝑀 ( 𝑋 ) also sits in an exact sequenceof spectra Ϙ ≥ 𝑚𝑀 ( 𝑋 ) → Ϙ s 𝑀 ( 𝑋 ) → hom H 𝑅 ( 𝑋, 𝜏 ≤ 𝑚 −1 H 𝑀 tC ) , and this can be used to estimate the gap between Ϙ ≥ 𝑚𝑀 and Ϙ s 𝑀 .Our next goal is to relate the above Poincaré structures to more classical notions of hermitian forms on 𝑅 -modules. For this, let Ch b (Proj( 𝑅 )) be the category of bounded chain complexes of finitely generatedprojective 𝑅 -modules, so that the ∞ -category D p ( 𝑅 ) can be identified with the ∞ -categorical localisationof Ch b (Proj( 𝑅 )) by the collection of quasi-isomorphisms. The inclusion Proj( 𝑅 ) ⊆ Ch b (Proj( 𝑅 )) as chain-complexes concentrated in degree determines a fully-faithful functor(89) Proj( 𝑅 ) → D p ( 𝑅 ) , 𝑃 ↦ 𝑃 [0] . We also point out that the category
Proj( 𝑅 ) is additive, and the inclusion (89) is additive in the sensethat it preserves direct sums. One can then show that (89) exhibits D p ( 𝑅 ) as the initial stable ∞ -categoryequipped with additive functor from Proj( 𝑅 ) . In particular, we may consider D p ( 𝑅 ) as the stable ∞ -categorygenerated from Proj( 𝑅 ) . It is also sometimes called the stable envelope of Proj( 𝑅 ) .The following definition is originally due to Eilenberg and MacLane [EM54, Theorem 9.11]. Definition.
Let C , D be additive ∞ -categories. We will say that a functor Ϙ ∶ C → D is polynomialof degree if its cross-effects B Ϙ ( 𝑋, 𝑌 ) (defined as in Definition 1.1.4 as the kernel of the split surjection Ϙ ( 𝑋 ⊕ 𝑌 ) → Ϙ ( 𝑋 ) ⊕ Ϙ ( 𝑌 ) ) preserves direct sums in each variable separately. We will denote by Fun ( C , D ) ⊆ Fun( C , D ) the full subcategory spanned by functors which are polynomial of degree functors and reduced, that is,zero object preserving.The following principal result will allow us to relate the above additive setting with the stable one forcategories of modules:4.2.12. Proposition.
Let 𝑅 be an associative ring. Then restriction along the inclusion Proj( 𝑅 ) ⊆ D p ( 𝑅 ) yields an equivalence of ∞ -categories Fun q ( D p ( 𝑅 )) ≃ ←←←←←←←→ Fun (Proj( 𝑅 ) op , S 𝑝 ) . Since the Eilenberg-MacLane embedding
H ∶ A 𝑏 ↪ S 𝑝 is fully-faithful and additive, Proposition 4.2.12implies in particular that if Ϙ proj ∶ Proj( 𝑅 ) op → A 𝑏 is a reduced polynomial functor of degree then thecomposed functor Proj( 𝑅 ) op → A 𝑏 → S 𝑝 extends to a quadratic functor Ϙ ∶ D p ( 𝑅 ) op → S 𝑝 in an essentiallyunique manner. We will refer to such an extension Ϙ ∶ D p ( 𝑅 ) → S 𝑝 as the non-abelian derived functor of Ϙ proj .4.2.13. Remark.
For connective objects 𝑋 ∈ D p ( 𝑅 ) one can express the value of the derived functor Ϙ asfollows. First note that such an 𝑋 can be represented by a non-negatively graded chain complex 𝐶 ∙ over Proj( 𝑅 ) , which in turn determines a simplicial 𝑅 -module using the Dold-Kan correspondence. Applyingthe functor Ϙ proj levelwise we obtain a cosimplicial abelian group Ϙ proj ( 𝐶 ∙ ) , which we can then re-translateinto a non-positively graded chain complex over ℤ , and consequently into a spectrum. This is the classicaldescription of non-abelian derived functors of Dold and Puppe [DP58].The proof of Proposition 4.2.12 will be given below. Before, let us explore some of its consequences. Astheir higher categorical counterparts, reduced degree polynomial functors Proj( 𝑅 ) op → A 𝑏 can be usedto encode various types of hermitian forms. To make this more explicit, consider for a projective module 𝑃 ∈ Proj( 𝑅 ) the abelian group Hom
𝑅⊗𝑅 ( 𝑃 ⊗ ℤ 𝑃 , 𝑀 ) of 𝑀 -valued 𝑅 -bilinear forms 𝛽 ∶ 𝑃 ⊗ ℤ 𝑃 → 𝑀 .This abelian group carries an involution which sends a form 𝛽 to the form ( 𝑣, 𝑢 ) ↦ 𝜎 ( 𝛽 ( 𝑢, 𝑣 )) , where 𝜎 ∶ 𝑀 → 𝑀 is the involution of 𝑀 . The C -orbits and C -fixed points of 𝜎 are then related via the norm map N C ∶ Hom 𝑅⊗𝑅 ( 𝑃 ⊗ ℤ 𝑃 , 𝑀 ) C → Hom
𝑅⊗𝑅 ( 𝑃 ⊗ ℤ 𝑃 , 𝑀 ) C [ 𝛽 ] ↦ 𝛽 ( 𝑣, 𝑢 ) + 𝜎𝛽 ( 𝑢, 𝑣 ) , which sends an orbit to the sum of its representatives.4.2.14. Definition.
Let 𝑅 be a ring and ( 𝑀, 𝜎 ) an invertible module with involution over 𝑅 . We definefunctors Proj( 𝑅 ) op → A 𝑏 by the formulas Ϙ sproj ( 𝑃 ) = Hom 𝑅⊗ ℤ 𝑅 ( 𝑃 ⊗ ℤ 𝑃 , 𝑀 ) C , Ϙ qproj ( 𝑃 ) = Hom 𝑅⊗𝑅 ( 𝑃 ⊗ ℤ 𝑃 , 𝑀 ) C , and Ϙ evproj ( 𝑃 ) = im [ Ϙ qproj ( 𝑃 ) N C2 ←←←←←←←←←←←←←←←←→ Ϙ sproj ( 𝑃 ) ] . For 𝑃 ∈ Proj( 𝑅 ) we will refer to these as the abelian groups of 𝜎 -symmetric , 𝜎 -quadratic , and 𝜎 -even formson 𝑃 , respectively. These functors are visibly reduced and the cross-effect of each of them is ( 𝑃 , 𝑄 ) ↦ Hom 𝑅⊗ ℤ 𝑅 ( 𝑃 ⊗ ℤ 𝑄, 𝑀 ) , which is additive in each variable separately. In particular, they are polynomialof degree .By definition, a 𝜎 -symmetric form is an 𝑅 -bilinear form 𝜙 ∶ 𝑃 ⊗ ℤ 𝑃 → 𝑀 such that 𝜙 ( 𝑏, 𝑎 ) = 𝜎 ( 𝜙 ( 𝑎, 𝑏 )) .On the other hand, the data of a 𝜎 -quadratic form, or a C -orbit [ 𝛽 ] ∈ Hom 𝑅⊗𝑅 ( 𝑃 ⊗ ℤ 𝑃 , 𝑀 ) , is equivalentto that of a pair ( 𝜙, 𝑞 ) , where 𝜙 ∈ Hom 𝑅⊗𝑅 ( 𝑃 ⊗ ℤ 𝑃 , 𝑀 ) C is a 𝜎 -symmetric form and 𝑞 ∶ 𝑃 → 𝑀 C is aset-theoretic function which satisfies ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 65 i) 𝑞 ( 𝑣 + 𝑢 ) − 𝑞 ( 𝑣 ) − 𝑞 ( 𝑢 ) = [ 𝜙 ( 𝑣, 𝑢 )] ∈ 𝑀 C for 𝑣, 𝑢 ∈ 𝑃 ;ii) 𝑞 ( 𝑟𝑣 ) = ( 𝑟 ⊗ 𝑟 ) 𝑞 ( 𝑣 ) for 𝑣 ∈ 𝑃 and 𝑟 ∈ 𝑅 ;iii) the image of 𝑞 ( 𝑣, 𝑣 ) under the norm map 𝑀 C → 𝑀 C is the C -fixed element 𝜙 ( 𝑣, 𝑣 ) for 𝑣 ∈ 𝑃 .To obtain this description, note that the abelian group of such pairs ( 𝜙, 𝑞 ) forms a reduced degree 2 polyno-mial functor Ϙ ′ ∶ Proj( 𝑅 ) → A 𝑏 , which receives a natural transformation Ϙ qproj ⇒ Ϙ ′ sending [ 𝛽 ] ∈ Ϙ qproj ( 𝑃 ) to the pair ( 𝜙, 𝑞 ) , where 𝜙 = N[ 𝛽 ] is the norm of 𝛽 and 𝑞 ( 𝑥 ) = [ 𝛽 ( 𝑥, 𝑥 )] ∈ 𝑀 C . One can then ver-ify in a straightforward manner that this natural transformation induces an isomorphism on cross-effectsand an isomorphism on the value on 𝑃 = 𝑅 , and is hence an isomorphism on every 𝑃 ∈ Proj( 𝑅 ) (seealso [Wal70, Theorem 1], where this argument is elaborated in the case where 𝑀 comes from a Wall anti-structure).When 𝑅 is commutative and 𝑀 = 𝑅 with trivial involution the above notion of a quadratic form identifieswith the usual one. In this case even forms are symmetric forms which admit a quadratic refinement in theclassical sense (which is not kept as part of the structure). For example, when 𝑀 = 𝑅 = ℤ with trivialinvolution then a symmetric bilinear form 𝑏 on 𝑃 admits a quadratic refinement if and only if 𝑏 ( 𝑥, 𝑥 ) ∈ 2 ℤ for all 𝑥 ∈ 𝑃 , hence the terminology “even forms”. In this case the quadratic refinement is even unique,though this is by no means the case in general. If 𝑅 is commutative and 𝑀 = 𝑅 with involution 𝜎 ( 𝑥 ) = − 𝑥 then 𝜎 -symmetric forms are anti-symmetric forms while the 𝜎 -even forms are the alternating ones. Fornon-commutative 𝑅 this way of viewing quadratic forms was first devised by Tits [Tit68] for central simplealgebras, and later generalized by Wall [Wal70] to arbitrary rings with anti-structure as above.4.2.15. Proposition.
The quadratic functors Ϙ ≥ 𝑀 , Ϙ ≥ 𝑀 and Ϙ ≥ 𝑀 on D p ( 𝑅 ) are canonically equivalent to thenon-abelian derived functors of Ϙ sproj , Ϙ evproj and Ϙ qproj , respectively. Notation.
In light of Proposition 4.2.15 we will denote the Poincaré structures Ϙ ≥ 𝑀 , Ϙ ≥ 𝑀 and Ϙ ≥ 𝑀 also by Ϙ gs 𝑀 , Ϙ ge 𝑀 and Ϙ gq 𝑀 , and refer to them as the genuine symmetric , genuine even and genuine quadratic Poincaré structures on 𝑅 associated to 𝑀 .4.2.17. Remark.
In [HS20] the fourth and ninth authors show that the Grothendieck-Witt groups (the onedefined in §2.4 as well as the higher ones we will define in Paper [II]) of derived Poincaré structures asabove can be described in terms of the group completion of the corresponding monoid of Poincaré formson projective modules. In light of Proposition 4.2.15 it then follows that for an invertible module withinvolution ( 𝑀, 𝜎 ) over 𝑅 the Grothendieck-Witt groups of the associated genuine symmetric, genuine evenand genuine quadratic Poincaré structures reproduce the classical 𝜎 -symmetric, 𝜎 -even and 𝜎 -quadraticGrothendieck-Witt groups of 𝑅 with coefficients in 𝑀 .4.2.18. Remark.
It follows from Proposition 4.2.15 and Corollary 3.4.11 that we have equivalences ofPoincaré ∞ -categories ( Mod 𝜔𝐴 , ( Ϙ gs 𝑀 ) [2] ) ≃ ←←←←←←←→ ( Mod 𝜔𝐴 , Ϙ ge− 𝑀 ) and ( Mod 𝜔𝐴 , ( Ϙ ge 𝑀 ) [2] ) ≃ ←←←←←←←→ ( Mod 𝜔𝐴 , Ϙ gq− 𝑀 ) . In Paper [II] we will show how this permits one to identify and prove the correct generalization of Karoubi’sfundamental theorem to the setting where is not invertible. We point out that while this generalizationpertains by Remark 4.2.17 to the classical theory encompassed in the polynomial functors Ϙ qproj , Ϙ evproj and Ϙ sproj on Proj( 𝑅 ) , the above equivalences of Poincaré ∞ -categories are only visible after deriving these intoPoincaré structures to D p ( 𝑅 ) .The proof of Proposition 4.2.15 will require the following lemma:4.2.19. Lemma.
For 𝑃 ∈ Proj( 𝑅 ) the canonical map (90) H Hom 𝑅⊗ ℤ 𝑅 ( 𝑃 ⊗ ℤ 𝑅, 𝑀 ) → hom H 𝑅⊗ H ℤ H 𝑅 (H 𝑃 ⊗ H ℤ H 𝑃 , H 𝑀 ) is an equivalence. In particular, the right hand side lies in the heart Sp ♡ and we obtain identifications Ϙ qproj ( 𝑃 ) ≅ 𝜋 Ϙ q 𝑀 ( 𝑃 [0]) and Ϙ sproj ( 𝑃 ) ≅ 𝜋 Ϙ s 𝑀 ( 𝑃 [0]) . Proof.
For 𝑃 ∈ Proj( 𝑅 ) the condition that (90) is an equivalence is closed under direct sums and retracts,and so it suffices to check it for 𝑅 = 𝑃 , where it can be identified with the identity map H 𝑀 ≃ H Hom 𝑅⊗ ℤ 𝑅 ( 𝑅 ⊗ ℤ 𝑅, 𝑀 ) → hom H 𝑅⊗ H ℤ H 𝑅 (H 𝑅 ⊗ H ℤ H 𝑅, H 𝑀 ) ≃ H 𝑀. We then get that Ϙ qproj ( 𝑃 ) = Hom 𝑅⊗ ℤ 𝑅 ( 𝑅 ⊗ ℤ 𝑅, 𝑀 ) C ≅ 𝜋 (hom H 𝑅⊗ H ℤ H 𝑅 (H 𝑅 ⊗ H ℤ H 𝑅, H 𝑀 )) C ≅ 𝜋 [hom H 𝑅⊗ H ℤ H 𝑅 (H 𝑅 ⊗ H ℤ H 𝑅, H 𝑀 ) hC ] ≅ Ϙ q 𝑀 ( 𝑃 [0]) , and similarly Ϙ sproj ( 𝑃 ) = Hom 𝑅⊗ ℤ 𝑅 ( 𝑅 ⊗ ℤ 𝑅, 𝑀 ) C ≅ 𝜋 (hom H 𝑅⊗ H ℤ H 𝑅 (H 𝑅 ⊗ H ℤ H 𝑅, H 𝑀 )) C ≅ 𝜋 [hom H 𝑅⊗ H ℤ H 𝑅 (H 𝑅 ⊗ H ℤ H 𝑅, H 𝑀 ) hC ] ≅ Ϙ s 𝑀 ( 𝑃 [0]) , as desired. (cid:3) Proof of Proposition 4.2.15.
By Lemma 4.2.19 we have that for 𝑃 ∈ Proj( 𝑃 ) the spectrum B 𝑀 ( 𝑃 [0] , 𝑃 [0]) belongs to S 𝑝 ♡ and so Ϙ q 𝑀 ( 𝑃 [0]) is connective and Ϙ s 𝑀 ( 𝑃 [0]) is coconnective. By Remark 4.2.10 we thenget that for 𝑚 = 0 , , the spectrum Ϙ ≥ 𝑚𝑀 ( 𝑃 [0]) is both connective and coconnective, and hence lies in theheart as well. Now consider the pair of maps 𝜋 Ϙ q 𝑀 ( 𝑃 [0]) → 𝜋 Ϙ ≥ 𝑚𝑀 ( 𝑃 [0]) → 𝜋 Ϙ s 𝑀 ( 𝑃 [0]) . By Remark 4.2.10 the first map above is an isomorphism when 𝑚 = 2 and the second map is an isomorphismwhen 𝑚 = 0 . Finally, when 𝑚 = 1 the same remark gives that the first map is surjective and the second isinjective. Invoking the last part of Lemma 4.2.19 we now get an identification of 𝜋 Ϙ ≥ 𝑚𝑀 ( 𝑃 [0]) for 𝑚 = 0 , , with Ϙ sproj ( 𝑃 ) , Ϙ evproj ( 𝑃 ) and Ϙ qproj ( 𝑃 ) respectively. By the uniqueness of Proposition 4.2.12 we may identify 𝜋 Ϙ ≥ 𝑚𝑀 ( 𝑃 [0]) for 𝑚 = 0 , , with the desired nonabelian derived functors. (cid:3) The remainder of this section is devoted to the proof of Proposition 4.2.12. Let Δ ≤ 𝑛 ⊆ Δ be the fullsubcategory spanned by those totally ordered sets with at most 𝑛 + 1 elements.4.2.20. Definition.
Let C and D be small ∞ -categories.i) A diagram 𝑝 ∶ Δ op → C is called finite if it is left Kan extended from its restriction to Δ op ≤ 𝑛 for some 𝑛 . A colimit over such a diagram is called a finite geometric realization . Dually, a diagram Δ → C is called finite if it is right Kan extended from its restriction to Δ ≤ 𝑛 for some 𝑛 . A limit over such adiagram is called a finite totalization .ii) An ∞ -category C is said to admit finite geometric realizations if every finite diagram Δ op → C in C admits a colimit. Dually, C is said to admit finite totalizations if every finite diagram Δ → C admits alimit.iii) A functor 𝑓 ∶ C → D is said to preserve finite geometric realizations if its sends colimit cones 𝑝 ∶ (Δ op ) ⊳ → C over finite diagrams 𝑝 ∶ Δ op → C to colimit cones in D . Here, it is not requiredthat 𝑓 ◦ 𝑝 ∶ Δ op → D remains finite in the above sense. Dually, a functor 𝑓 is said to preserve finitetotalizations if its sends limit cones 𝑝 ∶ (Δ op ) ⊲ → C over finite diagrams 𝑝 ∶ Δ → C to limit cones in D . Again, it is not required that 𝑓 ◦ 𝑝 ∶ Δ op → D remains finite.We will denote by Fun Δ opfin ( C , D ) ⊆ Fun( C , D ) the full subcategory spanned by the functors which preservefinite geometric realizations.4.2.21. Remark.
If an ∞ -category C admits finite colimits then it admits finite geometric realizations, sincethe ∞ -category Δ op ≤ 𝑛 is finite and colimits of left Kan extended diagrams 𝑋 ∶ Δ op → C can be calculatedon their restriction to Δ op ≤ 𝑛 . In addition, if C and D are ∞ -categories with finite colimits and 𝑓 ∶ C → D preserves finite colimits then 𝑓 sends finite diagrams 𝑋 ∶ Δ op → C to finite diagrams; indeed, this followsfrom the pointwise formula for left Kan extensions since for each [ 𝑚 ] ∈ Δ op the comma category (Δ op ≤ 𝑛 ) ∕ [ 𝑚 ] is finite. It then follows that such an 𝑓 also preserves finite geometric realizations. A similar statementholds for finite totalizations under the analogous assumptions concerning finite limits. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 67 Lemma.
Let C be an ∞ -category which admits finite colimits and D and ∞ -category which admitssifted colimits. Then restriction along the Yoneda embedding C → Ind( C ) induces an equivalence Fun sif (Ind( C ) , D ) → Fun Δ opfin ( C , D ) between the full subcategory of functors Ind( C ) → D which preserve sifted colimits on the left hand sideand functors C → D which preserve finite geometric realizations on the right.Proof. We first note that by the universal property of ind-categories [Lur09a, Proposition 5.3.5.10] we havethat
Ind( C ) admits filtered colimits and restriction along 𝜄 ∶ C ↪ Ind( C ) determines an equivalence Fun filt (Ind( C ) , D ) ≃ ←←←←←←←→ Fun( C , D ) , where the left hand side stands for the full subcategory of Fun(Ind( C ) , D ) spanned by the functors whichpreserves filtered colimits. It will hence suffice to show that if 𝑓 ∶ Ind( C ) → D is a functor which preservesfiltered colimits then 𝑓 preserves sifted colimits if and only if 𝑓 ◦ 𝜄 preserves finite geometric realizations. Tobegin, note that if 𝑓 ∶ Ind( C ) → D preserves sifted colimits then it preserves in particular finite geometricrealizations. Since the inclusion 𝜄 ∶ C → Ind( C ) preserves finite colimits it preserves finite diagrams andfinite geometric realizations by Remark 4.2.21. It then follows that in this case 𝑓 ◦ 𝜄 preserves finite geometricrealizations. To prove the other direction, let us now suppose that 𝑓 ∶ Ind( C ) → D is a functor whichpreserves filtered colimits such that 𝑓 ◦ 𝜄 preserves finite geometric realizations. We wish to show that 𝑓 preserves all sifted colimits. By [Lur09a, Corollary 5.5.8.17] it will suffice to show that 𝑓 preservesgeometric realizations. Let 𝑡 𝑛 ∶ Δ op ≤ 𝑛 → Δ op be the inclusion. Then every Δ op -diagram 𝜌 ∶ Δ op → Ind( C ) can be written as a sequential (and in particular filtered) colimit of finite diagrams of the form 𝜌 ≃ colim 𝑛 ( 𝑡 𝑛 ) ! ( 𝑡 𝑛 ) ∗ 𝜌. Since 𝑓 preserves filtered colimits it will suffice to prove that 𝑓 preserves finite geometric realizations. Nowlet 𝜌 𝑛 ∶ Δ op ≤ 𝑛 → Ind( C ) be a diagram indexed on Δ op ≤ 𝑛 . We claim that 𝜌 𝑛 can be written as a filtered colimit of Δ op ≤ 𝑛 -diagrams taking values in C . To see this, consider the smallest full subcategory E ⊆ Fun(Δ op ≤ 𝑛 , Ind( C )) which contains the left Kan extended functors ( 𝑖 𝑘 ) ! 𝜄 ( 𝑥 ) , where 𝑥 ∈ C is an object and 𝑖 𝑘 ∶ {[ 𝑘 ]} ↪ Δ op ≤ 𝑛 is the inclusion of the object [ 𝑘 ] , and closed under finite colimits. Then E forms a collection of compactgenerators for Fun(Δ op ≤ 𝑛 , Ind( C ))) and hence every diagram 𝜌 𝑛 ∶ Δ op ≤ 𝑛 → Ind( C ) is a filtered colimit of dia-grams in E . Since the mapping sets in Δ op ≤ 𝑛 are finite the functors ( 𝑖 𝑘 ) ! 𝜄 ( 𝑥 ) takes values in the image of C ,and hence factor through Δ op ≤ 𝑛 -diagrams in C . It will hence suffice to show that 𝑓 preserves finite geomet-ric realizations of simplicial diagrams which factor through 𝜄 ∶ C ↪ Ind( C ) . Indeed, since C admits finitegeometric realizations which are preserved by 𝜄 this follows from the assumption that 𝑓 ◦ 𝜄 preserves finitegeometric realizations. (cid:3) Lemma.
Any quadratic functor Ϙ ∶ C op → S 𝑝 preserves finite geometric realizations and finitetotalizations.Proof. It suffices to show the claim for the special cases where Ϙ is either exact or of the form 𝑥 ↦ B( 𝑥, 𝑥 ) for a bilinear functor B ∶ C op × C op → S 𝑝 , since these generate all quadratic functors under both limitsand colimits: indeed, a general quadratic functor Ϙ is both the fibre of a map from B Ϙ (− , −) hC to an exactfunctor and the cofibre of a map from an exact functor to B Ϙ (− , −) hC . The case where Ϙ is exact followsfrom Remark 4.2.21. We hence suppose that Ϙ ( 𝑥 ) = B( 𝑥, 𝑥 ) for some bilinear B , and let 𝑋 ∶ Δ op → C be afinite diagram. Since Δ is sifted we then have colim [ 𝑛 ]∈Δ Ϙ ( 𝑋 𝑛 ) ≃ colim [ 𝑛 ]∈Δ B( 𝑋 𝑛 , 𝑋 𝑛 ) ≃ colim [ 𝑛 ]∈Δ colim [ 𝑚 ]∈Δ B( 𝑋 𝑛 , 𝑋 𝑚 ) . Now for each fixed 𝑥 ∈ C the functor B( 𝑥, −) is exact and hence preserves finite geometric realizations byRemark 4.2.21. Similarly, for each fixed 𝑦 ∈ C the functor B(− , 𝑦 ) preserves finite geometric realizations.We hence get that colim [ 𝑛 ]∈Δ colim [ 𝑚 ]∈Δ B( 𝑋 𝑛 , 𝑋 𝑚 ) ≃ colim [ 𝑛 ]∈Δ B( 𝑋 𝑛 , colim [ 𝑚 ]∈Δ 𝑋 𝑚 ) ≃ B(colim [ 𝑛 ]∈Δ 𝑋 𝑛 , colim [ 𝑚 ]∈Δ 𝑌 𝑚 ) . Using again that Δ is sifted we consequently get B(colim 𝑛 𝑋 𝑛 , colim 𝑚 𝑌 𝑚 ) ≃ Ϙ (colim 𝑛 𝑋 𝑛 ) , as desired. (cid:3) Our strategy for the proof of Proposition 4.2.12 is to break the inclusion
Proj( 𝑅 ) ↪ D p ( 𝑅 ) into thecomposite of two inclusions Proj( 𝑅 ) ↪ D p ( 𝑅 ) ≥ ↪ D p ( 𝑅 ) , where D p ( 𝑅 ) ≥ ⊆ D p ( 𝑅 ) is the image under the localisation functor Ch b (Proj( 𝑅 )) → D p ( 𝑅 ) of the fullsubcategory Ch b ≥ (Proj( 𝑅 )) ⊆ Ch b (Proj( 𝑅 )) spanned by the complexes concentrated in non-negative de-grees. To make this strategy work we would like to have a notion of a (contravariant) quadratic functor from D p ( 𝑅 ) ≥ to spectra.Recall from Definition 1.1.1 that a functor Ϙ ∶ D p ( 𝑅 ) ≥ → S 𝑝 op is said to be -excisive if it sends stronglycocartesian cubes in D p ( 𝑅 ) ≥ to exact cubes in S 𝑝 op . We will denote by Fun ( D p ( 𝑅 ) ≥ ) ⊆ Fun( D p ( 𝑅 ) ≥ , S 𝑝 op ) op the full subcategory spanned by the reduced -excisive functors D p ( 𝑅 ) ≥ → S 𝑝 op .4.2.24. Remark.
It follows from [Lur17, Proposition 6.1.3.22] that the cross effect of any reduced -excisivefunctor Ϙ ∶ D p ( 𝑅 ) → S 𝑝 op preserve finite colimits in each variable separately.4.2.25. Lemma.
Restriction along the inclusion D p ( 𝑅 ) ≥ ⊆ D p ( 𝑅 ) yields an equivalence (91) Fun q ( D p ( 𝑅 )) ≃ ←←←←←←←→ Fun ( D p ( 𝑅 ) ≥ ) . Proof.
Consider the composite(92)
Fun ∗ ( D p ( 𝑅 ) op ≥ , S 𝑝 ) → Fun ∗ ( D p ( 𝑅 ) op , S 𝑝 ) P ←←←←←←←←←←→ Fun q ( D p ( 𝑅 )) where the first functor is given by right Kan extension along the inclusion 𝜄 ∶ D p ( 𝑅 ) op ≥ ↪ D p ( 𝑅 ) op (aprocedure which preserves reduced functors by the pointwise formula for Kan extensions) and the second bythe right adjoint P to the inclusion Fun q ( D p ( 𝑅 )) ⊆ Fun ∗ ( D p ( 𝑅 ) op , S 𝑝 ) described in Construction (1.1.26).Since (92) is a composite of right adjoints it is itself right adjoint to the composite(93) Fun q ( D p ( 𝑅 )) ↪ Fun ∗ ( D p ( 𝑅 ) op , S 𝑝 ) → Fun ∗ ( D p ( 𝑅 ) op ≥ , S 𝑝 ) . Since (93) factors through the full subcategory of quadratic functors the unit and counit of the adjunctionbetween (92) and (93) also yield an adjunction(94)
Fun ( D p ( 𝑅 ) ≥ ) ⟂ Fun q ( D p ( 𝑅 )) where the right adjoint is obtained by restricting the domain of (92) and the left adjoint is the restrictionfunctor (91) under consideration. We claim that the adjunction (94) is an equivalence. To begin, we firstshow that for Ϙ ∈ Fun ( D p ( 𝑅 ) ≥ ) the counit 𝜄 ∗ P ( 𝜄 ∗ Ϙ ) → Ϙ is an equivalence. We note that this counitis given itself by a composite(95) 𝜄 ∗ P ( 𝜄 ∗ Ϙ ) → 𝜄 ∗ 𝜄 ∗ Ϙ → Ϙ where the second map is the counit of 𝜄 ∗ ⊣ 𝜄 ∗ , which is an equivalence since 𝜄 is fully-faithful. The firstmap in turn is the one obtained by applying 𝜄 ∗ to the counit P ( 𝜄 ∗ Ϙ ) → 𝜄 ∗ Ϙ . We hence need to show that thecomponent(96) P ( 𝜄 ∗ Ϙ )( 𝑋 ) → 𝜄 ∗ Ϙ ( 𝑋 ) is an equivalence for 𝑋 in the image of the inclusion D p ( 𝑅 ) ≥ ⊆ D p ( 𝑅 ) . For this we recall that P is definedvia a sequential limit P ( R ) ∶= lim[ ... → T T ( R ) → T ( R ) → R ] where T ( R )( 𝑋 ) = Σ cof[B R (Σ 𝑋, Σ 𝑋 ) → R (Σ 𝑋, Σ 𝑋 )] , see Construction 1.1.26. It will then suffice toshow that for 𝑋 ∈ D p ( 𝑅 ) ≥ the sequence ΣB Ϙ (Σ 𝑋, Σ 𝑋 ) → Σ Ϙ (Σ 𝑋 ) → Ϙ ( 𝑋 ) is exact. Indeed, this follows by the exact same argument as the dual version of Lemma 1.1.19 (see Re-mark 1.1.20) in the case where 𝑧 and 𝑤 are zero objects, using that Ϙ is assumed to be reduced and -excisiveon D p ( 𝑅 ) ≥ .As the counit is an equivalence, to finish the proof it will suffice to show that 𝜄 ∗ is conservative, or equiva-lently (since its domain is stable), detects zero objects. In particular, we need to show that if Ϙ ∶ D p ( 𝑅 ) op → ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 69 S 𝑝 is a quadratic functor which vanishes on D p ( 𝑅 ) ≥ then Ϙ is the zero functor. Indeed, for such a Ϙ we willhave that ( T ) ◦ 𝑛 ( Ϙ ) vanishes on ( D p ( 𝑅 )) ≥ − 𝑛 ∶= Σ − 𝑛 D p ( 𝑅 ) ≥ and hence that Ϙ ≃ P ( Ϙ ) = lim[ ... → T T ( Ϙ ) → T ( Ϙ ) → Ϙ ] = 0 , since any 𝑋 ∈ D p ( 𝑅 ) lies in D p ( 𝑅 ) ≥ − 𝑛 for some 𝑛 . (cid:3) Lemma.
Restriction along the inclusion
Proj( 𝑅 ) ⊆ D p ( 𝑅 ) ≥ yields an equivalence Fun ( D p ( 𝑅 ) ≥ ) ≃ ←←←←←←←→ Fun (Proj( 𝑅 ) op , S 𝑝 ) Proof.
Let D ( 𝑅 ) ≥ ⊆ D ( 𝑅 ) denote the full subcategory spanned by the objects represented by non-negativelygraded complexes. By [Lur17, Proposition 1.3.3.14] restriction along the inclusion Proj( 𝑅 ) → D ( 𝑅 ) ≥ yields an equivalence Fun sif ( D ( 𝑅 ) op ≥ , S 𝑝 ) ≃ ←←←←←←←→ Fun(Proj( 𝑅 ) op , S 𝑝 ) , where the left hand side denotes the full subcategory of Fun( D ( 𝑅 ) op ≥ , S 𝑝 ) spanned by the functors whichpreserve sifted limits. It then follows from Lemma 4.2.22 that restriction along the inclusion Proj( 𝑅 ) → D p ( 𝑅 ) ≥ induces an equivalence(97) Fun Δ fin ( D p ( 𝑅 ) op ≥ , S 𝑝 ) ≃ ←←←←←←←→ Fun(Proj( 𝑅 ) op , S 𝑝 ) where on the left hand side we have the ∞ -category of functors D p ( 𝑅 ) op ≥ → S 𝑝 which preserve finitegeometric realizations. By Lemma 4.2.23 the latter contains Fun ( D p ( 𝑅 ) ≥ ) as a full subcategory. Wenow claim that under the equivalence (97) the reduced -excisive functors on the left hand side correspondto the functors that are reduced and polynomial of degree on the right hand side. First, since the inclusion Proj( 𝑅 ) → D p ( 𝑅 ) ≥ is additive and reduced -excisive functors have bi-exact cross effects (Remark 4.2.24)it follows that reduced -excisive functors restrict to reduced functors which are polynomial of degree .Conversely, suppose that R ∶ D p ( 𝑅 ) ≥ → S 𝑝 is a functor which preserves finite geometric realizationswhose restriction to Proj( 𝑅 ) is reduced and polynomial of degree . Since Proj( 𝑅 ) contains the zero objectof D p ( 𝑅 ) ≥ we have that R is reduced. To show that R is -excisive we need to show that it sends stronglycocartesian cubes in D p ( 𝑅 ) ≥ to exact cubes of spectra. For this, let 𝜌 ∶ (Δ ) → D p ( 𝑅 ) ≥ be a stronglycocartesian cube. We want to reduce to the case where 𝜌 takes values in the subcategory of finite projectivemodules and injections. Let us identify (Δ ) with the nerve of the poset P ([2]) of subsets of [2] = {0 , , .Let P ≤ ([2]) ⊆ P ([2]) be the full subposet spanned by {} , {0} , {1} , {2} . Since D p ( 𝑅 ) ≥ is the localisationof the cofibration category Ch b ≥ (Proj( 𝑅 )) it follows from [Cis19] that we can represent 𝜌 | P ≤ ([2]) by adiagram 𝜏 ∶ P ≤ ([2]) → Ch b ≥ (Proj( 𝑅 )) in which each map 𝑐 𝑖 ∶ 𝜏 ({}) → 𝜏 ({ 𝑖 }) is levelwise injective withprojective kernel. The Dold-Kan correspondence then associates to 𝜏 a diagram 𝜏 ′ ∶ P ≤ ([2]) → Proj( 𝑅 ) Δ op of simplicial 𝑅 -modules such that each of the maps 𝑐 𝑖 ∶ 𝜏 ({}) → 𝜏 ({ 𝑖 }) is levelwise injective with projectivecokernel (see [Qui06, §II.4.12]). In addition, since 𝜏 takes values in bounded complexes we can find an 𝑛 ≥ such that it takes values in complexes concentrated in degrees to 𝑛 . Under the Dold-Kan correspondencesuch complexes map to simplicial 𝑅 -modules which are left Kan extended from their restriction to Δ op ≤ 𝑛 .Switching the simplicial dimension with the P ≤ ([2]) -dimension we may conclude that 𝜏 | P ≤ ([2]) can bewritten as a finite geometric realization of a simplicial family of diagrams 𝜏 ′ 𝑛 ∶ P ≤ ([2]) → Proj( 𝑅 ) suchthat each 𝜏 ′ 𝑛 has the property that 𝜏 ′ 𝑛 ({}) → 𝜏 ′ 𝑛 ({ 𝑖 }) is a split injective map of projective modules. Left Kanextending from P ≤ ([2]) we obtain a representation of 𝜌 as a finite geometric realization of a simplicialfamily of strongly cocartesian cubes 𝜌 𝑛 ∶ P ([2]) → Proj( 𝑅 ) such that each 𝜌 𝑛 ({}) → 𝜌 𝑛 ({ 𝑖 }) is a splitinjective map of projective modules. Since R preserves finite geometric realizations it will suffice to showthat R sends each 𝜌 𝑛 to an exact cube of spectra. Now since Proj( 𝑅 ) is closed in D p ( 𝑅 ) ≥ under pushoutswith one leg split injective it follows that each 𝜌 𝑛 is a strongly cocartesian cubes in Proj( 𝑅 ) . More explicitly,we may pick projective modules 𝑋 𝑛 , 𝑌 𝑛 , 𝑌 𝑛 , 𝑌 𝑛 such that 𝜏 ′ 𝑛 ({}) = 𝑋 𝑛 and 𝜏 ′ 𝑛 ({ 𝑖 }) = 𝑋 𝑛 ⊕𝑌 𝑖𝑛 for 𝑖 = 0 , , ,in which case 𝜌 𝑛 ∶ P ([2]) → D p ( 𝑅 ) is given by [2] ⊇ 𝑆 ↦ 𝑋 𝑛 ⊕ [ ⊕ 𝑖 ∈ 𝑆 𝑌 𝑖𝑛 ] . To finish the proof we need to show that the resulting cube of spectra(98) 𝑆 ↦ R ( 𝑋 𝑛 ⊕ [ ⊕ 𝑖 ∈ 𝑆 𝑌 𝑖𝑛 ] ) is exact. Let B(− , −) be the cross-effect of R . The assumption that R | Proj( 𝑅 ) is polynomial of degree means that the restriction of B to Proj( 𝑅 ) × Proj( 𝑅 ) preserve direct sums in each variable separately. Thecube of spectra R ◦ 𝜌 𝑛 thus decomposes as a direct sum of four components 𝑆 ↦ R ( 𝑋 𝑛 ) ⊕ [ ⊕ 𝑖 ∈ 𝑆 R ( 𝑌 𝑖𝑛 ) ] ⊕ [ ⊕ 𝑖 ∈ 𝑆 B( 𝑋 𝑛 , 𝑌 𝑖𝑛 ) ] ⊕ [ ⊕ 𝑖<𝑗 ∈ 𝑆 B( 𝑌 𝑖𝑛 , 𝑌 𝑗𝑛 ) ] . The first component is constant and is hence an exact cube, and the second and third components are visiblyexact. Finally, the fourth component 𝑆 ↦ ⊕ 𝑖<𝑗 ∈ 𝑆 B( 𝑌 𝑖𝑛 , 𝑌 𝑗𝑛 ) is also an exact cube since it vanishes on 𝑆 ofsize smaller than and exhibits its value at 𝑆 = {0 , , as the product of its values on {0 , , {1 , and {0 , . We have thus showed that the cube of spectra (98) is exact, and so the proof is complete. (cid:3) Proof of Proposition 4.2.12.
Combine Lemma 4.2.25 and Lemma 4.2.26. (cid:3)
Group-rings and visible structures.
In this section we will consider group rings, and more generally,group ring spectra, which provide important examples of rings with anti-involutions, and whose varioustypes of L -groups play a principal role in surgery theory and manifold classification. In particular, we willshow how to construct modules with genuine involutions whose associated L -theory reproduces visible L -theory as studied in [Wei92], [Ran92] and [WW14].Recall that for an ordinary group 𝐺 equipped with a homomorphism 𝜒 ∶ 𝐺 → {−1 , , one may endowthe group ring ℤ 𝐺 with the associated 𝜒 -twisted involution 𝜏 𝜒 ∶ ℤ 𝐺 ≅ ←←←←←←←→ ( ℤ 𝐺 ) op , defined by sending 𝑔 ∈ ℤ 𝐺 to 𝜒 ( 𝑔 ) 𝑔 −1 . In what follows we will consider a generalization of this setup where 𝐺 is replaced by an E -group , that is, a group-like E -monoid in spaces, and instead of ℤ the coefficients are taken in the spherespectrum. More precisely, for such a 𝐺 the suspension spectrum 𝕊 [ 𝐺 ] ∶= Σ ∞+ 𝐺 inherits the structure ofan E -ring spectrum, which is called the group ring spectrum of 𝐺 . This E -ring spectrum is characterizedby the fact that its module spectra correspond to spectra with a 𝐺 -action, a notion which we will call here(naive) 𝐺 -spectra , and consequently also use the notation Mod 𝐺 ∶= Mod 𝕊 [ 𝐺 ] and Mod 𝜔𝐺 ∶= Mod 𝜔 𝕊 [ 𝐺 ] . Inthis section we consider data giving rise to Poincaré structures on Mod 𝜔 𝕊 [ 𝐺 ] . We also consider variants forthe group algebras 𝐴 [ 𝐺 ] ∶= 𝐴 ⊗ 𝕊 [ 𝐺 ] with coefficients in some E -ring spectrum 𝐴 . A closely relatedconstruction involving parameterized spectra over a space will be explored in §4.4.Recall that a spectrum 𝐸 ∈ S 𝑝 is called invertible if there exists a spectrum 𝐸 ′ such that 𝐸 ⊗ 𝐸 ′ ≃ 𝕊 .In this case 𝐸 is necessarily of the form Σ 𝑛 𝕊 for some 𝑛 ∈ ℤ and hom( 𝐸, 𝐸 ) ≃ 𝕊 . By a character of an E -group 𝐺 we will mean a pair ( 𝐸, 𝜒 ) where 𝐸 is an invertible spectrum and 𝜒 ∶ 𝐺 → Aut( 𝐸 ) ≃ gl ( 𝕊 ) is a homomorphism of E -groups, encoding an action of 𝐺 on 𝐸 .4.3.1. Construction.
Let 𝐺 be an E -group equipped with a character ( 𝐸, 𝜒 ) . We define a module withgenuine involution on the group ring 𝕊 [ 𝐺 ] as follows. First, since the diagonal map 𝐺 → 𝐺 × 𝐺 is naturallyequivariant with respect to the flip C -action on 𝐺 × 𝐺 and the trivial action on 𝐺 , it follows that the inductionfunctor Mod 𝐺 → Mod 𝐺 × 𝐺 refines to a functor Mod 𝐺 → [Mod 𝐺 × 𝐺 ] hC , where C acts on Mod 𝐺 × 𝐺 via its flip action on 𝐺 × 𝐺 . In particular, the 𝐺 × 𝐺 -spectrum 𝐸 [ 𝐺 − ] ∶= 𝐸 ⊗ 𝐺 𝕊 [ 𝐺 × 𝐺 ] = ( 𝐸 ⊗ 𝕊 [ 𝐺 × 𝐺 ]) h 𝐺 induced from the 𝐺 -spectrum 𝐸 is naturally a C -equivariant object of Mod 𝐺 × 𝐺 . Identifying 𝕊 [ 𝐺 ] ⊗ 𝕊 [ 𝐺 ] with 𝕊 [ 𝐺 × 𝐺 ] we then see that the Tate diagonal 𝕊 [ 𝐺 ] → 𝕊 [ 𝐺 × 𝐺 ] tC is given by the composite 𝕊 [ 𝐺 ] → 𝕊 [ 𝐺 × 𝐺 ] hC → 𝕊 [ 𝐺 × 𝐺 ] tC in which the first map is induced by the C -equivariant diagonal 𝐺 → 𝐺 × 𝐺 .In particular, the 𝐺 -action on 𝐸 [ 𝐺 − ] tC is induced by the 𝐺 -action on 𝐸 [ 𝐺 − ] restricted from 𝐺 × 𝐺 alongthe diagonal, and we may then consider the map of 𝐺 -spectra 𝜂 ∶ 𝐸 → 𝐸 [ 𝐺 − ] hC → 𝐸 [ 𝐺 − ] tC in which the first map is induced by the C -equivariant map 𝐸 = 𝐸 ⊗ 𝐺 𝕊 [ 𝐺 ] → 𝐸 ⊗ 𝐺 𝕊 [ 𝐺 × 𝐺 ] . Wethen call the resulting module with genuine involution ( 𝐸 [ 𝐺 − ] , 𝐸, 𝜂 ) the 𝜒 -twisted visible module. We ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 71 will denote the corresponding hermitian structure on Mod 𝜔𝐸 [ 𝐺 ] by Ϙ v 𝜒 and refer to it as the 𝜒 -twisted visiblestructure on Mod 𝜔 𝕊 [ 𝐺 ] .4.3.2. Lemma.
Let 𝑛 be such that 𝐸 ≃ Σ 𝑛 𝕊 as spectra. Then the ( 𝕊 [ 𝐺 ] ⊗ 𝕊 [ 𝐺 ]) -module 𝐸 [ 𝐺 − ] is in-vertible and equivalent to Σ 𝑛 𝕊 [ 𝐺 ] as an 𝕊 [ 𝐺 ] -module. Furthermore, under this equivalence the involutionon 𝐸 [ 𝐺 − ] corresponds to the 𝑛 -fold suspension of a ring anti-involution 𝜏 𝜒 ∶ 𝕊 [ 𝐺 ] → 𝕊 [ 𝐺 ] op , so that (Σ − 𝑛 𝐸 [ 𝐺 − ] , Σ − 𝑛 𝐸, Σ − 𝑛 𝜂 ) promotes 𝕊 [ 𝐺 ] to a genuine ring with involution. In particular, the visible struc-ture Ϙ v 𝜉 on Mod 𝜔 𝕊 [ 𝐺 ] is Poincaré.Proof. By Proposition 3.1.11 it will suffice to exhibit a C -equivariant map 𝑢 ∶ 𝕊 → Σ − 𝑛 𝐸 [ 𝐺 − ] whichfreely generates Σ − 𝑛 𝐸 [ 𝐺 − ] as an 𝕊 [ 𝐺 ] -module. Indeed, this is given by the C -equivariant map 𝕊 ≃ Σ − 𝑛 𝐸 ⊗ 𝐺 𝕊 [ 𝐺 ] → Σ − 𝑛 𝐸 ⊗ 𝐺 𝕊 [ 𝐺 × 𝐺 ] = Σ − 𝑛 𝐸 [ 𝐺 − ] , induced by the diagonal map 𝐺 → 𝐺 × 𝐺 . This map exhibits Σ − 𝑛 𝐸 [ 𝐺 − ] as a free 𝕊 [ 𝐺 ] -module on a singlegenerator since the shear map 𝐺 × 𝐺 → 𝐺 × 𝐺 (given informally by ( 𝑔, ℎ ) ↦ ( 𝑔, 𝑔ℎ ) ) is an equivalence bythe group-like property. (cid:3) Remark.
Unwinding the definitions, the involution 𝜏 𝜒 ∶ 𝕊 [ 𝐺 ] → 𝕊 [ 𝐺 ] op is the ring map induced bythe composed map 𝐺 → gl ( 𝕊 ) × 𝐺 op → gl ( 𝕊 [ 𝐺 ] op ) , in which the first map is given by 𝑔 ↦ ( 𝜒 ( 𝑔 ) , 𝑔 −1 ) . In particular, on 𝜋 𝕊 [ 𝐺 ] ≅ ℤ [ 𝜋 𝐺 ] this involution canbe written as ∑ 𝑖 𝑎 𝑖 𝑔 𝑖 ↦ ∑ 𝑖 𝑎 𝑖 𝜒 ( 𝑔 𝑖 ) 𝑔 −1 𝑖 , where 𝜒 ∶ 𝜋 𝐺 → 𝜋 gl ( 𝕊 ) = {1 , −1} is the induced homomorphism on 𝜋 .The following variant is also of interest:4.3.4. Variant.
Let 𝐺 be an E -group equipped with a character ( 𝐸, 𝜒 ) and let ( 𝐴, 𝑁, 𝛼 ) be an E -ring spec-trum with genuine involution. We may then obtain a module with genuine involution over 𝐴 [ 𝐺 ] ∶= 𝐴 ∧ 𝕊 [ 𝐺 ] by tensoring the visible structure ( 𝐸 [ 𝐺 − ] , 𝐸, 𝜂 ) of Construction 4.4.1 with ( 𝐴, 𝑁, 𝛼 ) using the monoidalstructure on genuine C -spectra (see Remark 3.2.3). Explicitly, this yields the module with genuine invo-lution ( 𝐴 𝜒 [ 𝐺 − ] , 𝑁 𝜒 , 𝛼 𝜒 ) ∶= ( 𝐴 ⊗ 𝐸 [ 𝐺 − ] , 𝑁 ⊗ 𝐸, 𝛼 𝜒 ) where 𝛼 𝜒 ∶ 𝑁 ⊗ 𝐸 → 𝑀 tC ⊗ 𝐸 [ 𝐺 − ] tC → ( 𝑀 ⊗ 𝐸 [ 𝐺 − ]) tC is the composite of 𝛼 ⊗ 𝜂 and the lax monoidal structure map of the Tate construction. We will denote thecorresponding hermitian structure on Mod 𝜔𝐴 [ 𝐺 ] by Ϙ v 𝜒 as well, and refer to it as the 𝜒 -twisted visible structure on Mod 𝜔𝐴 [ 𝐺 ] . It then follows from Lemma 4.3.2 that this hermitian structure is Poincaré and identifies up toa shift with the Poincaré structure associated to a genuine anti-involution on 𝐴 [ 𝐺 ] induced by the respectivegenuine anti-involutions of 𝕊 [ 𝐺 ] and 𝐴 .4.3.5. Example.
When 𝐺 is discrete a common choice of a twisting is via a sign homomorphism 𝜒 ∶ 𝐺 → {−1 , . This can be made into a 𝐺 -character ( 𝐸, 𝜒 ) once we fix what is meant by the sign action on 𝕊 . Here(at least) two equally natural options are possible: one can either take 𝐸 = 𝕊 𝜎 −1 to be the desuspension ofthe sign representation sphere, or take 𝐸 = 𝕊 𝜎 to be its inverse. These are generally not equivalent asspectra with C -action, see Warning 3.4.8.4.3.6. Example.
Let 𝐺 be an E -group equipped with a character ( 𝐸, 𝜒 ) . If ( 𝐴, , → 𝐴 tC ) is an E -ringspectrum with genuine involution associated to the quadratic genuine refinement of an anti-involution on 𝐴 , then the associated 𝜒 -twisted visible Poincaré structure on Mod 𝜔𝐴 coincides with the quadratic Poincaréstructure associated to the 𝜒 -twisted duality on Mod 𝜔𝐴 [ 𝐺 ] .4.3.7. Example.
Let 𝐺 be an E -group equipped with a character ( 𝐸, 𝜒 ) . If ( 𝐴, 𝐴 tC , id ∶ 𝐴 tC → 𝐴 tC ) is an E -ring spectrum with genuine involution associated to the symmetric genuine refinement of an anti-involution on 𝐴 , then the associated 𝜒 -twisted visible Poincaré structure on Mod 𝜔𝐴 [ 𝐺 ] is also called the visiblesymmetric Poincaré structure. When 𝐺 and 𝐴 are discrete this recovers the visible symmetric structuresof [Wei92]. It generally does not coincide with the associated symmetric Poincaré structure on Mod 𝜔𝐴 [ 𝐺 ] , except when the classifying space B 𝐺 is a finite space. Indeed, supposing for simplicity that 𝐸 = 𝕊 and 𝜒 is trivial, we can write the structure map as a composite(99) 𝐴 tC → (( 𝐴 [ 𝐺 ] ⊗ 𝐴 𝐴 [ 𝐺 ]) tC ) h 𝐺 → (( 𝐴 [ 𝐺 ] ⊗ 𝐴 𝐴 [ 𝐺 ]) h 𝐺 ) tC = 𝐴 [ 𝐺 − ] tC . Since B 𝐺 is finite taking 𝐺 -orbits is in particular finite colimit. This means, on the one hand, that taking 𝐺 -orbits commutes with the Tate construction (−) tC ∶ S 𝑝 𝐺 → S 𝑝 and hence the second map in (99) isan equivalence. For the same reason, it also means that the exact functor 𝑇 ∶ Mod 𝐴 → Mod 𝐴 given by 𝑇 ( 𝑋 ) ↦ ( 𝑋 ⊗ 𝐴 𝑋 ) tC commutes with 𝐺 -orbits (where 𝑋 means 𝑋 , but considered as a right 𝐴 -modulevia the involution on 𝐴 ), and hence the map (( 𝐴 [ 𝐺 ] ⊗ 𝐴 𝐴 [ 𝐺 ]) tC ) h 𝐺 → ( 𝐴 [ 𝐺 ] h 𝐺 ⊗ 𝐴 𝐴 [ 𝐺 ] h 𝐺 ) tC = 𝐴 tC isan equivalence. This map is inverse to the first map in (99), which is consequently an equivalence as well.4.3.8. Example.
When 𝐺 = Ω 𝑋 is the loop space of a pointed space ( 𝑋, 𝑥 ) then the data of a character ( 𝐸, 𝜒 )) for 𝐺 is equivalent to that of a local system 𝜉 → 𝑋 of invertible spectra on 𝑋 , whose fibre at 𝑥 isidentified with 𝐸 . The associated visible Poincaré structure on Mod 𝜔 𝕊 [Ω 𝑋 ] then recovers the 𝜉 -twisted visiblestructure studied in [WW14] (see [WW14, Corollary 8.2] to compare the two definitions). This structure ismore naturally studied from the perspective of parameterized spectra , a point of view which we will takeup in §4.4.4.3.9. Remark.
Let 𝐺 be an E -group equipped with a character ( 𝐸, 𝜒 ) . Consider the Poincaré struc-ture Ϙ u on S 𝑝 𝜔 associated to the module with genuine involution ( 𝕊 , 𝕊 , 𝕊 → 𝕊 tC ) . Since any stable ∞ -category is tensored over S 𝑝 the character 𝜒 induces an action of 𝐺 on 𝐸 ⊗ Ϙ u ∈ Fun q ( S 𝑝 𝜔 ) , and hence on ( S 𝑝 𝜔 , 𝐸 ⊗ Ϙ u ) ∈ Cat p∞ . It can be shown that the associated Poincaré ∞ -category (Mod 𝜔 𝕊 [ 𝐺 ] , Ϙ v 𝜒 ) can thenbe universally characterized as the quotient of ( S 𝑝 𝜔 , 𝐸 ⊗ Ϙ u ) by 𝐺 in Cat p∞ . Similarly, if ( 𝐴, 𝑁, 𝛼 ) is an E -ring spectrum with genuine anti-involution then (Mod 𝜔𝐴 [ 𝐺 ] , Ϙ v 𝜒 ) represents the quotient of the associated 𝐺 -action on (Mod 𝜔𝐴 , 𝐸 ⊗ Ϙ 𝛼𝐴 ) . Examples 4.3.6 and 4.3.7 above then reflect the fact that the quotient of aquadratic Poincaré ∞ -category is again quadratic, while the quotient of a symmetric Poincaré ∞ -categoryis not again symmetric. Indeed, this is consistent with the expected behavior in light of Proposition 7.2.17.4.3.10. Variant.
Construction 4.3.1 can be generalized by introducing the following additional pieces ofdata: a C -action 𝜏 ∶ C → Aut( 𝐺 ) on 𝐺 , an additional E -group 𝐻 , and a C -equivariant map 𝐻 → 𝐺 ,where C acts trivially on 𝐻 . We may consider such a structure as a genuine C -action on 𝐺 . The data ofa character ( 𝐸, 𝜒 ) for 𝐺 then needs to be promoted to that of C -equivariant character 𝜒 ∶ 𝐺 → Aut( 𝐸 ) (where C -acts trivially on the target), and the induced C -action on the restricted character 𝜒 | 𝐻 shouldbe equipped with a trivialization. To all this data one can associate a module with genuine involution ( 𝐸 [ 𝐺 − 𝜏 ] , 𝐸 [ 𝐺 ∕ 𝐻 ] , 𝜂 ) where 𝐸 [ 𝐺 − 𝜏 ] ∶= 𝐸 [ 𝐺 × 𝐺 ] h 𝐺 is the 𝐺 × 𝐺 -spectrum induced from 𝐸 , this timealong the 𝜏 -diagonal ( 𝜏, id) ∶ 𝐺 → 𝐺 × 𝐺 , and 𝐸 [ 𝐺 ∕ 𝐻 ] is the 𝐺 -module induced from the 𝐻 -character ( 𝐸, 𝜒 | 𝐻 ) along the map 𝐻 → 𝐺 . Since the map ( 𝜏, id) is C -equivariant with respect to flip action on 𝐺 × 𝐺 ,and 𝜒 is C -equivariant, the 𝐺 × 𝐺 -module 𝐸 [ 𝐺 − 𝜏 ] inherits an involution compatible with the flip involutionof 𝐺 × 𝐺 , and hence the structure of a module with involution over 𝕊 [ 𝐺 ] . At the same time, since the usualdiagonal 𝐺 → 𝐺 × 𝐺 is C -equivariant with respect to the trivial action on the domain, and coincides withthe 𝜏 -diagonal when restricted to 𝐻 , it induces a C -equivariant map 𝐸 ⊗ 𝐻 𝕊 [ 𝐺 ] → 𝐸 ⊗ 𝐺 𝕊 [ 𝐺 × 𝐺 ] withrespect to the trivial action on the domain. The composed map 𝜂 ∶ 𝐸 [ 𝐺 ∕ 𝐻 ] = 𝐸 ⊗ 𝐻 𝕊 [ 𝐺 ] → ( 𝐸 ⊗ 𝐺 𝕊 [ 𝐺 × 𝐺 ]) hC → ( 𝐸 ⊗ 𝐺 𝕊 [ 𝐺 × 𝐺 ]) tC = 𝐸 [ 𝐺 − 𝜏 ] tC then constitutes a structure map exhibiting ( 𝐸 [ 𝐺 𝜏 ] , 𝐸 [ 𝐺 ∕ 𝐻 ] , 𝜂 ) as a module with genuine involution over 𝕊 [ 𝐺 ] , yielding a hermitian structure Ϙ v 𝜒,𝜏 on Mod 𝜔𝐺 . The case of Construction 4.3.1 can be recovered ascorresponding to the trivial involution on 𝐺 with 𝐻 = 𝐺 and 𝐻 → 𝐺 the identity map. Arguing as in theproof of Lemma 4.3.2 we can check that this module with genuine involution is invertible and corresponds,up to a shift, to a genuine refinement of a suitable anti-involution on 𝕊 [ 𝐺 ] , whose underlying equivalence 𝕊 [ 𝐺 ] → 𝕊 [ 𝐺 ] op is induced by the map 𝐺 → gl ( 𝕊 ) × 𝐺 given by 𝑔 ↦ ( 𝜒 ( 𝑔 ) , 𝜏 ( 𝑔 ) −1 ) . As in Remark 4.3.9the resulting Poincaré ∞ -category (Mod 𝜔𝐺 , Ϙ v 𝜒,𝜏 ) can be characterized as a certain colimit in Cat p∞ . Thiscan be interpreted as reflecting the structure of Cat p∞ as a C -category, see §7.4, which admits quotients byactions of genuine C -groups. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 73 Parameterized spectra.
In this section we will discuss Poincaré structures on the ∞ -category ofcompact parameterized spectra over a space 𝑋 whose L -theory reproduces the visible L -theory of [WW14].We will then show how to construct visible signatures for Poincaré duality spaces in this setting.Let us begin by establishing some terminology. We write S 𝑝 𝑋 ∶= Fun( 𝑋, S 𝑝 ) for the ∞ -category offunctors 𝑋 → S 𝑝 , to which we will refer to as local systems of spectra or parameterized spectra over 𝑋 .We then denote by S 𝑝 𝜔𝑋 ⊆ S 𝑝 𝑋 the full subcategory spanned by the compact objects, which is our currentobject of interest. When 𝑋 is connected and pointed we may identify the latter with compact 𝕊 [Ω 𝑋 ] -modules, in which case we can consider various visible Poincaré structures as in §4.3, see Example 4.3.8.The point of view of parameterized spectra has however the advantage of working without a preferred basepoint, and not being restricted to connected spaces. The unpointed setting is more natural, for example,when the input is a Poincaré duality space, as arising in the surgery classification of manifolds.We will denote by Pic( 𝕊 ) ⊆ 𝜄 S 𝑝 the full subgroupoid spanned by the invertible spectra. By a sphericalfibration we will mean a local system 𝜉 ∶ 𝑋 → S 𝑝 which takes values in Pic( 𝕊 ) .4.4.1. Construction.
Let 𝑋 be a space and 𝜉 ∶ 𝑋 → Pic( 𝕊 ) a spherical fibration on 𝑋 . We associate to 𝜉 asymmetric bilinear functor on S 𝑝 𝜔𝑋 = Fun( 𝑋, S 𝑝 ) 𝜔 by setting(100) B 𝜉 ( 𝐿, 𝐿 ′ ) ∶= hom 𝑋 × 𝑋 ( 𝐿 ⊠ 𝐿, Δ ! ( 𝜉 )) , where the mapping spectra takes place in the ∞ -category Fun( 𝑋 × 𝑋, S 𝑝 ) , ⊠ is the exterior tensor product,and Δ ! is left Kan extension along the diagonal Δ ∶ 𝑋 → 𝑋 × 𝑋 . The associated symmetric and quadratichermitian structure on S 𝑝 𝜔𝑋 are then given by Ϙ s 𝜉 ( 𝐿 ) = hom 𝑋 × 𝑋 ( 𝐿 ⊠ 𝐿, Δ ! ( 𝜉 )) hC and Ϙ q 𝜉 ( 𝐿 ) = hom 𝑋 × 𝑋 ( 𝐿 ⊠ 𝐿, Δ ! ( 𝜉 )) hC , respectively. The above construction is functorial in 𝑋 in the following sense. Let us call a map of spaceswith spherical fibrations a pair ( 𝑓 , 𝜂 ) where 𝑓 ∶ 𝑋 → 𝑌 is a map between spaces equipped each sphericalfibrations 𝜉 𝑋 and 𝜉 𝑌 respectively, and 𝜂 ∶ 𝜉 𝑋 → 𝑓 ∗ 𝜉 𝑌 is a natural transformation. We may then associateto 𝑓 the corresponding left Kan extension functor 𝑓 ! ∶ S 𝑝 𝜔𝑋 → S 𝑝 𝜔𝑌 , which is well defined since left Kan extension preserves compact objects. The natural transformation 𝜂 theninduces a map B 𝜉 𝑋 ( 𝐿, 𝐿 ′ ) = hom 𝑋 × 𝑋 ( 𝐿 ⊠ 𝐿 ′ , Δ ! 𝜉 𝑋 ) 𝜂 ∗ ←←←←←←←←←→ hom 𝑋 × 𝑋 ( 𝐿 ⊠ 𝐿 ′ , Δ ! 𝑓 ∗ 𝜉 𝑌 ) → hom 𝑋 × 𝑋 ( 𝐿 ⊠ 𝐿 ′ , ( 𝑓 × 𝑓 ) ∗ Δ ! 𝜉 𝑌 ) ≃ hom 𝑌 × 𝑌 (( 𝑓 × 𝑓 ) ! ( 𝐿 ⊠ 𝐿 ′ ) , Δ ! 𝜉 𝑌 ) ≃ B 𝜉 𝑌 ( 𝑓 ! 𝐿, 𝑓 ! 𝐿 ′ ) which is natural in 𝐿 and 𝐿 ′ , so that we obtained an induced symmetric functor(101) ( S 𝑝 𝜔𝑋 , B 𝜉 𝑋 ) → ( S 𝑝 𝜔 , B 𝜉 𝑌 ) covering the left Kan extension functor 𝑓 ! .Our first goal is to show that the above construction gives a perfect bilinear functor and identify theassociated duality. For this, note that for a fixed local system 𝐿 ∈ S 𝑝 𝑋 the association 𝐿 ′ ↦ 𝐿 ⊠ 𝐿 ′ ∈ S 𝑝 𝑋 × 𝑋 is colimit preserving and hence admits a right adjoint hom ⊠ ( 𝐿, −) ∶ S 𝑝 𝑋 × 𝑋 → S 𝑝 𝑋 , which we can compute explicitly to be(102) hom ⊠ ( 𝐿, 𝑇 ) 𝑥 = hom 𝑋 × 𝑋 ( 𝑥 ! 𝕊 , hom ⊠ ( 𝐿, 𝑇 )) ≃ hom(
𝐿 ⊠ 𝑥 ! 𝕊 , 𝑇 ) ≃ hom(( 𝑗 𝑥 ) ! 𝐿, 𝑇 ) ≃ hom 𝑋 ( 𝐿, 𝑇 | 𝑋 ×{ 𝑥 } ) where 𝑥 ∶ ∗ → 𝑋 is the insertion of the point 𝑥 and 𝑗 𝑥 ∶ 𝑋 × { 𝑥 } → 𝑋 × 𝑋 is the corresponding insertionof the horizontal slice at height 𝑥 . Lemma.
For a spherical fibration 𝜉 , the bilinear functor (100) is non-degenerate with duality D 𝜉 𝐿 = hom ⊠ ( 𝐿, Δ ! 𝜉 ) . Furthermore, if ( 𝑓 , 𝜂 ) ∶ ( 𝑋, 𝜉 𝑋 ) → ( 𝑌 , 𝜉 𝑌 ) is a map of spaces with spherical fibrations such that 𝜂 ∶ 𝜉 𝑋 → 𝑓 ∗ 𝜉 𝑌 is an equivalence then the induced symmetric functor (101) is duality preserving.Proof. Indeed, by adjunction we have B 𝜉 ( 𝐿, 𝐿 ′ ) = hom 𝑋 × 𝑋 ( 𝐿 ⊠ 𝐿 ′ , Δ ! 𝜉 ) = hom 𝑋 ( 𝐿 ′ , hom ⊠ ( 𝐿, Δ ! 𝜉 )) , and so B 𝜉 𝑋 is non-degenerate. To see the second claim, suppose fix a map 𝑓 ∶ 𝑋 → 𝑌 and an equivalence 𝜂 ∶ 𝜉 𝑋 → 𝜉 𝑌 . Since S 𝑝 𝜔𝑋 is compactly generated by the collection of objects 𝑥 ! 𝕊 , for 𝑥 ∶ ∗ → 𝑋 a point, itsuffices to check that the induced map 𝑓 ! D 𝐿 → D 𝑓 ! 𝐿 is an equivalence for 𝐿 = 𝑥 ! 𝕊 . For this, it will sufficeto prove the claim for the maps 𝑥 ∶ ∗ → 𝑋 and 𝑓 ( 𝑥 ) ∶ ∗ → 𝑌 . In other words, we may as well assume that 𝑋 =∗ and 𝑓 is the inclusion of a point 𝑦 ∈ 𝑌 . Let 𝜉 𝑦 ∈ S 𝑝 be the value of 𝜉 𝑌 at 𝑦 , so that the goal becomesshowing that the canonical symmetric functor ( S 𝑝 𝜔 , B 𝜉 𝑦 ) → ( S 𝑝 𝜔𝑌 , B 𝜉 𝑌 ) is duality preserving. We now observe that if 𝐸 ∈ S 𝑝 𝜔 is a compact spectrum then we can factor the map 𝑦 ! D 𝜉 𝑦 ( 𝐸 ) → D 𝜉 𝑌 ( 𝑦 ! 𝐸 ) as the composite 𝑥 ! (D 𝜉 𝑌 ( 𝐸 )) = 𝑥 ! (D 𝕊 ( 𝐸 ) ⊗𝜉 𝑦 ) ≃ D 𝕊 ( 𝐸 ) ⊗𝑦 ! 𝜉 𝑦 → D 𝕊 ( 𝐸 ) ⊗ (Δ ! 𝜉 𝑌 ) | { 𝑦 }× 𝑌 ≃ hom ⊠ ( 𝑦 ! 𝐸, Δ ! 𝜉 𝑌 ) = D 𝜉 𝑌 ( 𝑦 ! 𝐸 ) where the fourth equivalence is by the formula in (102) and the third arrow is the Beck-Chevalley transfor-mation for the square { 𝑦 } { 𝑦 } × 𝑌𝑌 𝑌 × 𝑌 Δ where the top horizontal map picks the point ( 𝑦, 𝑦 ) ∈ { 𝑦 } × 𝑌 . This transformation is an equivalence sincethe square is cartesian. (cid:3) Corollary.
For a spherical fibration 𝜉 , the 𝜉 -twisted duality D 𝜉 𝐿 = hom ⊠ ( 𝐿, Δ ! 𝜉 ) is perfect. In particular, the associated symmetric and quadratic hermitian structures Ϙ s 𝜉 and Ϙ q 𝜉 are Poincaré.Proof. We need to show that the evaluation map 𝐿 → D op D 𝐿 is an equivalence for any 𝐿 ∈ S 𝑝 𝜔𝑋 . Since S 𝑝 𝜔𝑋 is compactly generated by the collection of objects 𝑥 ! 𝕊 , for 𝑥 ∶ ∗ → 𝑋 a point, it suffices to check thisfor 𝐿 = 𝑥 ! 𝕊 . Invoking Lemma 4.4.2 it will suffice to show that the duality D 𝜉 𝑥 on S 𝑝 𝜔 { 𝑥 } = S 𝑝 𝜔 is non-degenerate, that is, we may assume that 𝑋 is a point. But then for any invertible spectrum 𝐸 ∈ Pic( 𝕊 ) wehave that D 𝐸 (−) ≃ 𝐸 ⊗ D 𝕊 and is hence a perfect duality, the case if D 𝕊 being the usual Spanier-Whiteheadduality. (cid:3) Out next goal is to construct a Poincaré structure on S 𝑝 𝜔𝑋 , which in some sense interpolates between thequadratic and symmetric structures. This structure will be called the visible Poincaré structure. To constructit, we first need to identify the linear part of the of the symmetric Poincaré structure Ϙ s 𝜉 . By definition, it isgiven by the formula(103) 𝐿 ↦ hom 𝑋 × 𝑋 ( 𝐿 ⊠ 𝐿, Δ ! ( 𝜉 )) tC , which consequently constitutes an exact (contravariant) functor on S 𝑝 𝜔𝑋 , such functors are always repre-sented by an object in Ind( S 𝑝 𝜔𝑋 ) ≃ S 𝑝 𝑋 , and so there exists a (possibly non-compact paramterized spectrum 𝑁 ∶ 𝑋 → S 𝑝 such that hom 𝑋 × 𝑋 ( 𝐿 ⊠ 𝐿, Δ ! ( 𝜉 )) tC ≃ Map 𝑋 ( 𝐿, 𝑁 ) for 𝐿 ∈ S 𝑝 𝜔𝑋 . To identify it, it suffices to check the values of (103) at the generators 𝑥 ! 𝕊 of S 𝑝 𝜔𝑋 . Inparticular, 𝑁 is canonically identified with the parameterized spectrum 𝑥 ↦ hom 𝑋 × 𝑋 ( 𝑥 ! 𝕊 ⊠ 𝑥 ! 𝕊 , Δ ! 𝜉 ) tC = (Δ ! ( 𝜉 ) ( 𝑥,𝑥 ) ) tC , ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 75 or simply, 𝑁 = (Δ ∗ Δ ! 𝜉 ) tC . We thus obtain a natural transformation(104) hom 𝑋 × 𝑋 ( 𝐿 ⊠ 𝐿, Δ ! 𝜉 ) tC → hom 𝑋 ( 𝐿, (Δ ∗ Δ ! 𝜉 ) tC ) which is an equivalence for 𝐿 ∈ S 𝑝 𝜔𝑋 .4.4.4. Definition.
Let 𝑋 be space and 𝜉 ∶ 𝑋 → Pic( 𝕊 ) a spherical fibration on 𝑋 . We define the 𝜉 -twistedvisible Poincaré structure Ϙ v 𝜉 ∶ S 𝑝 𝜔𝑋 → S 𝑝 by the top pullback square Ϙ v 𝜉 ( 𝐿 ) hom 𝑋 ( 𝐿, 𝜉 )hom 𝑋 × 𝑋 ( 𝐿 ⊠ 𝐿, Δ ! 𝜉 ) hC hom 𝑋 ( 𝐿, (Δ ∗ Δ ! 𝜉 ) tC )hom 𝑋 × 𝑋 ( 𝐿 ⊠ 𝐿, Δ ! 𝜉 ) tC ≃ where the right vertical map is induced by the canonical map 𝜉 → (Δ ∗ Δ ! 𝜉 ) hC → (Δ ∗ Δ ! ( 𝜉 )) tC , which wecan also identify with the composite 𝜉 → 𝜉 tC → (Δ ∗ Δ ! ( 𝜉 )) tC (the first Tate construction being taken withrespect to the trivial action).4.4.5. Remark. If 𝑋 is connected with base point 𝑥 ∈ 𝑋 then S 𝑝 𝜔𝑋 is naturally equivalent to Mod 𝜔 Ω 𝑥 𝑋 .Under this equivalence, the visible Poincaré structure Ϙ v 𝜉 corresponds to the visible Poincaré structure ofExample 4.3.8.4.4.6. Remark.
Given a map ( 𝑓 , 𝜂 ) ∶ ( 𝑋, 𝜉 𝑋 ) → ( 𝑌 , 𝜉 𝑌 ) of spaces with spherical fibrations, Construc-tion 4.4.1 and Corollary 4.4.3 provide an induced hermitian functor ( S 𝑝 𝜔𝑋 , Ϙ s 𝜉 𝑋 ) → ( S 𝑝 𝜔𝑌 , Ϙ s 𝜉 𝑌 ) , which is Poincaré when 𝜂 is an equivalence. Using the fact that the maps 𝜉 → (Δ ∗ Δ ! 𝜉 ) hC → (Δ ∗ Δ ! ( 𝜉 )) tC are both natural in 𝜉 we may readily refine this to a hermitian functor ( S 𝑝 𝜔𝑋 , Ϙ v 𝜉 𝑋 ) → ( S 𝑝 𝜔𝑌 , Ϙ v 𝜉 𝑌 ) , which is again Poincaré under the same circumstances.4.4.7. Variant.
Let 𝑋 be a space with a spherical fibration 𝜉 ∶ 𝑋 → Pic( 𝕊 ) and let ( 𝐴, 𝑁, 𝛼 ) be a ring spec-trum with genuine involution. We may then form a Poincaré structure Ϙ v ,𝛼𝜉, on the ∞ -category Fun( 𝑋, Mod 𝐴 ) 𝜔 of compact local systems of 𝐴 -modules by the top pullback square Ϙ v ,𝛼𝜉 ( 𝐿 ) hom 𝑋 ( 𝐿, 𝜉 ⊗ 𝑁 )hom 𝑋 ( 𝐿, (Δ ∗ Δ ! 𝜉 ) tC ⊗ 𝐴 tC )hom 𝑋 × 𝑋 ( 𝐿 ⊠ 𝐿, Δ ! 𝜉 ⊗ 𝐴 ) hC hom 𝑋 ( 𝐿, Δ ∗ Δ ! 𝜉 ⊗ 𝐴 ) tC )hom 𝑋 × 𝑋 ( 𝐿 ⊠ 𝐿, Δ ! 𝜉 ⊗ 𝐴 ) tC ≃ Here we note that the C -equivariant structure of Δ ! 𝜉 ⊗ 𝐴 with respect to the flip action on 𝑋 × 𝑋 isinduced by the equivariant structure of Δ ! 𝑋 and the involution on 𝐴 . Arguing as in Lemma 4.4.2 andCorollary 4.4.3 one sees again that the hermitian ∞ -category (Fun( 𝑋, Mod 𝐴 ) 𝜔 , Ϙ v ,𝛼𝜉 ) is functorial in maps ( 𝑓 , 𝜂 ) ∶ ( 𝑋, 𝜉 𝑋 ) → ( 𝑌 , 𝜉 𝑌 ) of spaces with spherical fibrations and that Ϙ v ,𝛼𝜉 is again Poincaré with underlyingperfect duality 𝐿 ↦ hom ⊠𝐴 ( 𝐿, Δ ! ( 𝜉 ) ⊗ 𝐴 ) , where hom ⊠𝐴 ( 𝐿, −) denotes the right adjoint of the functor 𝐿 ⊠ (−) ∶ Fun( 𝑋, Mod 𝐴 ) → Fun( 𝑋 × 𝑋, Mod
𝐴⊗𝐴 ) . Remark.
In addition to its functoriality in maps of spaces with spherical fibrations, the constructionof Variant 4.4.7 is also functorial in maps in ( 𝐴, 𝑁, 𝛼 ) → ( 𝐵, 𝐾, 𝛽 ) of rings with genuine involution. In fact,the Poincaré ∞ -category (Fun( 𝑋, Mod 𝐴 ) 𝜔 , Ϙ v ,𝛼𝜉 ) depends functorially on the pair of Poincaré ∞ -categories ( S 𝑝 𝜔𝑋 , Ϙ v 𝜉 ) and (Mod 𝐴 , Ϙ 𝛼𝐴 ) : we may identify it with their tensor product ( S 𝑝 𝜔𝑋 , Ϙ v 𝜉 ) ⊗ (Mod 𝐴 , Ϙ 𝛼𝐴 ) ∈ Cat p∞ , aconstruction we will study in §5.For the remainder of this section we will show how to construct visible signatures for Poincaré dualityspaces in the present context. We take a purely homotopy theoretical approach to Poincaré duality spacesand their Spivak normal fibration following the strategy in [Kle01]. Let 𝑋 be a finite space (that is, aspace which can be realized by a simplicial set with only finitely many non-degenerate simplices). Theright Kan extension functor 𝑟 ∗ ∶ S 𝑝 𝑋 → S 𝑝 along 𝑟 ∶ 𝑋 → ∗ is given by taking the limit along 𝑋 , whichis in particular a finite limit and hence preserves colimits by the stability of S 𝑝 𝑋 . It is thus equivalent tothe functor 𝑟 ! (− ⊗ F 𝑋 ) for a unique F 𝑋 , which is then called the dualizing complex of 𝑋 . Here ⊗ standsfor the pointwise tensor product of 𝑋 -parameterized spectra. We say that 𝑋 is a Poincaré duality space ifthe parameterized spectrum F 𝑋 ∶ 𝑋 → S 𝑝 factors through the full subgroupoid Pic( 𝕊 ) ⊆ 𝜄 S 𝑝 of invertibleobjects. In this case we denote the resulting spherical fibration by 𝜈 ∶= F 𝑋 , and call it the Spivak normalfibration of 𝑋 . We observe that the invertibility of 𝜈 allows us to recast its universal property in a slightlydifferent form: indeed, for 𝐿 ∈ S 𝑝 𝑋 we may identify hom 𝑋 ( 𝜈, 𝐿 ) ≃ hom 𝑋 ( 𝕊 , 𝜈 −1 ⊗ 𝐿 ) ≃ 𝑟 ∗ ( 𝜈 −1 ⊗ 𝐿 ) ≃ 𝑟 ! ( 𝜈 ⊗ 𝜈 −1 ⊗ 𝐿 ) = 𝑟 ! 𝐿 and so we can equivalently characterize 𝜈 as the object corepresenting the functor 𝑟 ! ∶ S 𝑝 𝑋 → S 𝑝 . Moreprecisely, 𝜈 is equipped with a canonical map(105) 𝑐 ∶ 𝕊 → 𝑟 ! 𝜈 which exhibits it as representing the functor 𝑟 ! , in the sense that for every local system 𝐿 ∈ S 𝑝 𝑋 thecomposed map(106) hom 𝑋 ( 𝜈, 𝐿 ) → hom( 𝑟 ! 𝜈, 𝑟 ! 𝐿 ) 𝑐 ∗ ←←←←←←←←←→ 𝑟 ! 𝐿 is an equivalence. We will refer to (105) as the Thom-Pontryagin map. In terms of the universal property 𝑟 ∗ ( 𝐿 ) ≃ 𝑟 ! ( 𝐿 ⊗ 𝜈 ) it corresponds to the composite 𝕊 → 𝑟 ∗ ( 𝕊 𝑋 ) ≃ 𝑟 ! ( 𝜈 ) . In general, we will say that amap of the form 𝕊 → 𝑟 ! 𝜉 for 𝜉 a spherical fibration exhibits 𝜉 as the Spivak normal fibration of 𝑋 if thecomposite (106) is an equivalence for every 𝐿 ∈ S 𝑝 𝑋 .4.4.9. Example. If 𝑀 is a closed smooth manifold then the underlying space of 𝑀 is a Poincaré dualityspace. Furthermore, if 𝜄 ∶ 𝑀 ↪ ℝ 𝑁 is a smooth embedding with normal bundle 𝐸 = 𝑇 𝑀 ⟂ ⊆ 𝜄 ∗ 𝑇 ℝ 𝑁 and associated spherical fibration 𝜉 𝐸 , and 𝑀 ⊆ 𝑈 ⊆ ℝ 𝑁 is a chosen tubular neighborhood, then theThom-Pontryagin collapse map S 𝑁 → ℝ 𝑁 ∕( ℝ 𝑁 ⧵ 𝑈 ) ≃ 𝑀 𝐸 induces a map of spectra 𝕊 → Σ ∞− 𝑁 + 𝑀 𝐸 ≃ 𝑟 ! Σ − 𝑁 𝜉 𝐸 , that exhibits Σ − 𝑁 𝜉 𝐸 as the Spivak normal fibration of 𝑀 . Since 𝐸 ⊕ 𝑇 𝑀 = 𝜄 ∗ 𝑇 ℝ 𝑁 is a trivial 𝑁 -dimensional vector bundle we can identify Σ − 𝑁 𝜉 𝐸 ≃ 𝜉 −1 𝑇 𝑀 , where 𝜉 𝑇 𝑀 is the spherical fibration underlyingthe tangent bundle.4.4.10.
Remark.
Tensoring the Thom-Pontryagin map (105) with varying 𝐸 ∈ S 𝑝 we obtain a naturaltransformation 𝑐 ∗ ∶ id ⇒ 𝑟 ! ( 𝜈 ⊗ 𝑟 ∗ (−)) of functors from S 𝑝 to itself. This natural transformation then actsas a unit exhibiting 𝑟 ! ( 𝜈 ⊗ (−)) as right adjoint to 𝑟 ∗ . This is just another way of saying that 𝑐 encodesthe equivalence 𝑟 ! 𝜈 ⊗ (−) ≃ 𝑟 ∗ . This formulation has the advantage of easily extending to local systemswith arbitrary values. Indeed, suppose that E is a stable ∞ -category, so that E is canonically tensored over S 𝑝 𝜔 . We may then consider the restriction functor 𝑟 ∗ ∶ E → Fun( 𝑋, E ) together with the associated leftand right Kan extensions 𝑟 ! , 𝑟 ∗ ∶ Fun( 𝑋, E ) → E (these kan extensions exist because 𝑋 is finite and everystable ∞ -category admits finite limits and colimits). Tensoring the Thom-Pontryagin map with varyingobjects in E we obtain a natural transformation 𝑐 ∗ ∶ id ⇒ 𝑟 ! ( 𝜈 ⊗ 𝑟 ∗ (−)) of functors from E to itself. This ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 77 natural transformation then acts as a unit exhibiting 𝑟 ! ( 𝜈 ⊗ (−)) as right adjoint to 𝑟 ∗ . Indeed, for 𝐴 ∈ E and 𝐵 ∶ 𝑋 → E the composed map hom 𝑋 ( 𝑟 ∗ 𝐴, 𝐵 ) → hom E ( 𝑟 ! ( 𝜈 ⊗ 𝑟 ∗ 𝐴 ) , 𝑟 ! ( 𝜈 ⊗ 𝐵 )) ≃ hom E ( 𝑟 ! 𝜈 ⊗ 𝐴, 𝑟 ! ( 𝜈 ⊗ 𝐵 )) → hom E ( 𝐴, 𝑟 ! ( 𝜈 ⊗ 𝐵 )) is an equivalence, as can be seen by comparing it with the composed map hom 𝑋 ( 𝑟 ∗ 𝕊 , 𝐵 𝐴 ) → hom( 𝑟 ! 𝜈, 𝑟 ! ( 𝜈 ⊗ 𝐵 𝐴 )) → 𝑟 ! ( 𝜈 ⊗ 𝐵 𝐴 ) ≃ hom E ( 𝐴, 𝑟 ! ( 𝜈 ⊗ 𝐵 ))) where 𝐵 𝐴 is the local system 𝑥 ↦ hom( 𝐴, 𝐵 𝑥 ) , and we use the fact that 𝑟 ! is realized by a finite colimit andhence commutes with taking mapping spectra out of 𝐴 .Let 𝑝, 𝑞 ∶ 𝑋 × 𝑋 → 𝑋 denote the two projections. Applying Remark 4.4.10 to E = S 𝑝 𝑋 we obtain thatthe map(107) 𝑐 ⊗ 𝐿 ∶ 𝐿 → 𝑟 ! 𝜈 ⊗ 𝐿 ≃ 𝑝 ! ( 𝑞 ∗ 𝜈 ⊗ 𝑝 ∗ 𝐿 ) acts as unit exhibiting 𝑝 ! ( 𝑞 ∗ 𝜈 ⊗ (−)) as right adjoint to 𝑝 ∗ . Since 𝑞 ∗ 𝜈 is invertible tensoring with 𝑞 ∗ 𝜈 is left and right inverse to tensoring with 𝑞 ∗ 𝜈 −1 , and hence the above map also acts as a unit exhibiting 𝑝 ! ∶ S 𝑝 𝑋 × 𝑋 → S 𝑝 𝑋 as right adjoint to 𝑞 ∗ 𝜈 ⊗ 𝑝 ∗ (−) = 𝜈 ⊠ (−) . We consequently obtain a canonicalequivalence hom ⊠ ( 𝜈, −) ≃ 𝑝 ! (−) of functors S 𝑝 𝑋 × 𝑋 → S 𝑝 𝑋 . Evaluating at a spherical Δ ! 𝜉 for some spherical fibration 𝜉 we obtain a canon-ical equivalence D 𝜉 ( 𝜈 ) ≃ hom ⊠ ( 𝜈, Δ ! 𝜉 ) ≃ 𝑝 ! Δ ! 𝜉 ≃ 𝜉, and hence a canonical equivalence D 𝜉 ( 𝜉 ) ≃ 𝜈 for any spherical fibration 𝜉 on 𝑋 . In other words, for every spherical fibration 𝜉 on 𝑋 , the 𝜉 -twisted dualityswitches between 𝜉 and 𝜈 . Taking 𝜉 = 𝜈 we then get D 𝜈 ( 𝜈 ) ≃ 𝜈, that is, 𝜈 is equivalent to its own 𝜈 -twisted dual. We now claim that this equivalence is induced by a canonicalPoincaré form 𝑞 ∈ Ω ∞ Ϙ v 𝜈 ( 𝜈 ) . To see this, consider the pair of composable maps 𝑋 × 𝑋 𝑝 ←←←←←←→ 𝑋 𝑟 ←←←←←→ ∗ . Then we saw above that (107) exhibits 𝑝 ! ∶ S 𝑝 𝑋 × 𝑋 → S 𝑝 𝑋 as right adjoint to 𝜈 ⊠ (−) , while for the samereason the original Thom-Pontryagin map exhibits 𝑟 ! ∶ S 𝑝 𝑋 → S 𝑝 as right adjoint to 𝜈 ⊗𝑟 ∗ (−) ∶ S 𝑝 → S 𝑝 𝑋 .Composing the two adjunctions we then obtain that ( 𝑟𝑝 ) ! ∶ S 𝑝 𝑋 × 𝑋 → S 𝑝 is right adjoint to ( 𝜈⊠𝜈 ) ⊗ ( 𝑟𝑝 ) ∗ (−) .In particular, 𝜈 ⊠ 𝜈 is the Spivak normal bundle of 𝑋 × 𝑋 . Unwinding the definitions we see that theassociated Thom-Pontryagin map is 𝕊 ≃ 𝕊 ⊗ 𝕊 𝑐⊗𝑐 ←←←←←←←←←←←←←←←←→ 𝑟 ! 𝜈 ⊗ 𝑟 ! 𝜈 ≃ ( 𝑟𝑝 ) ! ( 𝜈 ⊠ 𝜈 ) . In particular, for every spherical fibration 𝜉 on 𝑋 the composed map(108) B 𝜉 ( 𝜈, 𝜈 ) = hom 𝑋 × 𝑋 ( 𝜈 ⊠ 𝜈, Δ ! 𝜉 ) → hom(( 𝑟𝑝 ) ! ( 𝜈 ⊠ 𝜈 ) , ( 𝑟𝑝 ) ! Δ ! 𝜉 ) ≃ hom( 𝑟 ! 𝜈 ⊗ 𝑟 ! 𝜈, 𝑟 ! 𝜉 ) ( 𝑐⊗𝑐 ) ∗ ←←←←←←←←←←←←←←←←←←←←←←←←←→ 𝑟 ! 𝜉 is an equivalence. Now since 𝑟𝑝 ∶ 𝑋 × 𝑋 → ∗ is (uniquely) C -equivariant with respect to the flip actionon 𝑋 × 𝑋 and the trivial action on ∗ the functor ( 𝑟𝑝 ) ! ∶ S 𝑝 𝑋 × 𝑋 → S 𝑝 refines to a C -equivariant functorwith respect to the corresponding induced maps. The canonical C -symmetric structure of the ( 𝑋 × 𝑋 ) -parameterized spectra 𝜈 ⊠ 𝜈 and ( 𝑟𝑝 ) ! Δ ! 𝜉 then induces a C -action on the corresponding spectra ( 𝑟𝑝 ) ! 𝜈 ≃ 𝑟 ! 𝜈 ⊗ 𝑟 ! 𝜈 and ( 𝑟𝑝 ) ! Δ ! 𝜉 . In the case of 𝑟 ! 𝜈 ⊗ 𝑟 ! 𝜈 this simply recovers the corresponding flip action, while for ( 𝑟𝑝 ) ! Δ ! 𝜉 , the identification 𝑟 ◦ 𝑝 ◦ Δ ≃ 𝑟 as C -equivariant functors 𝑋 → ∗ (with trivial actions on both sides)implies that the induced action on ( 𝑟𝑝 ) ! Δ ! 𝜉 ≃ 𝑟 ! 𝜉 is in fact trivial. Finally, the map 𝑐 ⊗ 𝑐 ∶ 𝕊 → 𝑟 ! 𝜈 ⊗ 𝑟 𝜈 is also canonically C -equivariant. We may hence view (108) as a sequence of spectra with C -actionsand C -equivariant maps between them, where we start with the C -action on B 𝜉 ( 𝜈, 𝜈 ) associated to thesymmetric structure on B 𝜉 and finish with the trivial C -action on 𝑟 ! 𝜉 . We hence obtain a C -equivariantequivalence B 𝜉 ( 𝜈, 𝜈 ) ≃ 𝑟 ! 𝜉 with C acting trivially on 𝑟 ! 𝜉 . In particular, C acts essentially trivially on B 𝜉 ( 𝜈, 𝜈 ) ≃ hom( 𝜈, D 𝜉 ( 𝜈 )) .Taking 𝜉 = 𝜈 we then obtain:4.4.11. Corollary.
The equivalence 𝜈 → D 𝜈 𝜈 constructed above canonically refines to a self-dual equiva-lence, yielding in particular a symmetric Poincaré form 𝑞 s ∈ Ϙ s 𝜈 ( 𝜈 ) . We would like to lift the symmetric form 𝑞 s from Ϙ s 𝜈 ( 𝜈 ) to a visible form 𝑞 v ∈ Ϙ s 𝜈 . For this, we note thatsince 𝜈 corepresents the functor 𝑟 ! (−) we have an equivalence hom( 𝜈, 𝜉 ) ≃ 𝑟 ! 𝜉 , and so we can write thepullback square defining the visible Poincaré structure evaluated at 𝜈 as Ϙ v 𝜉 ( 𝜈 ) 𝑟 ! 𝜉𝑟 ! 𝜉 hC 𝑟 ! 𝜉 tC where the C -fixed points and Tate construction are performed with respect to the trivial action on 𝑟 ! 𝜉 , andthe right vertical map is given by the composite 𝑟 ! 𝜉 → 𝑟 ! ( 𝜉 tC ) → ( 𝑟 ! 𝜉 ) tC .Taking 𝜉 = 𝜈 we then obtain:4.4.12. Corollary.
Then the symmetric Poincaré form of Corollary 4.4.11 canonically lifts to a visiblePoincaré form 𝑞 v ∈ Ϙ v 𝜈 ( 𝜈 ) . The Poincaré object ( 𝜈, 𝑞 v ) ∈ Pn( S 𝑝 𝜔𝑋 , Ϙ v 𝜈 ) is called the visible signature of the Poincaré duality space 𝑋 .5. M ONOIDAL STRUCTURES AND MULTIPLICATIVITY
In this section we will show that the tensor product of stable ∞ -categories refines to give symmetricmonoidal structures on the ∞ -categories Cat h∞ and Cat p∞ . We give a careful analysis of algebra objects in Cat p∞ , and use it to show that many examples of interest, such as the symmetric and genuine symmetricPoincaré structures on perfect derived categories of commutative rings, admit such an algebra structure.The significance of this fact is that for an algebra object in Cat p∞ , the L -groups and the Grothendieck-Wittgroup inherit the structure of rings. We will show in Paper [IV] that this phenomenon extends to an E ∞ -structure on the level of Grothendieck-Witt- and L -theory spectra , a structure which plays a key role in thestudy of these invariants in the commutative setting.This section is organized as follows. In §5.1 we define the tensor product of hermitian and Poincaré ∞ -categories and give a formula for the linear and bilinear parts of the hermitian structure on the tensorproduct in term of the linear and bilinear parts of the individual terms. In §5.2 we show that this operationorganizes into a symmetric monoidal structure on the ∞ -categories Cat h∞ and Cat p∞ , such that the forgetfulfunctors Cat p∞ → Cat h∞ → Cat ex∞ are symmetric monoidal. In §5.3 we analyse what it means for a hermitianor Poincaré ∞ -category to be an algebra with respect to this structure, and use this analysis in §5.4 in orderto identify various examples of interest of symmetric monoidal Poincaré ∞ -categories. Finally, in §7.5 weshow that the Grothendieck-Witt group and zero’th L -group are symmetric monoidal functors on Cat p∞ , andhence their values on a given symmetric monoidal Poincaré ∞ -category are rings. Similarly, we show thatin this case the collection of all L -groups acquires the structure of a graded-commutative ring.5.1. Tensor products of hermitian ∞ -categories. In this first part we define the tensor product of her-mitian ∞ -categories on the level of objects and show that the tensor product of Poincaré ∞ -categories isagain Poincaré.We begin by recalling the tensor product of stable ∞ -categories due to Lurie.5.1.1. Construction.
Applying the construction of [Lur17, §4.8.1] with respect to the collection K of finitesimplicial sets yields a symmetric monoidal structure on the ∞ -category Cat rex∞ , whose objects are small ∞ -categories with finite colimits and whose morphisms are functors which preserve finite colimits. For apair of ∞ -categories with finite colimits C and C ′ , their tensor product C ⊗ C ′ ∈ Cat rex∞ is equipped with afunctor C × C ′ → C ⊗ C ′ which preserves finite colimits in each variable, and is initial with this property.Now the ∞ -category Cat ex∞ embeds fully-faithful in
Cat rex∞ , and is furthermore a reflective subcategory: aleft adjoint to the inclusion
Cat ex∞ ⊆ Cat rex∞ is given by tensoring with S 𝑝 f . To see this, observe that for an ∞ -category C with finite colimits, the tensor product of C and S 𝑝 f yields an ∞ -category with finite colimits ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 79 C ⊗ S 𝑝 f ∈ Cat rex∞ which is a module over S 𝑝 f . Since S 𝑝 f is a self-dual object it acts via a self-adjointfunctor and hence C ⊗ S 𝑝 f is pointed. In addition, since the object Σ 𝕊 ∈ S 𝑝 f is invertible we get thatsuspension is invertible on C ⊗ S 𝑝 and so the latter is furthermore stable. To see that this operation givesa left adjoint to the inclusion Cat rex∞ ⊆ Cat ex∞ we note that for every stable ∞ -category D the restrictionfunctor Fun rex ( C ⊗ S 𝑝 f , D ) = Fun rex ( C , Fun rex ( S 𝑝 f , D )) = Fun rex ( C , Fun ex ( S 𝑝 f , D )) → Fun rex ( C , D ) is an equivalence by Lemma 4.1.2. It then follows from [Lur17, Proposition 4.1.7.4] that the symmetricmonoidal structure ⊗ descends to Cat ex∞ . In particular, the unit of
Cat ex∞ is given by S 𝑝 f , and if C , C ′ are stablethen C ⊗ C ′ is universal among stable ∞ -categories receiving a bilinear functor 𝛽 ∶ C × C ′ → C ⊗ C ′ : given astable ∞ -category D , restriction along 𝛽 induces an equivalence between the ∞ -category of exact functorsfrom C ⊗ C ′ → D and the ∞ -category of bilinear functors C × C ′ → D . We will refer to E ∞ -algebra objectsin (Cat ex∞ ) ⊗ as stably symmetric monoidal ∞ -categories. Concretely, this means a symmetric monoidal ∞ -category whose underlying ∞ -category is stable and the tensor product is exact in each variable.We now refine this construction to the level of hermitian ∞ -categories.5.1.2. Construction.
For a pair of hermitian ∞ -categories ( C , Ϙ ) and ( C ′ , Ϙ ′ ) , we define their tensor product ( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ ) ∶= ( C ⊗ C ′ , Ϙ ⊗ Ϙ ′ ) to be the hermitian ∞ -category whose underlying stable ∞ -category is the tensor product of the underlyingstable ∞ -categories, and whose hermitian structure Ϙ ⊗ Ϙ ′ ∶= P 𝛽 ! ( Ϙ ⊠ Ϙ ′ ) ∶ C ⊗ C ′ → S 𝑝 is obtained by applying the 2-excisive approximation functor of Construction 1.1.26 to the left Kan extensionalong 𝛽 ∶ C × C ′ → C ⊗ C ′ of the ‘external’ tensor product Ϙ ⊠ Ϙ ′ ∶ C op × C ′op Ϙ × Ϙ ′ ←←←←←←←←←←←←←←←←←←←→ S 𝑝 × S 𝑝 ⊗ ←←←←←←←←←→ S 𝑝. Here we note that 𝛽 ! ( Ϙ ⊠ Ϙ ′ ) is already reduced since Ϙ ⊠ Ϙ ′ is reduced and 𝛽 preserve zero objects, and soConstruction 1.1.26 is applicable to it.Note that the tensor product ( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ ) = ( C ⊗ C ′ , Ϙ ⊗ Ϙ ′ ) carries by design a similar universalproperty to the tensor product C ⊗ C ′ of stable ∞ -categories: for any hermitian ∞ -category ( C ′′ , Ϙ ′′ ) , mapsfrom the tensor product ( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ ) to ( C ′′ , Ϙ ′′ ) correspond to bilinear maps 𝑏 ∶ C × C ′ → C ′′ togetherwith a natural transformation Ϙ ⊠ Ϙ ′ → 𝑏 ∗ Ϙ ′′ , i.e. a natural transformation in the square C op × C ′op S 𝑝 × S 𝑝 C ′′op S 𝑝 . 𝑏 Ϙ × Ϙ ′ ⊗ Ϙ ′′ Our next goal is to identify the linear and bilinear parts of the hermitian structure on ( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ ) in more explicit terms. To this end, let L Ϙ , L Ϙ ′ denote the linear parts and B Ϙ , B Ϙ ′ the bilinear parts of thehermitian structures Ϙ and Ϙ ′ , respectively. The functor L Ϙ ⊠ L Ϙ ′ ∶ C op × C ′op L Ϙ ×L Ϙ ′ ←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ S 𝑝 × S 𝑝 ⊗ ←←←←←←←←←→ S 𝑝 is then bilinear, and therefore extends along 𝛽 to a linear functor L Ϙ ⊗ L Ϙ ′ ∶ C op ⊗ C ′op → S 𝑝 in an essentiallyunique manner. Similarly, the multilinear functor B Ϙ ⊠ B Ϙ ′ ∶ C op × C op × C ′op × C ′op B Ϙ ×B Ϙ ′ ←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ S 𝑝 × S 𝑝 ⊗ ←←←←←←←←←→ S 𝑝 extends to a bilinear functor B Ϙ ⊗ B Ϙ ′ ∶ ( C op ⊗ C ′op ) × ( C op ⊗ C ′op ) → S 𝑝. The symmetric structures of B Ϙ and B Ϙ ′ then determine a C -fixed structure on B Ϙ ⊠ B Ϙ ′ with respect tothe C -action which permutes the two C op -coordinates and the two C ′op -coordinates. This structure thendescends to a symmetric structure on B Ϙ ⊗ B Ϙ ′ by its universal characterization. Proposition.
For hermitian ∞ -categories ( C , Ϙ ) , ( C ′ , Ϙ ′ ) there is a canonical pullback square (109) Ϙ ⊗ Ϙ ′ L Ϙ ⊗ L Ϙ ′ [ (B Ϙ ⊗ B Ϙ ′ ) Δ ] hC [ (B Ϙ ⊗ B Ϙ ′ ) Δ ] tC of functors C op ⊗ C ′op → S 𝑝 . In particular the linear part of Ϙ ⊗ Ϙ ′ is given by L Ϙ ⊗ L Ϙ ′ and its symmetricbilinear part by B Ϙ ⊗ B Ϙ ′ .Proof. By definition the functor Ϙ ⊗ Ϙ ′ = P 𝛽 ! ( Ϙ ⊠ Ϙ ′ ) is characterized by the fact that for every quadraticfunctor Ϙ ′′ ∶ C op ⊗ C ′op → S 𝑝 the natural map Nat( Ϙ ⊗ Ϙ ′ , Ϙ ′′ ) → Nat( Ϙ ⊠ Ϙ ′ , 𝛽 ∗ Ϙ ′′ ) is an equivalence. Using this universal mapping property the commutative square (109) is then obtainedfrom the external square in the diagram Ϙ ⊠ Ϙ ′ L Ϙ ⊠ L Ϙ ′ [ (B Ϙ ) Δ ] hC ⊠ [ (B Ϙ ′ ) Δ ] hC [ (B Ϙ ) Δ ] tC ⊠ [ (B Ϙ ′ ) Δ ] tC [ (B Ϙ ⊠ B Ϙ ′ ) Δ ] hC [ (B Ϙ ⊠ B Ϙ ′ ) Δ ] tC , where the top square is obtained by taking the external product of the classifying squares of Ϙ and Ϙ ′ , andthe bottom square witnesses the lax symmetric monoidal structure of the homotopy fixed points functor andthe projection to the Tate construction. To show that the resulting square (109) is a pullback square we needto show that the induced map(110) Nat ( (L Ϙ ⊗ L Ϙ ′ ) × [(B Ϙ ⊗ B Ϙ ′ ) Δ ] tC2 [(B Ϙ ⊗ B Ϙ ′ ) Δ ] hC , Ϙ ′′ ) → Nat( Ϙ ⊠ Ϙ ′ , 𝛽 ∗ Ϙ ′′ ) is an equivalence for any quadratic functor Ϙ ′′ . Let us analyze both sides of the map (110). We startwith the following claim (see §1.3 for the terminology of homogeneous and cohomogenous and their basicproperties): Claim 1:
Nat( Ϙ ⊠ Ϙ ′ , 𝛽 ∗ Ϙ ′′ ) = 0 if either Ϙ is exact and Ϙ ′′ is cohomogeneous or Ϙ ishomogeneous and Ϙ ′′ is exact.To see this claim we note that for a fixed object 𝑐 ′ ∈ C ′ we have that the space of natural transformations Nat( Ϙ (−) ⊗ Ϙ ′ ( 𝑐 ′ ) , Ϙ ′′ ( 𝛽 (− , 𝑐 ′ ))) (natural in (−) ) vanishes under these assumptions since left hand functor is exact (resp. homogeneous) andthe right hand functor is cohomogenous (resp. exact). But the space Nat( Ϙ ⊠ Ϙ ′ , 𝛽 ∗ Ϙ ′′ ) can be written asa limit of these spaces over the twisted arrow category of C ′ so that the claim follows. We also have thefollowing claim Claim 2:
Nat ( (L Ϙ ⊗ L Ϙ ′ ) × [(B Ϙ ⊗ B Ϙ ′ ) Δ ] tC2 [(B Ϙ ⊗ B Ϙ ′ ) Δ ] hC , Ϙ ′′ ) = 0 if either Ϙ is exactand Ϙ ′′ is cohomogeneous or Ϙ is homogeneous and Ϙ ′′ is exact.which follows by the same argument as above since under the assumptions the pullback is either L Ϙ ⊗ L Ϙ ′ or [(B Ϙ ⊗ B Ϙ ′ ) Δ ] hC .Together the last two claims show that the map (109) is an equivalence under the assumption that either Ϙ is exact and Ϙ ′′ is cohomogeneous or Ϙ is homogeneous and Ϙ ′′ is exact. Now Ϙ can be fit in an exactsequence between a homogeneous functor and an exact functor, and Ϙ ′′ can be fit in an exact sequencebetween an exact and cohomogeneous functors. Since both sides of (109) are natural and exact in Ϙ , Ϙ ′ and Ϙ ′′ we can thus assume without loss of generality that Ϙ and Ϙ ′′ are both exact or Ϙ is homogeneous and Ϙ ′′ is cohomogeneous. Since everything is symmetric in Ϙ and Ϙ ′ we can also make the same reduction in thisvariable so that we only need to show the fact that (109) is a pullback under the following assumptions: ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 81 Claim 3:
Either all three functors Ϙ , Ϙ ′ , Ϙ ′′ are exact or Ϙ and Ϙ ′ are homogeneous and Ϙ ′′ is cohomogeneous.In the first case the statement unwinds to the universal property of C ⊗ C ′ and in the second case it unwinds(also using the universal property) to the statement that maps from a homgenous functor to a cohomogenousfunctor are equivalent to maps between the associated symmetric bilinear functors. (cid:3) Corollary. If ( C , Ϙ ) and ( C ′ , Ϙ ′ ) are Poincaré with duality functors D Ϙ , D Ϙ ′ then ( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ ) isPoincaré with duality functor D Ϙ ⊗ D Ϙ ∶ ( C ⊗ C ) op = C op ⊗ C op → C ⊗ C .Proof. We get from Proposition 5.1.3 that the cross effect of the quadratic functor on C ⊗ C ′ is given by B Ϙ ⊗ B Ϙ ′ , which coincides with the left Kan extension of B Ϙ ⊠ B Ϙ ′ along the map 𝛽 × 𝛽 ∶ ( C op × C ′op ) × ( C op × C ′op ) → ( C op ⊗ C ′op ) × ( C op ⊗ C ′op ) , where [B Ϙ ⊠ B Ϙ ′ ]( 𝑥, 𝑥 ′ , 𝑦, 𝑦 ′ ) = B Ϙ ( 𝑥, 𝑦 ) ⊗ B Ϙ ′ ( 𝑥 ′ , 𝑦 ′ ) = hom C ( 𝑥, D Ϙ 𝑦 ) ⊗ hom C ′ ( 𝑥 ′ , D Ϙ ′ 𝑦 ′ ) . We want to show that this is represented by the functor D Ϙ ⊗ D Ϙ ′ . Now the left Kan extension along 𝛽 × 𝛽 can be computed by composing left Kan extensions along 𝛽 × id and id × 𝛽 . Then for 𝑦 ∈ C , 𝑦 ′ ∈ C ′ wethen have [( 𝛽 × id) ! B Ϙ ⊠ B Ϙ ′ ] | C ⊗ C ′ ×{ 𝑦 }×{ 𝑦 ′ } = 𝛽 ! [B Ϙ (− , 𝑦 ) ⊗ B Ϙ ′ (− , 𝑦 ′ )] = 𝛽 ! [hom C (− , D Ϙ 𝑦 ) ⊗ hom C ′ (− , D Ϙ ′ 𝑦 ′ )] = hom C ⊗ C ′ (− , D Ϙ 𝑦 ⊗ D Ϙ ′ 𝑦 ′ ) , and the left Kan extension along 𝛽 of the functor ( 𝑦, 𝑦 ′ ) ↦ D Ϙ 𝑦 ⊗ D Ϙ ′ 𝑦 ′ is D Ϙ ⊗ D Ϙ ′ . (cid:3) Proposition.
Recall the universal Poincaré ∞ -category ( S 𝑝 f , Ϙ u ) of §4.1. Then we have a naturalequivalence ( C , Ϙ ) ⊗ ( S 𝑝 f , Ϙ u ) ≃ ( C , Ϙ ) for ( C , Ϙ ) ∈ Cat h∞ .Proof. We first note that S 𝑝 f is the unit with respect to the tensor product of stable ∞ -categories (seeConstruction 5.1.1), and that ( S 𝑝 f ) op ≃ S 𝑝 f through Spanier Whitehead duality. By definition, the linearpart of Ϙ u ∶ ( S 𝑝 f ) op → S 𝑝 is Spanier Whitehead duality D and the bilinear part is the composite ( S 𝑝 f ) op × ( S 𝑝 f ) op ⊗ 𝕊 ←←←←←←←←←←←←←→ ( S 𝑝 f ) op D ←←←←←←←←→ S 𝑝 f , which also corresponds to Spanier Whitehead duality ( S 𝑝 f ) op → S 𝑝 f under the equivalence ( S 𝑝 f ) op ⊗ ( S 𝑝 f ) op ≃ ( S 𝑝 f ) op . Now using Proposition 5.1.3 the claim is reduced to the statement that for any exactfunctor L ∶ C op → S 𝑝 the functor C op = ( C ⊗ S 𝑝 f ) op → C op ⊗ ( S 𝑝 f ) op L ⊗ D ←←←←←←←←←←←←←←←←←←←→ S 𝑝 ⊗ S 𝑝 ⊗ ←←←←←←←←←→ S 𝑝 is equivalent to L . Indeed, the composite C op = ( C ⊗ S 𝑝 f ) op → C op ⊗ ( S 𝑝 f ) op sends 𝑐 ∈ C op to 𝑐 ⊗ 𝕊 . (cid:3) Example.
Let
𝐴, 𝐵 be E -ring spectra equipped with modules with genuine involutions ( 𝑀 𝐴 , 𝑁 𝐴 , 𝛼 ) and ( 𝑀 𝐵 , 𝑁 𝐵 , 𝛽 ) respectively (see §3.2). As discussed in the proof of Theorem 3.2.13, it follows from [Lur17,Theorem 4.8.5.16 and Remark 4.8.5.19] that the exact functors(111) Mod f 𝐴 ⊗ Mod f 𝐵 → Mod f 𝐴⊗ 𝕊 𝐵 and Mod 𝜔𝐴 ⊗ Mod 𝜔𝐵 → Mod 𝜔𝐴⊗ 𝕊 𝐵 , induced by the bilinear functor ( 𝑋, 𝑌 ) ↦ 𝑋 ⊗𝑌 , both yield an equivalence
Mod 𝐴 ⊗ S 𝑝 Mod 𝐵 ≃ ←←←←←←←→ Mod 𝐴⊗ 𝕊 𝐵 upon passing to Ind-categories, where ⊗ S 𝑝 denotes the tensor product of stable presentable ∞ -categories.In particular, the functors (111) are necessarily fully-faithful inclusions which are dense, that is, every objectin the target is a retract of an object in the domain. It then follows that the functor on the left hand side is anequivalence since its image contains 𝐴 ⊗ 𝐵 and its target contains no proper stable subcategories with thatproperty. On the right hand side we only have that
Mod 𝜔𝐴 ⊗ Mod 𝜔𝐵 is some full subcategory of Mod 𝐴⊗ 𝕊 𝐵 ,intermediate between Mod f 𝐴⊗ 𝕊 𝐵 and Mod 𝜔𝐴⊗ 𝕊 𝐵 . It then follows from Proposition 5.1.3 below that (Mod f 𝐴 , Ϙ 𝛼𝑀 𝐴 ) ⊗ (Mod f 𝐵 , Ϙ 𝛽𝑀 𝐵 ) ≃ (Mod f 𝐴⊗ 𝕊 𝐵 , Ϙ 𝛼⊗ 𝕊 𝛽𝑀 𝐴 ⊗ 𝕊 𝑀 𝐵 ) where the reference map on the right hand side is the composite 𝛼 ⊗ 𝕊 𝛽 ∶ 𝑁 𝐴 ⊗ 𝕊 𝑁 𝐵 → 𝑀 tC 𝐴 ⊗ 𝕊 𝑀 tC 𝐵 → ( 𝑀 𝐴 ⊗ 𝕊 𝑀 𝐵 ) tC , obtained using the lax monoidal structure of the Tate construction. In the case of perfect modules we onlyhave in general that (Mod 𝜔𝐴 , Ϙ 𝛼𝑀 𝐴 ) ⊗ (Mod 𝜔𝐵 , Ϙ 𝛽𝑀 𝐵 ) is equivalent to a full subcategory of Mod 𝜔𝐴⊗ 𝕊 𝐵 , equippedwith the hermitian structure restricted from Ϙ 𝛼⊗ 𝕊 𝛽𝑀 𝐴 ⊗ 𝕊 𝑀 𝑁 .5.2. Construction of the symmetric monoidal structure.
In this section we will show that the notion oftensor product constructed in §5.1 above can be enhanced to symmetric monoidal structures on
Cat h∞ andon Cat p∞ . The construction is somewhat technical and can be skipped on a first read.5.2.1. Construction.
For an ∞ -category D we will denote by (Cat ∞ ) ∕∕ D → Cat ∞ the cartesian fibration classified by the functor Cat op∞ → Cat ∞ C ↦ Fun( C , D ) . We will refer to (Cat ∞ ) ∕∕ D as the lax slice over D . The objects of (Cat ∞ ) ∕∕ D are given by functors C → D and the morphisms by diagrams C DC ′ 𝑓 𝑝𝑞 filled by a non-invertible 2-cell 𝑝 ⇒ 𝑞𝑓 . The actual slice (Cat ∞ ) ∕ D is a non-full subcategory of (Cat ∞ ) ∕∕ D which contains all objects but only those 1-morphisms for which the natural transformation 𝑝 ⇒ 𝑞𝑓 is anequivalence.5.2.2. Remark.
The ∞ -category (Cat ∞ ) ∕∕ D can be characterized by the following universal mapping prop-erty: the data of a functor from E to (Cat ∞ ) ∕∕ D is equivalent to the data of a diagram in Cat ∞ of the form(112) C DE 𝑝 with 𝑝 a cocartesian fibration. In this description functors from E to the actual slice (Cat ∞ ) ∕ D correspond todiagrams as above where the functor C → D send 𝑝 -cocartesian lifts to equivalences in D . This descriptionalso uniquely determines (Cat ∞ ) ∕∕ D since it describes the represented functor Ho(Cat ∞ ) op → Set .5.2.3.
Lemma.
Let D be a symmetric monoidal ∞ -category. Then (Cat ∞ ) ∕∕ D admits a symmetric monoidalrefinement (Cat ∞ ) ⊗ ∕∕ D with the following properties:i) The tensor product of 𝑓 ∶ C → D and 𝑔 ∶ C ′ → D in (Cat ∞ ) ∕∕ D is given by the composite C × C ′ 𝑓 × 𝑔 ←←←←←←←←←←←←←←←←←→ D × D ⊗ ←←←←←←←←←→ D , and the tensor unit by the functor pt → D corresponding to the tensor unit of D .ii) The forgetful functor (Cat ∞ ) ∕∕ D → Cat ∞ admits a symmetric monoidal refinement (Cat ∞ ) ⊗ ∕∕ D → Cat ×∞ with respect to the cartesian symmetric monoidal structure on Cat ∞ .iii) For any ∞ -operad O ⊗ , the space of O -algebras in (Cat ∞ ) ⊗ ∕∕ D is naturally equivalent to the space ofdiagrams of ∞ -operads C ⊗ D ⊗ O ⊗𝑝 ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 83 where 𝑝 is a cocartesian fibration of ∞ -operads (see [Lur17, Definition 2.1.2.13] ). In other words,a pair consisting of an O -monoidal ∞ -category C together with a lax O -monoidal functor from C to D , where the latter is consider as an O -monoidal ∞ -category by pullback along the terminal map of ∞ -operads O → E ∞ . Remark.
Property iii) of Lemma 5.2.3 determines the symmetric monoidal ∞ -category (Cat ∞ ) ⊗ ∕∕ D uniquely. Indeed, a symmetric monoidal ∞ -category is uniquely determined by its underlying ∞ -operad,and Property iii) determines the functor Ho(Op ∞ ) op → Set represented by the underlying ∞ -operad (Cat ∞ ) ⊗ ∕∕ D . One can in fact show that the first two properties ofthe lemma are direct consequences of the third (see the arguments in the proof of Lemma 5.2.3 below). Proof of Lemma 5.2.3.
For a fixed symmetric monoidal ∞ -category D with underlying ∞ -operad D ⊗ , weconsider the ∞ -category X whose objects are given by diagrams of ∞ -operads of the form(113) C ⊗ D ⊗ O ⊗𝑝 where 𝑝 is a cocartesian fibration of ∞ -operads. The morphisms in X are those maps of such diagramswhich are the identity on D ⊗ and preserve 𝑝 -cocartesian arrows in the C ⊗ component. Projecting to the O ⊗ -component defines a functor X → Op ∞ which is a cartesian fibration classified by some functor Op op∞ → Cat ∞ . Postcomposing the latter with the groupoid core functor we obtain a functor 𝑞 ∶ Op op∞ → S sending an ∞ -operad O to the space consisting of pairs of a cocartesian fibration 𝑝 ∶ C ⊗ → O ⊗ of ∞ -operads together with a map C ⊗ → D ⊗ of ∞ -operads. We now claim that the functor 𝑞 is representable byan ∞ -operad. To see this we use that Op ∞ is a presentable ∞ -category by [Lur17, Section 2.1.4] so thatwe have to check that the functor 𝑞 preserves limits by [Lur09a, Proposition 5.5.2.2]. This can be seen asfollows: a cocartesian fibration C ⊗ → O ⊗ = colim 𝑖 O ⊗𝑖 over a colimit of ∞ -operads is classified by a mapof ∞ -operads 𝜒 ∶ colim 𝑖 O ⊗𝑖 → Cat ×∞ where Cat ×∞ is equipped with the structure of an ∞ -operad inducedby the cartesian symmetric monoidal on Cat ∞ (see [Lur17, Remark 2.4.2.6]). The space of such functorsis thus a limit of the spaces of maps of ∞ -operads 𝜒 𝑖 ∶ O ⊗𝑖 → Cat ×∞ . In particular the space of cocartesianfibrations over colim 𝑖 O ⊗𝑖 is the limit of the spaces of cocartesian fibrations over O ⊗𝑖 . In addition, for everycocartesian fibration C ⊗ → O ⊗ of ∞ -operads the natural map(114) colim 𝑖 C ⊗𝑖 → C ⊗ is an equivalence of ∞ -operads, where C ⊗𝑖 ∶= C ⊗ × O ⊗ O ⊗𝑖 is the corresponding fibre product (computedin Op ∞ ). This follows from the fact that the functor C ⊗ × O ⊗ − ∶ (Op ∞ ) ∕ O ⊗ → (Op ∞ ) ∕ O ⊗ commutes with colimits of ∞ -operads since it has a right adjoint given by the relative Day convolution Fun O ( C , −) ⊗ , see [Lur17, Construction 2.2.6.7 and Remark 2.2.6.8]. The colimit description of (114) thenimplies that the space of maps C ⊗ → D ⊗ of ∞ -operads is given the limit of the space of maps C ⊗𝑖 → D ⊗ .Together this shows that the functor 𝑞 preserves limits and is thus representable. We denote the representingobject by (Cat ⊗ ∞ ) ∕∕ D . We note that by Remark 5.2.2 the underlying ∞ -category ( (Cat ⊗ ∞ ) ∕∕ D ) ⟨ ⟩ identifieswith (Cat ∞ ) ∕∕ D ; indeed, when O ⊗ is the image of an ∞ -category E under the full inclusion Cat ∞ ⊆ Op ∞ ,the data of a diagram as in (113) with 𝑝 a cocartesian fibration of ∞ -operads reduces to that of a diagram ofthe form (112), with 𝑝 a cocartesian fibration of ∞ -categories. To show (Cat ⊗ ∞ ) ∕∕ D is a symmetric monoidalstructure on (Cat ∞ ) ∕∕ D we will need a more explicit description of the multi-mapping spaces in (Cat ⊗ ∞ ) ∕∕ D . Let 𝐶 𝑛 (sometimes called the 𝑛 -corolla) be the ∞ -operad freely generated by a single 𝑛 -ary operation 𝑥 , … , 𝑥 𝑛 → 𝑥 with colours 𝑥 , ..., 𝑥 𝑛 and 𝑥 .The space of maps 𝐶 ⊗𝑛 → (Cat ⊗ ∞ ) ∕∕ D is then by the definingproperty of (Cat ⊗ ∞ ) ∕∕ D the classifying space of pairs of a 𝐶 𝑛 -monoidal ∞ -category C and a lax 𝐶 𝑛 -monoidalfunctor C → D . As the ∞ -operad 𝐶 𝑛 is free a 𝐶 𝑛 -monoidal ∞ -category is simply given by a sequence { C , ..., C 𝑛 ; C } of ∞ -categories together with a functor 𝛼 ∶ C × … × C 𝑛 → C , and a lax 𝐶 𝑛 -monoidalfunctor from this to D corresponds to collection of functors { 𝑓 ∶ C → D , … , 𝑓 𝑛 ∶ C 𝑛 → D ; 𝑓 ∶ C → D } together with a transformation in the square C × … × C 𝑛 D × … × DC D . 𝛼 𝑓 ×…× 𝑓 𝑛 ⊗𝑓 As a result we find that the corresponding multi-mapping space, which can be identified with the pullback
Mul (Cat ⊗ ∞ ) ∕∕ 𝐷 ( 𝑓 , ..., 𝑓 𝑛 ; 𝑓 ) Map Op ∞ ( 𝐶 𝑛 , (Cat ⊗ ∞ ) ∕∕ 𝐷 ) ∏ 𝑖 ev 𝑥𝑖 ×ev 𝑥 pt − ( 𝑓 , ..., 𝑓 𝑛 , 𝑓 ) ∏ 𝑛 +1 Map Op ∞ ( T riv ⊗ , (Cat ⊗ ∞ ) ∕∕ D ) , is given by the spaces of maps in (Cat ∞ ) ∕∕ D from the object(115) C × … × C 𝑛 𝑓 ×…× 𝑓 𝑛 ←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ D × … × D ⊗ ←←←←←←←←←→ D to the object 𝑓 ∶ C → D . In particular, this multi-mapping space is corepresented by the object (115) and sothe ∞ -operad (Cat ∞ ) ⊗ ∕∕ D is corepresentable in the sense of [Lur17, Definition 6.2.4.3], that is, the functor (Cat ∞ ) ⊗ ∕∕ D → Fin ∗ is a locally cocartesian fibration. To see that it is actually cocartesian, i.e. (Cat ∞ ) ⊗ ∕∕ 𝐷 issymmetric monoidal, we have to additionally verify that the induced maps from [Lur17, Example 6.2.4.9]are equivalences. But this is clear in the case at hand.Finally, let us verify that (Cat ∞ ) ⊗ ∕∕ D satisfies the required Properties i)-iii). Indeed, Property iii) issatisfied by construction and Property i) follows from the explicit description of multi-mapping spacesabove. To prove Property ii) we note that the ∞ -operad Cat ×∞ represents the functor Op op∞ → S which sendsan ∞ -operad O to the space of cocartesian fibrations E → O (this follows from [Lur17, Remark 2.4.2.6]).The functor (Cat ∞ ) ∕∕ 𝐷 → Cat ∞ which forgets the map refines to a transformation of represented functors Op op∞ → S (again given by forgetting the map to D ⊗ ). Thus we get a lax symmetric monoidal structure onthe functor (Cat ∞ ) ∕∕ D → Cat ∞ and by the description of the tensor product given above it follows that thisfunctor is actually symmetric monoidal as opposed to merely lax symmetric monoidal. (cid:3) Remark.
It is also possible to give a direct construction of the symmetric monoidal ∞ -category (Cat ∞ ) ⊗ ∕∕ D as follows. Let LaxAr ⊆ (Cat ∞ ) ∕Δ be the full subcategory spanned by the cartesian fibrations M → Δ . We note that such a cartesian fibration encodes the data of an functor (Δ ) op → Cat ∞ , corre-sponding to an arrow M → M in Cat ∞ , where M 𝑖 ∶= M × Δ Δ { 𝑖 } is the fibre of M over 𝑖 ∈ {0 , .By definition the morphisms in LaxAr(Cat ∞ ) simply correspond to functors M → M ′ over Δ , and theseare not required to preserve cartesian edges. As a result, morphisms in LaxAr correspond to lax naturaltransformations of arrows, that is, to squares which commute up to a specified transformation. We thenendow
LaxAr with the cartesian monoidal structure
LaxAr × , which is simply given by fibre product over Δ (since cartesian fibrations are closed under fibre products) and we define (Cat ∞ ) ∕∕ D to be the fibre ofthe functor 𝑓 ∶ LaxAr(Cat ∞ ) → Cat ∞ [ 𝑀 → Δ ] ↦ 𝑀 This is in fact the nerve of an ordinary operad which can for example be seen using the theory of dendroidal sets, but we shall notneed this fact here.
ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 85 over D . Since 𝑓 is product preserving and D is an E ∞ -monoid object in Cat ∞ the fibre (Cat ∞ ) ∕∕ D inheritsa symmetric monoidal structure, which we denote by (Cat ∞ ) ⊗ ∕∕ D . We now claim that the underlying ∞ -operad of (Cat ∞ ) ⊗ ∕∕ D represents the same functor described in Property iii), and hence identifies with theconstruction given above. To see this, let us first identify the functor represented by LaxAr × . For O an ∞ -operad , O -algebra objects in LaxAr × correspond to O -monoid, which are simply functors O ⊗ → LaxAr in which certain diagrams are cartesian. But the ∞ -category of functors O ⊗ → LaxAr embeds in the ∞ -category of functors O ⊗ → (Cat ∞ ) ∕Δ , and the latter correspond via unstraightening to cocartesianfibrations 𝑝 ∶ E → O ⊗ equipped with a map E → Δ which sends 𝑝 -cocartesian edges to equivalences. Thecondition that the associated functor O ⊗ → (Cat ∞ ) ∕Δ lands in LaxAr corresponds in these terms to thecondition that for every 𝑥 in O ⊗ the restricted map E 𝑥 → Δ is a cartesian fibration. By the dual of [Lur09a]Corollary 4.3.1.15 this is equivalent to saying that E → Δ is a cartesian fibrations whose cartesian edgesall map to equivalences in O ⊗ . Straightening over Δ , this data is equivalent to that of a map E → E of ∞ -categories over O ⊗ such that the maps E → O ⊗ and E → O ⊗ are cocartesian fibations. The monoidcondition is then equivalent to the condition that for 𝑖 = 0 , the cocartesian fibration E 𝑖 → O ⊗ exhibits E 𝑖 as an O -monoidal ∞ -category and that the functor E → E preserves inert maps. We hence get that thedata of an O -algebra object in LaxAr × is equivalent to that of a pair of O -monoidal ∞ -categories E , E equipped with a lax monoidal functor E → E . We may then conclude that the data of an O -algebra objectin (Cat ∞ ) ∕∕ 𝐷 is equivalent to that of an O -monoidal ∞ -category E equipped with a lax monoidal functor E → D .We will now apply the construction of Lemma 5.2.3 to the category D = S 𝑝 of spectra, equipped withits symmetric monoidal structure given by the tensor product of spectra, and form the pullback along theautoequivalence (−) op ∶ Cat ∞ → Cat ∞ . More precisely we define a symmetric monoidal ∞ -category ( (Cat ∞ ) op∕∕ S 𝑝 ) ⊗ as the pullback ( (Cat ∞ ) op∕∕ S 𝑝 ) ⊗ ( (Cat ∞ ) ∕∕ S 𝑝 ) ⊗ Cat ×∞ Cat ×∞≃(−) op ≃ Objects of this symmetric monoidal ∞ -category are given by pairs ( C , Ϙ ) consisting of an ∞ -category C and a functor Ϙ ∶ C op → S 𝑝 . Morphisms ( C , Ϙ ) ⊗ … ⊗ ( C 𝑛 , Ϙ 𝑛 ) → ( C ′ , Ϙ ′ ) in ( (Cat ∞ ) op∕∕ S 𝑝 ) ⊗ are given by pairs ( 𝑓 , 𝜂 ) of a functor 𝑓 ∶ C ×…× C 𝑛 → C ′ and a natural transformation 𝜂 ∶ Ϙ ⊠ … ⊠ Ϙ 𝑛 ⇒ Ϙ ′ ◦ 𝑓 op . This description also holds for 𝑛 = 0 .5.2.6. Construction.
We define the ∞ -operad (Cat h∞ ) ⊗ as the suboperad of ( (Cat ∞ ) op∕∕ S 𝑝 ) ⊗ spanned bythose objects (( C , Ϙ ) , ..., ( C 𝑛 , Ϙ 𝑛 )) ∈ ( (Cat ∞ ) op∕∕ S 𝑝 ) ⊗ such that each C 𝑖 is stable and each Ϙ 𝑖 is quadratic,and those maps ( 𝑓 , 𝜂 ) as above such that 𝑓 ∶ C × … × C 𝑛 → C ′ is exact in each variable.We note that the underlying ∞ -category of (Cat h∞ ) ⊗ is indeed given by Cat h∞ since it is, essentially bydefinition, the Grothendieck construction of the functor C ↦ Fun q ( C ) which is a subcategory of the laxslice (Cat ∞ ) op∕∕ S 𝑝 .By definition we have that the composed lax monoidal functor (Cat h∞ ) ⊗ → ( (Cat ∞ ) op∕∕ S 𝑝 ) ⊗ → Cat ×∞ factors through the suboperad (Cat h∞ ) ⊗ → (Cat ex∞ ) ⊗ ⊆ Cat ×∞ spanned by the tuples of stable ∞ -categories and tuples of functors which are exact in each variable.5.2.7. Theorem. i) The ∞ -operad (Cat h∞ ) ⊗ is symmetric monoidal with tensor product given by the tensor product ofConstruction 5.1.2 and tensor unit given by ( S 𝑝 f , Ϙ u ) . ii) The resulting map 𝑝 ⊗ ∶ (Cat h∞ ) ⊗ → (Cat ex∞ ) ⊗ is a cocartesian fibration of ∞ -operads and in particulara symmetric monoidal functor.iii) This symmetric monoidal structure on Cat h∞ restricts to a symmetric monoidal structure on the sub-category Cat p∞ .Proof. By design, the hermitian ∞ -category ( C , Ϙ ) ⊗ ( C , Ϙ ) of Construction 5.1.2 corepresents the binarymulti-mapping space functor Mul
Cat h∞ ( ( C , Ϙ ) , ( C , Ϙ ); − ) ∶ Cat h∞ → S . Similarly, the object ( S 𝑝 f , Ϙ u ) corepresents the nullary operations Mul
Cat h∞ ( ∅; − ) ∶ Cat h∞ → S since the morphisms (pt , 𝕊 ) → ( S 𝑝 f , Ϙ u ) exhibits the target as the initial hermitian ∞ -category under (pt , 𝕊 ) by Proposition 4.1.3. These constructions thus produce locally cocartesian lifts for the active arrows ⟨ ⟩ → ⟨ ⟩ and ⟨ ⟩ → ⟨ ⟩ , and more generally for every active arrow 𝛼 ∶ ⟨ 𝑛 ⟩ → ⟨ 𝑚 ⟩ whose fibres are of size ≤ . Since the latter generate all maps in Com , to prove i) it will now suffice to show that these locallycocartesian lifts are cocartesian. Taking into account the decomposition of mapping spaces in ∞ -operadsit will be enough to verify that the induced maps Mul
Cat h∞ ( ( C , Ϙ ) ⊗ ( C , Ϙ ) , ( C , Ϙ ) , … , ( C 𝑛 , Ϙ 𝑛 ); − ) → Mul
Cat h∞ ( ( C , Ϙ ) , … , ( C 𝑛 , Ϙ 𝑛 ); − ) and Mul
Cat h∞ ( ( S 𝑝 f , Ϙ u ) , ( C , Ϙ ) , … , ( C 𝑛 , Ϙ 𝑛 ); − ) → Mul
Cat h∞ ( ( C , Ϙ ) , … , ( C 𝑛 , Ϙ 𝑛 ); − ) are equivalences of functors Cat h∞ → S . We will give the argument for the first assertion, the second workssimilar. The first assertion unwinds to the statement that natural transformations from the functor ( P 𝛽 ! ( Ϙ ⊠ Ϙ ) ) ⊠ Ϙ ⊠ Ϙ 𝑛 ∶ ( C ⊗ C ) × C × … × C 𝑛 → S 𝑝 to any functor ( C ⊗ C ) × C × … × C 𝑛 → S 𝑝 pulled back from a quadratic functor C ⊗ … ⊗ C 𝑛 → S 𝑝 are equivalent to natural transformations from Ϙ ⊠ Ϙ ⊠ Ϙ ⊠ Ϙ 𝑛 ∶ C × C × C × … × C 𝑛 → S 𝑝 to the restriction of the same functor. After fixing objects 𝑥 ∈ C , 𝑥 ∈ C , ... it is certainly true that thespace of natural transformations between the restricted functors along ( C ⊗ C ) → ( C ⊗ C ) × C × … × C 𝑛 ( 𝑥 ⊗ 𝑥 ) ↦ (( 𝑥 ⊗ 𝑥 ) , 𝑥 , ..., 𝑥 𝑛 ) C × C → C × C × C × … × C 𝑛 ( 𝑥 , 𝑥 ) ↦ ( 𝑥 , 𝑥 , 𝑥 , ..., 𝑥 𝑛 ) agree by the universal properties of left Kan extension 𝛽 ! and 2-excisive approximation P . The claim thenfollows since the space of transformations is a limit over these restricted spaces.To see ii) first observe that the operad map 𝑝 ⊗ ∶ (Cat h∞ ) ⊗ → (Cat ex∞ ) ⊗ preserves cocartesian edges, as isvisible by the explicit formula for the tensor product above. In particular, it is a symmetric monoidal functor.Since the functor on underlying ∞ -categories 𝑝 ∶ Cat h∞ → Cat ex∞ is a cocartesian fibration by Corollary 1.4.2it now follows from [Lur09a, Proposition 2.4.2.11] that 𝑝 ⊗ is a locally cocartesian fibration. To show that 𝑝 ⊗ is a cocartesian fibration one needs to additionally verify that for every arrow in 𝛼 ∶ ⟨ 𝑛 ⟩ → ⟨ 𝑚 ⟩ in Com ⊗ , the associated transition functor 𝛼 ! ∶ (Cat h∞ ) ⊗ ⟨ 𝑛 ⟩ → (Cat h∞ ) ⊗ ⟨ 𝑚 ⟩ sends locally 𝑝 ⊗ ⟨ 𝑛 ⟩ -cocartesian edges tolocally 𝑝 ⊗ ⟨ 𝑚 ⟩ -cocartesian edges (indeed, by the explicit description of locally 𝑝 ⊗ -cocartesian edges providedin [Lur09a, Proposition 2.4.2.11], this would imply that these are closed under composition, and are henceall 𝑝 ⊗ -cocartesian by [Lur09a, Proposition 2.4.2.8]). Unwinding the definitions and using Corollary 1.4.2we observe that this statement is straightforward when 𝛼 is inert, and for 𝛼 active amounts to verifying that ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 87 for every commutative square of the form C × ... × C 𝑛 D × ... × D 𝑛 C ⊗ ... ⊗ C 𝑛 D ⊗ ... ⊗ D 𝑛𝑓 × ... × 𝑓 𝑛 𝛽 𝛽𝑓 ⊗...⊗𝑓 𝑛 and every collection of quadratic functors Ϙ 𝑖 ∈ Fun q ( C 𝑖 ) , the natural map ( 𝑓 ⊗ ... ⊗ 𝑓 𝑛 ) ! P 𝛽 ! [ Ϙ ⊠ ... ⊠ Ϙ 𝑛 ] → P 𝛽 ! [ 𝑓 Ϙ ⊠ ... ⊠ 𝑓 𝑛 ! Ϙ 𝑛 ] . Since 𝑓 ⊗ ... ⊗ 𝑓 𝑛 is exact, restriction along it preserves quadratic functors and hence left Kan extensionalong it commutes with P . We may consequently identify the above map with the image under P 𝛽 ! of themap ( 𝑓 ⊠ ... ⊠ 𝑓 𝑛 ) ! [ Ϙ ⊠ ... ⊠ Ϙ 𝑛 ] → 𝑓 Ϙ ⊠ ... ⊠ 𝑓 𝑛 ! Ϙ 𝑛 . The latter is then easily seen to be an equivalence by the pointwise formula for left Kan extension and thefact that tensor products of spectra commute with colimits in each variable.For Assertion iii) about
Cat p∞ , since every equivalence between Poincaré ∞ -categories belongs to Cat p∞ it suffice to check that the tensor product of (Cat h∞ ) ⊗ preserves Poincaré ∞ -categories, which is Corol-lary 5.1.4, and that the tensor unit ( S 𝑝 f , Ϙ u ) is Poincaré, which was already observed in Example 1.2.15. (cid:3) Corollary.
The functors
Pn ∶ Cat p∞ → S and He ∶ Cat h∞ → S admit canonical lax symmetric monoidalstructures.Proof. By Theorem 5.2.7 and Proposition 4.1.3 both of these functors are corepresented by the respectivetensor units, so that the result immediately follows from [Nik16, Corollary 3.10]. (cid:3)
We now point out that both (Cat p∞ ) ⊗ ↪ (Cat h∞ ) ⊗ ↪ ( (Cat ∞ ) op∕∕ S 𝑝 ) ⊗ are subcategory inclusions, and hence induce subcategory inclusions(116) Alg O (Cat p∞ ) ↪ Alg O (Cat h∞ ) ↪ Alg O ((Cat ∞ ) op∕∕ S 𝑝 ) on the level of algebras for every ∞ -operad O . By Lemma 5.2.3 an O -algebra in ( (Cat ∞ ) op∕∕ S 𝑝 ) ⊗ consistsof an O -monoidal ∞ -category C equipped with a lax O -monoidal functor Ϙ ∶ C op → S 𝑝 . By construction,such an O -algebra belongs to the essential image of Alg O (Cat h∞ ) if and only if the following conditionshold:i) for every colour 𝑡 ∈ O the corresponding ∞ -category C 𝑡 is stable and the functor Ϙ 𝑡 ∶ C op 𝑡 → S 𝑝 isquadratic;ii) for every multi-map 𝛼 ∶ { 𝑡 , … , 𝑡 𝑛 } → 𝑡 ′ in O the induced functor 𝛼 ∗ ∶ C 𝑡 × … × C 𝑡 𝑛 → C 𝑡 ′ is exact in each variable.In addition, such an O -algebra is further in the essential image of Alg O (Cat p∞ ) if and only if for every colour 𝑡 ∈ O , the corresponding hermitian ∞ -category ( C 𝑡 , Ϙ 𝑡 ) is Poincaré, and for every multi-map 𝛼 ∶ { 𝑡 , … , 𝑡 𝑛 } → 𝑡 ′ in O and each tuple 𝑥 ∈ C 𝑡 , … , 𝑥 𝑛 ∈ C 𝑡 𝑛 of objects the corresponding hermitian functor 𝛼 ∗ ∶ ( C 𝑡 , Ϙ 𝑡 ) ⊗ … ⊗ ( C 𝑡 𝑛 , Ϙ 𝑡 𝑛 ) → ( C 𝑡 ′ , Ϙ 𝑡 ′ ) is Poincaré.5.2.9. Notation.
For an ∞ -operad O , we will refer to O -algebra objects ( C , Ϙ ) in Cat h∞ ) with respect to thesymmetric monoidal structure of Theorem 5.2.7 as O -monoidal hermitian ∞ -categories , and similarly to O -algebra objects in Cat p∞ as O - monoidal Poincaré ∞ -categories . We will then refer to the hermitian(resp. Poincaré) structure Ϙ as an O -monoidal hermitian (resp. Poincaré) structure. When O = Com wewill replace as customary the term O -monoidal by symmetric monoidal . Day convolution of hermitian structures.
In this section we will analyse in more explicit termssymmetric monoidal hermitian and Poincaré structures over a fixed stably symmetric monoidal ∞ -category C , and show that they can be encoded in terms of their linear and bilinear parts. To this end, recall that fortwo symmetric monoidal ∞ -categories E , D , there is an associated ∞ -operad Fun( E , D ) ⊗ with underlying ∞ -category Fun( E , D ) , called the Day convolution ∞ -operad, see [Gla16], [Lur17, §2.2.6], and [Day70]for the classical counterpart. It is characterized by the following universal property: there is an evaluationmap of ∞ -operads ev ∶ E ⊗ × Com ⊗ Fun( E , D ) ⊗ → D ⊗ , refining the usual evaluation map, such that for every ∞ -operad O ⊗ , the composed map Alg O (Fun( E , D )) → Alg E × Com O ( E × Com
Fun( E , D )) ev ∗ ←←←←←←←←←←←←←→ Alg E × Com O ( D ) is an equivalence of ∞ -categories. Here, we may identify Alg E × Com O ( D ) ≃ Alg E × Com O ∕ O ( D × Com O ) withthe ∞ -category of lax O -monoidal functors from E to D , both considered as O -monoidal ∞ -categories bypulling back along O → Com . In particular, for O ⊗ = Com ⊗ we get an equivalence between commutativealgebra objects in Fun( E , D ) and lax monoidal functors E → D . On the other hand, taking O ⊗ to bethe underlying ∞ -operad of a symmetric monoidal ∞ -category C , we get that lax monoidal functors C → Fun( E , D ) correspond to lax monoidal functors C × E → D .By [Lur17, Corollary 2.2.6.12] the multi-mapping space in Fun( E , D ) ⊗ from a collection { 𝜑 𝑖 ∶ E → D } 𝑖 =1 ,...,𝑛 to 𝜓 ∶ E → D is given by the space of natural transformations in the square E × … × E D × … ×
DE D . ⊗ 𝜑 ×…× 𝜑 𝑛 ⊗𝜓 If E is small, D admits small colimits, and the tensor product in D preserves small colimits in each variable,then Fun( E , D ) ⊗ is a symmetric monoidal ∞ -category, with tensor product 𝜑 ⊗ 𝜑 ∶ E → D given by theleft Kan extension of 𝜑 ⊠ 𝜑 ∶ E × E → D × D → D along E × E → E (see [Lur17, Proposition 2.2.6.16]).Furthermore, in this case Fun( E , D ) admits small colimits and the Day convolution product preserves smallcolimits in each variable [Gla16, Lemma 2.13].5.3.1. Remark.
Comparing universal properties we see that if C , E are small symmetric monoidal ∞ -categories and D is an ∞ -category with small colimits which is endowed with a symmetric monoidal struc-ture which preserves colimits in each variable, then there is a natural equivalence of symmetric monoidal ∞ -categories Fun( C × E , D ) ≃ Fun( C , Fun( E , D )) , where the left hand side is equipped with the Day convolution structure, and the right hand side with thetwice nested Day convolution structure.We now relate this concept to the constructions we made in the previous sections. Given a small sym-metric monoidal ∞ -category C , consider the pullback square of ∞ -operads Fun( C op , S 𝑝 ) ⊗ E ∞ ((Cat ∞ ) op∕∕ S 𝑝 ) ⊗ Cat ⊗ ∞ C refining the pullback square of ∞ -categories which identifies Fun( C op , S 𝑝 ) as the fibre of the cartesianfibration (Cat ∞ ) op∕∕ D → Cat ∞ over C ∈ Cat ∞ . We then have the following:5.3.2. Lemma.
Let C be a small stably symmetric monoidal ∞ -category. Then the ∞ -operad Fun( C op , S 𝑝 ) is a symmetric monoidal ∞ -category. Furthermore, we may identify the symmetric monoidal structure on Fun( C op , S 𝑝 ) with Day convolution . In particular, for an ∞ -operad O , the data of an O -algebra structureon Ϙ ∈ Fun( C op , S 𝑝 ) corresponds, naturally in O , to that of a lax O -monoidal refinement of Ϙ , where C and S 𝑝 are considered as O -monoidal ∞ -categories by pulling back along O → Com . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 89 Remark.
It follows from Lemma 5.3.2 that the tensor product of Ϙ , Ϙ ′ ∈ Fun( C op , S 𝑝 ) is given by theleft Kan extension of the functor Ϙ ⊠ Ϙ ′ ∶ C op × C op → S 𝑝 along C op × C op → C op . Proof.
Since ( (Cat ∞ ) op∕∕ D ) ⊗ → Cat ×∞ is a cocartesian fibration it follows that Fun( C op , S 𝑝 ) ⊗ → E ∞ isalso a cocartesian fibration, so that Fun( C op , S 𝑝 ) ⊗ is a symmetric monoidal ∞ -category. By the descriptionof algebra objects in ( (Cat ∞ ) op∕∕ D ) ⊗ given in Lemma 5.2.3iii) and the fact that taking algebra objects iscompatible with limits of ∞ -operads we get that O -algebras in Fun( C op , S 𝑝 ) ⊗ are given by lax O -monoidalfunctors C op → S 𝑝 , and this description is natural in O , which allows us to identify the monoidal structureon Fun( C op , S 𝑝 ) with Day convolution. (cid:3) We now wish to understand in similar terms the hermitian context, in which one considers quadratic functors C op → S 𝑝 . For this, let us consider the setting of Day convolution when the target D is a presentablysymmetric monoidal ∞ -category, that is, it is presentable and the tensor product preserves small colimits ineach variable. In this case, for a small ∞ -category E , the functor category Fun( E , D ) is again presentablysymmetric monoidal with respect to Day convolution. Suppose now that I = { 𝑝 𝛼 ∶ 𝐾 ⊳𝛼 → E } is a smallcollection of diagrams in E . Let Fun I ( E , D ) ⊆ Fun( E , D ) be the full subcategory spanned by those functors E → D which send every diagram in I to a limit diagram. We claim that Fun I ( E , D ) is an accessiblelocalisation of Fun( E , D ) . Indeed, choose a small set of generators 𝑇 for D , and for 𝛼 ∈ I let us denote by 𝑝 𝛼 ∶= ( 𝑝 𝛼 ) | 𝐾 𝛼 the corresponding restriction. Then for every 𝑎 ∈ 𝑇 and 𝛼 ∈ I the diagram 𝑝 𝛼 induces amap 𝑠 𝛼,𝑎 ∶ colim 𝐾 𝛼 𝑗𝑝 op 𝛼 ⊗ 𝑎 → 𝑗𝑝 𝛼 (∗) ⊗ 𝑎, where 𝑗 ∶ E op → Fun( E , S ) is the Yoneda embedding and ⊗ denotes the canonical bifunctor Fun( E , S ) × D → Fun( E , D ) induced levelwise by the tensoring of the presentable ∞ -category D over spaces. Let 𝑆 = { 𝑠 𝛼,𝑎 } 𝛼 ∈ I ,𝑎 ∈ 𝑇 . We may then identify Fun I ( E , D ) ⊆ Fun( E , D ) with the full subcategory spanned bythe 𝑆 -local objects. Since 𝑆 is a set it follows from [Lur09a, Proposition 5.5.4.15] that Fun I ( E , D ) is alsopresentable and its inclusion admits a left adjoint 𝐿 ∶ Fun( E , D ) → Fun I ( E , D ) . We then have that 𝐿 exhibits Fun I ( E , D ) as the localisation of Fun( E , D ) by the set of maps 𝑆 . Since 𝐿 isa left adjoint functor we also refer to it as a left Bousfield localisation .We now consider the above setup in the context of Day convolution. Recall that in general, a left Bous-field localisation functor 𝐿 ∶ A → B , with fully-faithful right adjoint 𝑅 ∶ B ↪ A , is said to be compatible with respect to a given symmetric monoidal structure A ⊗ on A , if for every 𝑓 ∶ 𝑥 → 𝑦 in A such that 𝐿 ( 𝑓 ) is an equivalence, and every 𝑧 ∈ A , the map 𝐿 ( 𝑧 ⊗ 𝑓 ) is again an equivalence. By [Lur17, Propo-sition 2.2.1.9] the ∞ -category B then inherits a symmetric monoidal structure B ⊗ such that 𝐿 refines toa symmetric monoidal functor 𝐿 ⊗ ∶ A ⊗ → B ⊗ and 𝑅 refines to a fully-faithful inclusion of ∞ -operads 𝑅 ⊗ ∶ B ⊗ → A ⊗ . In addition, the symmetric monoidal functor 𝐿 ⊗ exhibits B ⊗ as universal among sym-metric monoidal ∞ -categories receiving a symmetric monoidal functor from A which inverts the maps in-verted by 𝐿 . Indeed, the symmetric monoidal functor 𝐿 ⊗ must factor through such a symmetric monoidallocalisation by [Lur17, Proposition 4.1.7.4], and the resulting comparison between the two symmetricmonoidal ∞ -categories under A ⊗ is an equivalence since it is an equivalence on the level of underlying ∞ -categories. We will consequently refer to 𝐿 ⊗ as a symmetric monoidal Bousfield localisation .5.3.4. Lemma.
Given E , D and I be as above, let us denote by Fun I ( E , D ) ⊗ ⊆ Fun( E , D ) ⊗ the full subop-erad spanned by Fun I ( E , D ) . Suppose that I is closed under post-composition with 𝑥 ⊗ (−) ∶ E → E forevery 𝑥 ∈ E , that is, if 𝑝 𝛼 ∈ I then ( 𝑥 ⊗ (−)) ◦ 𝑝 𝛼 is also in I . Then the left Bousfield localisation functor 𝐿 is compatible with Day convolution, and hence extends to a symmetric monoidal localisation Bousfieldlocalisation functor 𝐿 ⊗ ∶ Fun( E , D ) ⊗ → Fun I ( E , D ) ⊗ . In particular,
Fun I ( E , D ) ⊗ inherits a symmetric monoidal ∞ -category, universally obtained from Fun( E , D ) by inverting the set of maps 𝑆 . Remark.
In the situation of Lemma 5.3.4, the tensor product ⊗ I in Fun I ( E , D ) ⊗ can be expressed interms of the tensor product ⊗ Day in Fun( E , D ) ⊗ and the localisation functor. Explicitly, the tensor productof 𝜑, 𝜓 ∈ Fun I ( E , D ) is given by 𝐿 ( 𝜑 ⊗ Day 𝜓 ) . Remark.
In the situation of Lemma 5.3.4, since Day convolution preserves small colimits in each vari-able and 𝐿 preserves colimits it follows from Remark 5.3.5 that the localized tensor product on Fun I ( E , D ) also preserves small colimits in each variable. Proof of Lemma 5.3.4.
Let 𝑊 be the collection of all maps in Fun( E , D ) whose image under 𝐿 is an equiv-alence. We need to show that 𝑊 is closed under Day convolution against objects, that is, to show that for 𝜏 ∈ 𝑊 and 𝜑 ∈ Fun( E , D ) , we have that 𝜏 ⊗ Day 𝜑 is again in 𝑊 . By [Lur09a, Proposition 5.5.4.15] we havethat 𝑊 is generated as a strongly saturated class by the set 𝑆 = { 𝑠 𝛼,𝑎 } above, and so it will suffice to showthat 𝑆 is closed under Day convolution against 𝜑 ∈ Fun( E , D ) . Since Day convolution preserves colimitsin each variable we may as well check this for a generating set of Fun( E , D ) . Such a generating set is given,for example, by the functors of the form 𝑗 ( 𝑥 ) ⊗ 𝑎 , for 𝑥 ∈ E and 𝑎 ∈ 𝑇 , where 𝑗 is the Yoneda embeddingas above. Since 𝑗 is symmetric monoidal ([Gla16, §3] or [Lur17, Corollary 4.8.1.12 and Remark 4.8.1.13])and using again that Day convolution preserves colimits in each variable, the closure of 𝑆 under Day con-volution with these generators now follows from our condition that I is closed under post-composition with 𝑥 ⊗ (−) for every 𝑥 ∈ E . (cid:3) We now apply the above ideas in the context of quadratic functors. For a given stable ∞ -category C , the ∞ -category Fun q ( C ) sits in a diagram(117) Fun q ( C ) Fun( C op , S 𝑝 ) Δ Cat h∞ ( (Cat ∞ ) op∕∕ D ) ⊗ Cat ∞ . C in which all squares are pullbacks. We then refine this diagram to a diagram of ∞ -operads(118) Fun q ( C ) ⊗ Fun( C op , S 𝑝 ) ⊗ E ∞ (Cat h∞ ) ⊗ ( (Cat ∞ ) op∕∕ S 𝑝 ) ⊗ Cat ⊗ ∞ . C by extending the lower and right part and defining the the ∞ -operads Fun( C op , S 𝑝 ) ⊗ , Fun q ( C ) ⊗ as therespective pullbacks. Since taking algebra objects is compatible with limits in the target it follows that foran ∞ -operad O we have a pullback square Alg O (Fun q ( C )) { C }Alg O (Cat h∞ ) Alg O (Cat ∞ ) , and so O -algebras in Fun q ( C ) correspond to O -monoidal hermitian ∞ -categories refining the underlying O -monoidal ∞ -category of C . On the other hand, since Fun q ( C ) ⊆ Fun( C op , S 𝑝 ) is a full inclusion and themonoidal structure on the latter identifies with Day convolution by Lemma 5.3.2, we may identify the dataof an O -algebra structure on a given quadratic functor Ϙ ∶ C op → S 𝑝 with that of a lax O -monoidal structure.We will refer to such Ϙ as O -monoidal hermitian refinements of C .5.3.7. Corollary.
Let C be a small stably symmetric monoidal ∞ -category. Then the full inclusion of ∞ -operads Fun q ( C ) ⊗ ⊆ Fun( C op , S 𝑝 ) ⊗ admits a symmetric monoidal left adjoint exhibiting Fun q ( C ) ⊗ as a symmetric monoidal localisation of Fun( C op , S 𝑝 ) ⊗ . In particular, the ∞ -operad Fun q ( C ) ⊗ is a symmetric monoidal ∞ -category. Remark.
It follows from Corollary 5.3.7 that the tensor product of Ϙ , Ϙ ′ ∈ Fun q ( C ) is given by theirDay convolution followed by an application of the left adjoint to the inclusion Fun q ( C ) ⊆ Fun( C op , S 𝑝 ) .Since Ϙ , Ϙ ′ are in particular reduced the result of this left Kan extension is reduced and hence the left adjointin question can be implemented via the -excisive approximation of Construction 1.1.26. Comparing this ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 91 description with Construction 5.1.2 we may equivalently describe the tensor product of Ϙ , Ϙ ′ ∈ Fun q ( C ) asthe left Kan extension of the quadratic functor Ϙ ⊗ Ϙ ′ ∈ Fun q ( C ⊗ C ) along C op ⊗ C op → C op . Proof of Corollary 5.3.7.
This is a particular case of Lemma 5.3.4 since the condition of being a quadraticfunctor is equivalent to that of being reduced and 2-excisive (Proposition 1.1.13) which in turn can beformulated as sending a suitable set of diagrams in C op (consisting of the constant diagram on and allstrongly exact -cubes) to limit diagrams. (cid:3) Our next goal is to understand the monoidal structure on
Fun q ( C ) ⊗ in terms of decomposition into linearand bilinear components, as described in Corollary 5.3.14 below. Towards this end, we begin with thefollowing direct application of Lemma 5.3.4:5.3.9. Corollary.
Let C be a small stably symmetric monoidal ∞ -category C . Then the following holds:i) Then full suboperad Fun ex ( C op , S 𝑝 ) ⊗ ⊆ Fun( C op , S 𝑝 ) ⊗ is a symmetric monoidal localisation of theDay convolution product on Fun( C op , S 𝑝 ) .ii) The full suboperad Fun b ( C ) ⊗ ⊆ Fun( C op × C op , S 𝑝 ) ⊗ is a symmetric monoidal localisation of the Dayconvolution product on Fun( C op × C op , S 𝑝 ) .Proof. In case i) we apply Lemma 5.3.4 with respect to the collection of diagrams in C op consisting of theconstant diagram on and all exact squares. For ii) we apply Lemma 5.3.4 with respect to the collection ofdiagrams in C op × C op consisting of the constant diagrams on ( 𝑥, and (0 , 𝑥 ) for all 𝑥 ∈ C op and all squaresof the form { 𝑥 } × 𝜎 and 𝜎 × { 𝑥 } where 𝜎 ∶ Δ × Δ → C op is an exact square. (cid:3) Remark.
As in Remark 5.3.5, the tensor product of L , L ′ ∈ Fun ex ( C op , S 𝑝 ) is obtained by taking theirDay convolution L ⊗ Day L ′ and applying to it the left adjoint to the inclusion Fun ex ( C op , S 𝑝 ) ⊆ Fun( C op , S 𝑝 ) .Similarly, in ii) the tensor product of B , B ′ ∈ Fun b ( C ) is obtained by taking their Day convolution B ⊗ Day B ′ as functors C op × C op → S 𝑝 , and then applying to it the left adjoint to the inclusion Fun b ( C ) ⊆ Fun( C op × C op , S 𝑝 ) .5.3.11. Remark.
It follows from Remark 5.3.6 that the tensor products on
Fun q ( C ) , Fun ex ( C op , S 𝑝 ) and Fun b ( C ) of Corollaries 5.3.7 and 5.3.9 all preserve small colimits in each variable.Combining Corollary 5.3.7 and Corollary 5.3.9 we obtain5.3.12. Corollary.
The linear part functor L (−) refines to a symmetric monoidal localisation functor L ⊗ (−) ∶ Fun q ( C ) → Fun ex ( C op , S 𝑝 ) ⊗ . We would like to establish a similar property for the bilinear part functor. To this end, consider thecommutative diagram(119)
Fun b ( C ) BiFun( C ) Fun ∗ ( C op × C op , S 𝑝 ) Fun( C op × C op , S 𝑝 )Fun q ( C ) Fun ∗ ( C ) Fun ∗ ( C ) Fun( C ) in which the horizontal arrows are the relevant inclusions and the vertical arrows are all induced by restric-tion along the diagonal C op → C op × C op . An application of Lemma 5.3.4 shows that all ∞ -categoriesin this diagram inherits a symmetric monoidal structure from the Day convolution product on the functor ∞ -categories in the right most column, and such that all horizontal inclusions admit symmertic monoidalleft adjoints, which are also localisation functors. In addition, the right most vertical functor, given by re-striction along the diagonal C op → C op × C op , admits a left adjoint via the corresponding left Kan extension.Since the diagonal is symmetric monoidal so it the corresponding left Kan extension. In addition, sincethe diagonal admits itself a two sided adjoint via the direct sum functor ⊕ ∶ C op × C op → C op this left Kanextension is just given restriction along ⊕ . By the universal property of all the appearing localisation itthen follows that the arrows in (119) admit symmetric monoidal left adjoints and we consequently obtain a diagram of symmetric monoidal ∞ -categories and symmetric monoidal functors(120) Fun b ( C ) ⊗ BiFun( C ) ⊗ Fun ∗ ( C op × C op , S 𝑝 ) ⊗ Fun( C op × C op , S 𝑝 ) ⊗ Fun q ( C ) ⊗ Fun ∗ ( C ) ⊗ Fun ∗ ( C ) ⊗ Fun( C ) ⊗ B (−) in which all horizontal functors are symmetric monoidal localisations and all vertical functors are givenby restriction along ⊕ ∶ C op × C op → C op followed by the projection to the relevant full subcategory of Fun ∗ ( C op × C op , S 𝑝 ) . In particular, the top middle horizontal arrow in (120) is the bi-reduction functor ofLemma 1.1.3 and the second from the left vertical functor in (120) is the cross-effect functor of Defini-tion 1.1.4. Since the cross-effect of any quadratic functor is bilinear this formula also holds for the left mostvertical arrow, that is, we may identify it with the bilinear part functor B (−) . Arguing as in the proof ofLemma 1.1.9 we see that this symmetric monoidal refinement of B (−) is C -equivariant with respect to theflip action on Fun b ( C ) ⊗ and the trivial action on Fun q ( C ) , and consequently refines to a symmetric monoidalfunctor(121) B ⊗ (−) ∶ Fun q ( C ) ⊗ → Fun s ( C ) ⊗ , where the target is endowed with the symmetric monoidal structure obtained by taking the C -fixed pointsof Fun b ( C ) ⊗ in the ∞ -category of symmetric monoidal ∞ -categories.Now recall from that the symmetric bilinear part functor B (−) ∶ Fun q ( C ) → Fun s ( C ) is also a left Bousfield localisation functor, since it admits a fully-faithful right adjoint (in fact, it admitsfully-faithful adjoints from both sides). We now verify that the same holds for its symmetric monoidalrefinement just constructed:5.3.13. Lemma.
The symmetric monoidal functor (121) is a symmetric monoidal localisation functor.Proof.
It will suffice to show that the localisation functor B (−) is compatible with the symmetric monoidalstructure (the resulting comparison between the corresponding localisation and B ⊗ (−) is then necessarily anequivalence since it is an equivalence on the level of underlying ∞ -categories). Now the maps that are sentto equivalences by B (−) are exactly the maps whose cofibre is exact. Since left Kan extensions along exactfunctors preserve exact functors (Lemma 1.4.1) it will suffice to show that if Ϙ , Ϙ ′ ∈ Fun q ( C ) are such that Ϙ ′ is exact then Ϙ ⊗ Ϙ ′ ∈ Fun q ( C ⊗ C ) (defined as in Construction 5.1.2) is exact. Indeed, its bilinear partis B Ϙ ⊗ B Ϙ ′ = 0 by Proposition 5.1.3. (cid:3) We now combine the symmetric monoidal structures of Corollaries 5.3.7 and 5.3.9 to enhance the clas-sification of hermitian structures of Corollary 1.3.12 to a symmetric monoidal one:5.3.14.
Corollary (Monoidal classification of hermitian structures) . The pullback square of Corollary 1.3.12refines to a pullback square in Op ∞ (122) Fun q ( C ) ⊗ Ar(Fun ex ( C op , S 𝑝 )) ⊗ Fun s ( C ) ⊗ Fun ex ( C op , S 𝑝 ) ⊗ , 𝜏 ⊗ B ⊗ (−) in which all corners are symmetric monoidal ∞ -categories and the vertical functors are symmetric monoidal,and lax symmetric monoidal functors. Here the left vertical arrow is (121) , the bottom right corner is en-dowed with the symmetric monoidal structure of corollary 5.3.9, and the arrow category carries the corre-sponding pointwise symmetric monoidal structure. Concretely, this means that for a given ∞ -operad O , the data of an O -monoidal hermitian refinementof a stably symmetric monoidal C can be identified with that of a triple B , L , 𝛼 ∶ L ⇒ [B Δ ] hC ) where B ∶ C op × C op → S 𝑝 is a lax O -monoidal symmetric bilinear functor, L ∶ C op → S 𝑝 is a lax O -monoidalexact functor, and L ⇒ [B Δ ] tC is a lax O -monoidal transformation. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 93 Proof of Corollary 5.3.14.
Recall from Remark 1.3.7 that the pair of fully-faithful exact functors
Fun s ( C ) Ϙ s(−) ←←←←←←←←←←←←←←←←→ Fun q ( C ) ← Fun ex ( C op , S 𝑝 ) form a stable recollement . We may then invoke the theory of monoidal recollements as developed in [QS19]to promote our classification of hermitian structures to a monoidal one. More precisely, combining Corol-lary 5.3.12 and Lemma 5.3.13 it follows that the above stable recollement is symmetric monoidal in thesense of [QS19, Definition 1.19]. The symmetric monoidal refinement of the classification square is then aconsequence of [QS19, Proposition 1.26]. (cid:3) Recall from Definition 1.2.8 that for a given stable ∞ -category C , let us denote by Fun p ( C ) is the non-full subcategory of Fun q ( C ) consisting of the the Poincaré structures on C and the duality preserving naturaltransformations between them. We the refine this inclusion to a (non-full) inclusion of ∞ -operads by ex-tending (118) to a commutative diagram of ∞ -operads(123) Fun p ( C ) ⊗ Fun q ( C ) ⊗ Fun( C op , S 𝑝 ) ⊗ E ∞ (Cat p∞ ) ⊗ (Cat h∞ ) ⊗ ( (Cat ∞ ) op∕∕ S 𝑝 ) ⊗ Cat ⊗ ∞ . C in which all squares are pullbacks. For an ∞ -operad O , the data of an O -algebra structure on a given Poincaréstructure Ϙ ∈ Fun p ( C ) then corresponds to that of an O -monoidal structure on ( C , Ϙ ) ∈ Cat p∞ which refinesthe underlying O -monoidal structure of C .5.3.15. Construction.
Applying Remark 5.3.1 and the uniqueness of localized symmetric monoidal struc-tures we deduce that the equivalence
Fun b ( C ) ≃ Fun ex ( C op , Fun ex ( C op , S 𝑝 )) = Fun ex ( C op , Ind( C )) , refines to a symmetric monoidal equivalence Fun b ( C ) ⊗ ≃ Fun ex ( C op , Ind( C )) ⊗ , where the the domain is endowed with the localized Day convolution product and the target with the cor-responding nested localized Day convolution. Passing to non-degenerate bilinear functors we obtain anequivalence(124) Fun nb ( C ) ⊗ ≃ Fun ex ( C op , C ) ⊗ between the full suboperad of Fun b ( C ) ⊗ spanned by the non-degenerate bilinear functors and the full sub-operad of the Day convolution ∞ -operad C op → C spanned by the exact functors. We note that the lattermay fail to be a symmetric monoidal ∞ -category since C generally does not admit small colimits. To avoidconfusion, let us try to make the equivalence (124) more explicit. Given non-degenerate bilinear functors B , ..., B 𝑛 and B ′ , a multi-map {B , ..., B 𝑛 } → B ′ in Fun nb ( C ) ⊗ is given by a natural transformation(125) ( 𝑚 op × 𝑚 op ) ! [B ⊠ … ⊠ B 𝑛 ] = ( ̃𝑚 op × ̃𝑚 op ) ! [B ⊗ … ⊗ B 𝑛 ] → B ′ , where 𝑚 ∶ C × … × C → C is the symmetric monoidal product of C , and ̃𝑚 ∶ C ⊗ … ⊗ C → C is the exactfunctor induced by it. Here we point out that the domain in (125) is generally not non-degenerate, whichis why Fun nb ( C ) ⊗ is usually not a symmetric monoidal ∞ -category. By adjunction we may equivalentlydescribe the data of (125) via a natural transformation in the square(126) [ C op ⊗ … ⊗ C op ] × [ C op ⊗ … ⊗ C op ] S 𝑝 ⊗ … ⊗ S 𝑝 C op × C op S 𝑝 . 𝑚 op × 𝑚 op (B , … , B 𝑛 ) ⊗𝛽 B ′ By Corollary 5.1.4 (and its proof), the bilinear functor B ⊗ … ⊗ B 𝑛 is non-degenerate and represented by D B ⊗ … D B 𝑛 ∶ C op ⊗ … ⊗ C op → C ⊗ … ⊗ C , where D B 𝑖 ∶ C op → C is the exact functor representing thenon-degenerate bilinear functor B 𝑖 . The corresponding multi-map on the right hand side of (124) is thengiven by the map 𝑚 ! [D B ⊠ … ⊠ D B 𝑛 ] = ̃𝑚 ! [D ⊗ … ⊗ D 𝑛 ] → D B ′ induced by to the natural transformation in the square(127) C op ⊗ … ⊗ C op C ⊗ … ⊗ CC op C . 𝑚 op D ⊗ … ⊗ D 𝑛 𝑚𝜏 D ′ associated to (126) by Lemma (1.2.4).We now define Fun pb ( C ) ⊗ ↪ Fun nb ( C ) ⊗ to be the non-full suboperad spanned by the perfect bilinearfunctors and those multi-maps between whose corresponding natural transformations (127) is an equiva-lence. Similarly, we define Fun ps ( C ) ⊗ to be the suboperad of the symmetric monoidal ∞ -category Fun s ( C ) ⊗ =[Fun b ( C ) ⊗ ] hC sitting in the pullback square Fun ps ( C ) ⊗ Fun s ( C ) ⊗ Fun pb ( C ) Fun b ( C ) ⊗ . We note that by construction, the underlying ∞ -categories of Fun pb ( C ) ⊗ and Fun ps ( C ) ⊗ are the subcat-egories Fun pb ( C ) and Fun ps ( C ) of Fun b ( C ) and Fun s ( C ) respectively, spanned by the perfect (symmetric)bilinear functors respectively and duality preserving (symmetric) transformations between them.5.3.16. Lemma.
The pullback square
Fun p ( C ) Fun q ( C )Fun pb ( C ) Fun b ( C ) , refines to a pullback square of ∞ -operads Fun p ( C ) ⊗ Fun q ( C ) ⊗ Fun pb ( C ) ⊗ Fun b ( C ) ⊗ , Proof.
Since
Fun pb ( C ) ⊗ ↪ Fun b ( C ) ⊗ is a suboperad this follows from the fact that a hermitian functor of ( ̃𝑚, 𝜂 ) ∶ ( C , Ϙ ) ⊗ … ⊗ ( C , Ϙ 𝑛 ) → ( C , Ϙ ′ ) lying over the monoidal product ̃𝑚 ∶ C ⊗ … ⊗ C → C of C , is Poincaré exactly when the correspondingnatural transformation ̃𝑚 (D Ϙ ⊗ … ⊗ D Ϙ 𝑛 ) → D Ϙ ′ ̃𝑚 op is an equivalence, where we have used the identification of the duality on a Poincaré tensor product ofCorollary 5.1.4. (cid:3) Corollary (Monoidal classification of Poincaré structures) . The pullback square of Corollary 1.3.13refines to a pullback square in Op ∞ (128) Fun p ( C ) ⊗ Ar(Fun ex ( C op , S 𝑝 )) ⊗ Fun ps ( C ) ⊗ Fun ex ( C op , S 𝑝 ) ⊗ , 𝜏 ⊗ B ⊗ (−) in which all corners are symmetric monoidal ∞ -categories and the vertical functors are symmetric monoidal,and lax symmetric monoidal functors. Corollary.
Let C be a stably symmetric monoidal ∞ -category, O an ∞ -operad and Ϙ ∶ C → S 𝑝 alax O -monoidal quadratic functor, so that B Ϙ and L Ϙ inherit lax O -monoidal structures by Corollary 5.3.14.Then the corresponding O -monoidal hermitian ∞ -category ( C , Ϙ ) is O -monoidal Poincaré if and only if thefollowing holds: ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 95 i) The underlying hermitian ∞ -category ( C , Ϙ ) is Poincaré.ii) The lax O -monoidal structure on D Ϙ ∶ C op → C induced from that of B Ϙ via Construction 5.3.15 is(strongly) O -monoidal. In particular, D Ϙ is an equivalence of O -monoidal ∞ -categories. Concretely, Corollary 5.3.18 implies that for a stably symmetric monoidal ∞ -category C , providing acompatible O -monoidal Poincaré structure on C is equivalent to providing a self-dual equivalence of O -monoidal ∞ -categories D ∶ C op ≃ ←←←←←←←→ C , a lax O -monoidal exact functor L ∶ C op → S 𝑝 , and a lax O -monoidaltransformation L(−) ⇒ [hom(− , D(−))] tC . Proof of Corollary 5.3.18.
In light of Corollary 5.3.17 it will suffice to show that for a lax O -monoidal bilin-ear form B , the corresponding O -algebra object in Fun b ( C ) ⊗ lies in the non-full subcategoy Alg O (Fun pb ( C )) ⊆ Alg O (Fun b ( C )) if and only if B is perfect the associated lax O -monoidal structure on D B is strongly laxmonoidal. To see this, note that under the equivalence (124) of Construction 5.3.15, the suboperad Fun pb ( C ) ⊗ corresponds to the suboperad Fun ◦ ( C op , C ) ⊗ ⊆ Fun ex ( C op , C ) spanned the colours corresponding to equiv-alences D ∶ C op ≃ ←←←←←←←→ C and the multi-maps {D , … D 𝑛 } → D ′ corresponding to natural transformations ̃𝑚 (D ⊗ … ⊗ D 𝑛 ) ⇒ D ′ ̃𝑚 op . which are equivalences. We now observe that by the universal property of the tensor product of stable ∞ -categories, such a natural transformation of exact functors is an equivalence if and only if it is sent to anequivalence of multi-linear maps after pre-composing both sides with the functor C × … × C → C ⊗ … ⊗ C . We may hence equivalently defined this suboperad
Fun ex ( C op , C ) to be spanned by the colours correspondingto equivalences and the multi-maps {D , … D 𝑛 } → D ′ corresponding to natural equivalences 𝑚 (D ⊠ … ⊠ D 𝑛 ) ⇒ D ′ 𝑚 op , where 𝑚 is the composite C × … × C → C ⊗ … ⊗ C ≃ ←←←←←←←→ C . It then follows directly from the definitions thatalgebra objects in this suboperad correspond to equivalences D ∶ C op → C equipped with strong monoidalstructures, as desired. (cid:3) Examples.
In this section we want to give some examples of algebras and modules in
Cat p∞ .5.4.1. Example.
Consider the Poincaré ∞ -category ( S 𝑝 f , Ϙ u ) . It is the tensor unit of Cat p∞ thus admits thestructure of a commutative algebra. This algebra structure is given by the usual smash product on S 𝑝 f andthe canonical commutative algebra structure on the universal quadratic functor Ϙ u induced by its being theimage under the symmetric monoidal left adjoint Fun(( S 𝑝 f ) op , S ) → Fun q (( S 𝑝 f ) op , S ) of the Yoneda imageof 𝕊 ∈ S 𝑝 f .5.4.2. Example.
Since the forgetful functor 𝑈 ∶ Cat p∞ → Cat ex∞ is symmetric monoidal by Theorem 5.2.7every symmetric monoidal Poincaré ∞ -category ( C , Ϙ ) yields a stably symmetric monoidal ∞ -category C upon forgetting Ϙ . On the other direction, we will show in §7.2 that the formation of hyperbolic categories C ↦ Hyp( C ) (see §2.2) gives both a left and a right adjoint to 𝑈 , and hence Hyp is both lax and oplaxmonoidal, see Remark 7.2.22 below. It then follows in particular that if C is a stably symmetric monoidal ∞ -category then Hyp( C ) inherits a symmetric monoidal structure. Unwinding the definitions, this structureis given explicitly by the operation ( 𝑥, 𝑦 ) ⊗ ( 𝑥 ′ , 𝑦 ′ ) = ( 𝑥 ⊗ 𝑥 ′ , 𝑦 ⊗ 𝑦 ′ ) with the canonical lax monoidalstructure on Ϙ hyp ( 𝑥, 𝑦 ) = hom C ( 𝑥, 𝑦 ) . Similarly, if ( C , Ϙ ) is a symmetric monoidal Poincaré ∞ -categorythen the Poincaré functor f gt ∶ ( C , Ϙ ) → Hyp( C ) sending 𝑥 to ( 𝑥, D 𝑥 ) (see §2.2) is canonically a symmetric monoidal Poincaré functor, since f gt acts as theunit of the symmetric monoidal adjunction 𝑈 ⊣
Hyp , see Remark 7.2.21. On the other hand, the Poincaréfunctor hyp ∶ Hyp( C ) → ( C , Ϙ ) sending ( 𝑥, 𝑦 ) to 𝑥 ⊕ D 𝑦 (see §2.2) is not symmetric monoidal in general,though we will show below that it is a morphism of ( C , Ϙ ) -module objects in Cat p∞ .5.4.3. Example.
We will show below (see Remark 7.3.18) that the association ( C , Ϙ ) ↦ Ar( C , Ϙ ) from Cat p∞ to Cat p∞ (see §2.3) refines to a lax symmetric monoidal functor. It then follows that if ( C , Ϙ ) is a Poincaré ∞ -category then Ar( C , Ϙ ) inherits a symmetric monoidal structure. On the level of underlying objects this is given simply by the levelwise product of arrows: [ 𝑥 → 𝑦 ] ⊗ [ 𝑥 ′ → 𝑦 ′ ] = [ 𝑥 ⊗ 𝑥 ′ → 𝑦 ⊗ 𝑦 ′ ] . The Poincaréfunctor id ∶ ( C , Ϙ ) → Ar( C , Ϙ ) 𝑥 ↦ [ 𝑥 = 𝑥 ] then inherits the structure of a symmetric monoidal Poincaré functor by its role as a unit of a symmetricmonoidal adjunction, see Remark 7.3.17.We now consider the case of modules over ring spectra studied in §3. For this, we will specialize to thecase where 𝐴 is an E ∞ -ring spectrum, to which we will refer simply as a commutative ring spectrum . Inthis case, the ∞ -category Mod 𝜔𝐴 carries a symmetric monoidal structure given by tensoring over 𝐴 . Wethen consider the data needed in order to promote (Mod 𝜔𝐴 ) ⊗ to a symmetric monoidal hermitian or Poincaré ∞ -category.To begin, by Corollary 5.3.7 and Remark 5.3.8, the ∞ -category Fun q (Mod 𝜔𝐴 ) inherits a symmetricmonoidal structure, given by Day convolution followed by -excisive approximation, such that symmet-ric monoidal hermitian refinements of Mod 𝜔𝐴 correspond to algebra objects Ϙ ∈ Alg E ∞ (Fun q (Mod 𝜔𝐴 )) . ByTheorem 3.2.13 we have a natural equivalence Fun q (Mod 𝜔𝐴 ) ≃ Mod N 𝐴 between the ∞ -category of hermit-ian structures on Mod 𝜔𝐴 and that of modules with genuine involution over 𝐴 , and so by transport of structurethe symmetric monoidal structure on Fun q (Mod 𝜔𝐴 ) induces one on Mod N 𝐴 .5.4.4. Notation.
For a commutative ring spectrum 𝐴 and an ∞ -operad O , we will refer to O -algebra objectin Mod N 𝐴 with respect to the above symmetric monoidal structure as O -algebras with genuine involutionover 𝐴 . Note that this is compatible with the terminology of Example 3.2.8.As described in the proof of Corollary 5.3.14, the recollement ( Fun s (Mod 𝜔𝐴 ) , Fun ex ((Mod 𝜔𝐴 ) op , S 𝑝 ) ) on Fun q (Mod 𝜔𝐴 ) is compatible with the symmetric monoidal structure and hence the same holds for the rec-ollement (Mod hC 𝐴⊗𝐴 , Mod 𝐴 )) on Mod N 𝐴 . It then follows that the square Mod N 𝐴 Ar(Mod 𝐴 )Mod hC 𝐴⊗ 𝑘 𝐴 Mod 𝐴 t(−) tC2 refines to a square of symmetric monoidal ∞ -categories and lax symmetric monoidal functors, such thatthe equivalence equivalence (71) becomes a symmetric monoidal one. To identify the resulting symmetricmonoidal structure on the individual components Mod hC 𝐴⊗𝐴 , Mod 𝐴 we have the following:5.4.5. Lemma.
Let 𝐴 be a commutative ring spectrum.i) Under the equivalence Fun ex ((Mod 𝜔𝐴 ) op , S 𝑝 ) ≃ Mod 𝐴 the symmetric monoidal structure on the lefthand side induced by Day convolution corresponds to the symmetric monoidal structure ⊗ 𝐴 on Mod 𝐴 .ii) Under the equivalence Fun b (Mod 𝜔𝐴 ) ≃ Fun ex ((Mod 𝜔𝐴⊗𝐴 ) op , S 𝑝 ) ≃ Mod 𝐴⊗𝐴 the symmetric monoidalstructure on the left hand side induced by Day convolution corresponds to the symmetric monoidalstructure ⊗ 𝐴⊗𝐴 on Mod
𝐴⊗𝐴 .Proof.
We begin with i). By Corollary [Lur17, Corollary 4.8.1.14] there is a unique monoidal structureon
Ind(Mod 𝜔𝐴 ) which preserves filtered colimits in each variable and such that the inclusion Mod 𝜔𝐴 ↪ Ind(Mod 𝜔𝐴 ) is symmetric monoidal. These two properties are satisfied by the tensor product on Mod 𝐴 ,and by Remark 5.3.11 the first property holds for the localized the localized Day convolution product.To finish the proof it is hence left to verify that for a stable ∞ -category E the stable Yoneda embedding E → Fun ex ( E op , S 𝑝 ) admits a symmetric monoidal structure. Indeed, it follows from [Nik16, Proposition6.3] that the stable Yoneda embedding coincides with the composite E → Fun( E op , S ) Σ ∞+ ◦ (−) ←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ Fun( E op , S 𝑝 ) → Fun ex ( E op , S 𝑝 ) , where the first arrow is the ordinary Yoneda embedding, which is naturally symmetric monoidal [Gla16, §3],the second is post-composition with the symmetric monoidal suspension infinity functor Σ ∞+ , and the lastone is the left adjoint of the inclusion Fun ex ( E op , S 𝑝 ) ⊆ Fun( E op , S ) , which is symmetric monoidal byCorollary 5.3.9. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 97 We now prove ii). In light of i), it will suffice to show the localized Day convolution structures on
Fun b (Mod 𝜔𝐴 ) and Fun ex (Mod 𝜔𝐴⊗𝐴 ) coincide. For this we consider the commutative square Fun ex (Mod 𝜔𝐴⊗𝐴 ) Fun b (Mod 𝐴 )Fun(Mod 𝜔𝐴⊗𝐴 ) Fun((Mod 𝜔𝐴 ) op × (Mod 𝜔𝐴 ) op , S 𝑝 ) ≃ where the horizontal maps are induced by restriction along the bifunctor ( 𝑋, 𝑌 ) ↦ 𝑋⊗𝑌 from
Mod 𝜔𝐴 × Mod 𝜔𝐴 to Mod 𝜔𝐴 , and the vertical maps are the relevant full inclusions. Passing to left adjoints and using the uni-versal property of localisation we obtain a commutative square Fun ex (Mod 𝜔𝐴⊗𝐴 ) Fun b (Mod 𝐴 )Fun(Mod 𝜔𝐴⊗𝐴 ) Fun((Mod 𝜔𝐴 ) op × (Mod 𝜔𝐴 ) op , S 𝑝 ) ≃ in which the vertical maps are induced by left Kan extension along ( 𝑋, 𝑌 ) ↦ 𝑋 ⊗𝑌 , followed by projectionto
Fun ex (Mod 𝜔𝐴⊗𝐴 ) in case of the top arrow. In particular, all arrows carry a natural symmetric monoidalstructure, and so the top horizontal equivalence identifies the localized Day convolution structures on bothsides. (cid:3) Corollary.
Let 𝐴 be a commutative ring spectrum and O an ∞ -operad. Then the data of an O -monoidal hermitian refinement of (Mod 𝜔𝐴 ) ⊗ corresponds to a triple ( 𝐵, 𝐶, 𝛼 ∶ 𝐶 → 𝐵 tC ) where 𝐵 ∈Alg hC 𝐴⊗𝐴 is an O - ( 𝐴 ⊗ 𝐴 ) -algebra equipped with a symmetry with respect to the flip action on 𝐴 ⊗ 𝐴 , 𝐶 isan O - 𝐴 -algebra, and 𝛼 is a map of O - 𝐴 -algebras. Corollary.
Let 𝐴 be a commutative ring spectrum and O an ∞ -operad. Suppose that O is unital.Then 𝐵 , an O -algebra with genuine involution over 𝐴 , determines an O -monoidal Poincaré structure on (Mod 𝜔𝐴 ) ⊗ if and only if the underlying O - 𝐴 -algebra of 𝐵 (with respect to either of the two componentinclusions 𝐴 → 𝐴 ⊗ 𝐴 , this makes no difference due to the symmetry of 𝐵 ) is initial. Remark.
In the situation of Corollary 5.4.7, the initiality condition on 𝐵 can be more explicitlyformulated by saying that for every colour 𝑡 ∈ O the composed map 𝐴 → 𝐴 ⊗ 𝐴 → 𝐵 𝑡 associated to theessentially unique null-operation {} → 𝑡 and either of the two component inclusions 𝐴 → 𝐴 ⊗ 𝐴 , is anequivalence.
Proof of Corollary 5.4.7.
Let Ϙ 𝛼𝐵 be the O -monoidal hermitian structure on Mod 𝜔𝐴 associated to the O -algebra with genuine involution ( 𝐵, 𝐶, 𝛼 ∶ 𝐶 → 𝐵 tC ) . By Corollary 5.3.18 this hermitian structure is O -monoidal Poincaré if and only if Ϙ 𝛼𝐵 is Poincaré and the associated lax O -monoidal functor D 𝐵 = hom 𝐴 (− , 𝐵 ) is O -monoidal. This condition requires in particular that the map Mod 𝜔𝐴 = 𝐴 → hom 𝐴 ( 𝐴, 𝐵 ) = 𝐵 in question is an equivalence. On the other hand, when this condition holds we get from Proposition 3.1.11that ( 𝐵, 𝐶, 𝛼 ) is induced by an anti-involution on 𝐴 , and hence in particular Ϙ 𝛼𝐵 is Poincaré by Example 3.1.6.In addition, in this case we may write the duality as a composition of the standard duality D 𝐴 and the C -action on Mod 𝜔𝐴 induced by the involution on 𝐵 = 𝐴 , both of which are O -monoidal. (cid:3) Taking O = Com in Corollary 5.4.7, we note that the data of a symmetric commutative ( 𝐴 ⊗ 𝐴 ) -algebrawhose underlying 𝐴 -algebra is initial is the same as the data of a C -action on 𝐴 as a commutative algebra.We hence get that symmetric monoidal Poincaré refinements of (Mod 𝜔𝐴 ) ⊗ correspond to the data of a C -action on 𝐴 as a commutative ring spectrum, together with a map of commutative 𝐴 -algebras 𝐶 → 𝐴 tC .Here 𝐴 tC is considered as a commutative 𝐴 -algebra via the Tate Frobenius 𝐴 → 𝐴 tC . We will refer tosuch a triple ( 𝐴, 𝐶, 𝐶 → 𝐴 tC ) as commutative algebra with genuine involution .5.4.9. Examples.
Let 𝐴 ∈ Alg hC E ∞ be a commutative ring spectrum equipped with an involution. We thenhave the following examples of genuine refinements of interest: i) The map id ∶ 𝐴 tC → 𝐴 tC determines a commutative algebra with genuine involution ( 𝐴, 𝐴 tC , id) ,and hence a symmetric monoidal refinement of the associated symmetric Poincaré structure Ϙ s 𝐴 on Mod 𝜔𝐴 .ii) If 𝐴 is connective then the commutative 𝐴 -algebra map 𝜏 ≥ 𝐴 tC → 𝐴 tC determines a commuta-tive algebra with genuine involution ( 𝐴, 𝐴 tC , 𝑡 ) , and hence a symmetric monoidal refinement of thePoincaré ∞ -category (Mod 𝜔𝐴 , Ϙ ≥ 𝐴 ) of Example 3.2.7. When 𝐴 is discrete, this is the genuine symmetric Poincaré structure, see 4.2.iii) The twisted Tate Frobenuis map t ∶ 𝐴 → ( 𝐴 ⊗ 𝕊 𝐴 ) tC → 𝐴 tC , which in this context is also theunit map of the commutative 𝐴 -algebra 𝐴 tC , determines a commutative algebra with genuine in-volution ( 𝐴, 𝐴 tC , t) , and hence a symmetric monoidal refinement of the Poincaré structure Ϙ t 𝐴 ofExample 3.2.11.We now consider some examples where O is not the commutative operad, but instead the operad MCom whose algebras are pairs ( 𝐴, 𝑀 ) of a commutative algebra and a module over it [Lur17, Definition 3.3.3.8].In particular, the data of an O -monoidal hermitian (resp. Poincaré) ∞ -category consists of a pair (( C , Ϙ ) , ( E , Ψ)) where ( C , Ϙ ) is a symmetric monoidal hermitian (resp. Poincaré) ∞ -category and ( E , Ψ) is a module over ( C , Ϙ ) in Cat h∞ (resp. Cat p∞ ). Consider the case where E is C considered as a module over itself, so that Ψ isa module over Ϙ in Fun q ( C ) ⊗ . By Corollary 5.3.14 the data of such a Ϙ -module corresponds to a choice of B Ϙ -module B ∈ Fun s ( C ) , an L Ϙ -module L ∈ Fun ex ( C op , S 𝑝 ) , and an L Ϙ -module map 𝛼 ∶ L → [B Δ ] tC . Forexample, we may always take B = B Ϙ considered as a module over itself. In this case, if ( C , Ϙ ) is a Poincarésymmetric monoidal Poincaré ∞ -category then all the structure maps in the MCom -algebra structure on (( C , Ϙ ) , ( C , Ϙ 𝛼 B )) are Poincaré, since on the level of the underlying bilinear forms these maps come from thestructure of ( C , Ϙ ) as a module over itself. Similarly, any commutative triangle L L ′ [B Δ Ϙ ] tC 𝛼 𝛽 of L Ϙ -modules in Fun ex ( C op , S 𝑝 ) induces a morphism ( C , Ϙ 𝛼 B ) → ( C , Ϙ 𝛽 B ) of ( C , Ϙ ) -module objects in Cat p∞ .5.4.10. Example.
Let ( C , Ϙ ) be a symmetric monoidal hermitian ∞ -category with underlying bilinear part B = B Ϙ . Then the associated quadratic hermitian ∞ -category ( C , Ϙ qB ) is canonically a module over ( C , Ϙ ) ,and the natural hermitian functor(129) ( C , Ϙ qB ) → ( C , Ϙ ) is a map of ( C , Ϙ ) -module objects in Cat h∞ . In addition, if ( C , Ϙ ) is Poincaré then ( C , Ϙ qB ) is a module over ( C , Ϙ ) in Cat p∞ and the corresponding Poincaré functor (129) is a morphism of ( C , Ϙ ) -module objects in Cat p∞ .When C = Mod 𝜔𝐴 for some E ∞ -algebra 𝐴 and Ϙ = Ϙ 𝛼𝐴 for some C -action on 𝐴 and map 𝐴 -algebras 𝛼 ∶ 𝐶 → 𝐴 tC then Corollary 5.4.7 tells us that the notion of a L Ϙ -module in Fun ex ( C , S 𝑝 ) is equivalent tothat of a 𝐶 -module 𝐷 equipped with a 𝐶 -module map to 𝐴 tC .5.4.11. Example.
Let 𝐴 be a connective commutative ring spectrum equipped with an involution. Thenfor every 𝑚 ∈ ℤ the truncated Poincaré structure (Mod 𝜔𝐴 , Ϙ ≥ 𝑚𝐴 ) ∈ Cat p∞ of Example 3.2.7 is canonicallya module over the symmetric monoidal Poincaré ∞ -category (Mod 𝜔𝐴 , Ϙ ≥ 𝐴 ) of Example 5.4.9ii). Indeed,this follows from the above since 𝜏 ≥ 𝑚 𝐴 tC is canonically a module over the algebra 𝜏 ≥ 𝐴 tC for every 𝑚 .Similarly, the canonical Poincaré functors (Mod 𝜔𝐴 , Ϙ ≥ 𝑚𝐴 ) → (Mod 𝜔𝐴 , Ϙ ≥ 𝑚 −1 𝐴 ) refines to a map of (Mod 𝜔𝐴 , Ϙ ≥ 𝐴 ) -modules. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 99
6. T
HE CATEGORY OF P OINCARÉ CATEGORIES
In the present section we study the global structural properties of the ∞ -categories Cat p∞ and Cat h∞ . Wewill begin in §6.1 by showing that Cat p∞ and Cat h∞ have all limits and colimits, and that those are preservedby the inclusion Cat p∞ ↪ Cat h∞ and by the forgetful functor Cat h∞ → Cat ex∞ . In §6.2 we will prove that thesymmetric monoidal structures on
Cat p∞ and Cat h∞ constructed in §5.2 are closed, that is, they are equippedwith a compatible notion of internal mapping objects, which we refer to as internal functor categories . Thisimplies, in particular, that these monoidal structures preserve colimits in each variable separately. We willthen show in §6.3 and §6.4 that Cat h∞ is tensored and cotensored over Cat ∞ in a manner compatible with itsclosed symmetric monoidal structure. Taken together, these properties imply that Cat h∞ (just like Cat ∞ ) canbe viewed as an (∞ , -category, that is furthermore enriched over itself. Though this point of view doesnot directly extend to Cat p∞ , in some special cases, such as poset of faces of finite simplicial complexes,the tensor and cotensor constructions do preserve Poincaré ∞ -categories. We will prove this in the final§6.6, after dedicating §6.5 to the role played by imposing certain finiteness conditions on the (co)tensoring ∞ -category.In addition to its conceptual aspect, we will also make concrete use of the material of this section in avariety of contexts, including §7.3 in the present paper, and later in Paper [II], notably via the Q -constructionwhich is used in the definition of the Grothendieck-Witt spectrum, and the dual Q -construction which is usedin establishing its universal property.6.1. Limits and colimits.
In this section we will prove that
Cat h∞ and Cat p∞ have all small limits andcolimits and that the functors Cat p∞ → Cat h∞ and Cat h∞ → Cat ex∞ preserve these limits and colimits. Sincethe former is conservative it follows automatically that it also detects limits and colimits. We will thenverify a few (co)limit-related results which will be useful later on, including the semi-additivity of
Cat h∞ and Cat p∞ , and the fact that the functors Pn and Fm preserve filtered colimits.We begin by recording the following statement, to which we could not find a reference in this form:6.1.1. Proposition.
The ∞ -category Cat ex∞ admits all small limits and colimits. Limits and filtered colimitsare preserved by the inclusion
Cat ex∞ → Cat ∞ .Proof. The statement for limits is [Lur17, Theorem 1.1.4.4] and for filtered colimits is given by [Lur17,Proposition 1.1.4.6]. For general colimits we consider the fully-faithful embedding of
Cat ex∞ inside the ∞ -category Cat rex∞ of ∞ -categories with finite colimits and right exact functors between them. The latter admitssmall colimits by [Lur09a, Lemma 6.3.4.4]. On the other hand, the embedding Cat ex∞ ⊆ Cat rex∞ admits a leftadjoint given by tensoring with S 𝑝 f , see Construction 5.1.1. It then follows that Cat ex∞ has small colimitsobtained by forming colimits in
Cat rex∞ and then tensoring with S 𝑝 f . (cid:3) Proposition.
The ∞ -category Cat h∞ admits all small limits and colimits, and these are preserved bythe forgetful functor 𝜋 ∶ Cat h∞ → Cat ex∞ .Proof.
By construction the forgetful functor 𝜋 ∶ Cat h∞ → Cat ex∞ is a cartesian fibration classified by thefunctor C ↦ Fun q ( C ) . By Corollary 1.4.2 we have that 𝜋 is also a cocartesian fibration and by Remark 1.1.15the fibres Fun q ( C ) of 𝜋 admits small limits and colimits. Given an exact functor 𝑓 ∶ C → D the associatedcartesian transition functor 𝑓 ∗ ∶ Fun q ( D ) → Fun q ( C ) preserves limits and the cocartesian transition functor 𝑓 ! ∶ Fun q ( C ) → Fun q ( D ) preserves colimits (indeed 𝑓 ! is left adjoint to 𝑓 ∗ ). Now the base Cat ex∞ of thisbicartesian fibration admits small limits and colimits by Proposition 6.1.1. It then follows from [Lur09a,4.3.1.11] and [Lur09a, 4.3.1.5.(2)] that
Cat h∞ admits all small limits and colimits and that these are preservedby 𝜋 . (cid:3) Remark.
To make the content (and proof) of Proposition 6.1.2 more explicit, let 𝐾 be a simplicialset and 𝑝 ∶ 𝐾 → Cat h∞ a diagram. Then the limit of 𝑝 is computed as follows: one first extends the diagram 𝑞 ∶= 𝜋𝑝 ∶ 𝐾 → Cat ex∞ of stable ∞ -categories to a limit diagram 𝑞 ∶ 𝐾 ⊲ → Cat ex∞ . Let C ∞ = 𝑞 (∞) bethe image of the cone point (which we denote by the symbol ∞ ), so that 𝑞 exhibits C ∞ as the limit of 𝑞 in Cat ex∞ . Interpreting 𝑞 as a natural transformation with target 𝑞 ∶ 𝐾 → Cat ex∞ domain the constant diagram 𝐾 → Cat ex∞ with value C , we may lift it to a pointwise 𝜋 -cartesian natural transformation to 𝑝 ∶ 𝐾 → Cat h∞
00 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE with target 𝑝 and domain some diagram 𝑝 ∞ ∶ 𝐾 → Cat h∞ which is concentrated in the fibre of C ∞ . In otherwords, 𝑝 ∞ encodes a diagram 𝑝 ∞ ∶ 𝐾 → Fun q ( C ∞ ) . The hermitian ∞ -category ( C , lim 𝐾 𝑝 ∞ ) is then thelimit of the original diagram 𝑝 . Somewhat informally, though more explicitly, we may describe the diagram 𝑝 ∞ via the formula 𝑘 ↦ 𝑟 ∗ 𝑘 Ϙ 𝑘 , where ( C 𝑘 , Ϙ 𝑘 ) is the hermitian ∞ -category associated to 𝑘 by the diagram 𝑝 and 𝑟 𝑘 ∶ C ∞ → C 𝑘 is the exact functor associated to the map ∞ → 𝑘 in 𝐾 ⊲ by the limit diagram 𝑝 . Since Fun q ( C ∞ ) is closed in Fun( C op , S 𝑝 ) under limits the limit of 𝑝 ∞ can also be computed in Fun( C op , S 𝑝 ) , thatis, object-wise. Similarly, in order to compute the colimit of 𝑝 we first extend 𝑞 = 𝜋𝑝 to a colimit diagram 𝑞 ∶ 𝐾 ⊳ → Cat ex∞ . Let C ∞ = 𝑞 (∞) be the image of the cone point, and 𝑖 𝑘 ∶ C 𝑘 → C ∞ the exact functorassociated to the arrow 𝑘 → ∞ in 𝐾 ⊳ . Then as above we can “push” the diagram 𝑝 to the fibre over C ,yielding a diagram 𝑝 ∞ ∶ 𝐾 → Fun q ( C ∞ ) given by the formula 𝑘 ↦ ( 𝑖 𝑘 ) ! Ϙ 𝑘 . The colimit of 𝑝 in Cat h∞ isthen given by ( C , colim 𝐾 𝑝 ∞ ) = ( C ∞ , colim 𝑘 ∈ 𝐾 ( 𝑖 𝑘 ) ! Ϙ 𝑘 ) , where the colimit of 𝑝 ∞ is computed in Fun q ( C ∞ ) . This last colimit can also be computed in Fun( C op , S 𝑝 ) since Fun q ( C ) is closed in Fun( C op , S 𝑝 ) under colimits (see Remark 1.1.15).6.1.4. Proposition.
The ∞ -category Cat p∞ has small limits and colimits and the inclusion Cat p∞ → Cat h∞ preserves small limits and colimits. The proof of Proposition 6.1.4 will require the following lemma.6.1.5.
Lemma.
Let 𝑝 ∶ 𝐾 ⊳ → Cat ex∞ be a colimit diagram of stable ∞ -categories and let 𝑓 ∶ C ∞ ∶= 𝑝 (∞) → D be an exact functor to a cocomplete stable ∞ -category. Then the canonical map colim 𝑘 ∈ 𝐾 ( 𝑖 𝑘 ) ! 𝑖 ∗ 𝑘 𝑓 → 𝑓 is an equivalence, where 𝑖 𝑘 ∶ C 𝑘 ∶= 𝑝 ( 𝑘 ) → C ∞ is the map associated to 𝑘 → ∞ by 𝑝 .Proof. For any diagram 𝑝 ∶ 𝐾 ⊳ → Cat ex∞ , the functor 𝑖 ∗ ∶ Fun ex ( C ∞ , D ) → lim 𝑘 ∈ 𝐾 Fun ex ( C 𝑘 , D ) sending 𝑓 to { 𝑓 ◦ 𝑖 𝑘 } 𝑘 ∈ 𝐾 has a left adjoint given by 𝑖 ! ∶ { 𝑓 𝑘 } 𝑘 ∈ 𝐾 ↦ colim 𝑘 ∈ 𝐾 ( 𝑖 𝑘 ) ! 𝑓 𝑘 . Indeed, this follows for example by the general formula of [HY17, Theorem B]. Our claim is then equivalentto the statement that the counit 𝑖 ! 𝑖 ∗ 𝑓 → 𝑓 is an equivalence if 𝑝 is a colimit diagram. But Fun ex (− , D ) ∶ (Cat ex∞ ) op → Cat ex∞ is a right adjoint (infact, its own right adjoint) and so it preserves all limits. Hence 𝑖 ∗ is an equivalence of ∞ -categories and inparticular the counit is an equivalence of exact functors. (cid:3) Proof of 6.1.4.
We begin with the case of colimits. Let 𝑝 ∶ 𝐾 → Cat p∞ be a diagram. By Proposition 6.1.2we may find a colimit diagram 𝑝 ∶ 𝐾 ⊳ → Cat h∞ in Cat h∞ extending (the image of) 𝑝 . We will show that theimage of 𝑝 is contained in Cat p∞ and forms a colimit diagram there. Let ( C ∞ , Ϙ ∞ ) = 𝑝 (∞) be the image ofthe cone point and for 𝑘 ∈ 𝐾 let ( 𝑖 𝑘 , 𝜂 𝑘 ) ∶ ( C 𝑘 , Ϙ 𝑘 ) → ( C ∞ , Ϙ ∞ ) be the hermitian functor associated to themap 𝑘 → ∗ in 𝐾 ⊳ by 𝑝 . In particular, the collection of natural transformations 𝜂 ad 𝑘 ∶ ( 𝑖 𝑘 ) ! Ϙ 𝑘 → Ϙ ∞ exhibits Ϙ ∞ as the colimit of the diagram 𝑘 ↦ ( 𝑖 𝑘 ) ! Ϙ 𝑘 . Our argument proceeds in two steps: Step 1.
The hermitian ∞ -category ( C ∞ , Ϙ ∞ ) is Poincaré, and all the hermitian functors ( 𝑖 𝑘 , 𝜂 𝑘 ) ∶ ( C 𝑘 , Ϙ 𝑘 ) → ( C ∞ , Ϙ ∞ ) are Poincaré. Let D 𝑘 ∶ C op 𝑘 → C 𝑘 be the duality associated to Ϙ 𝑘 . Let 𝑞 ∶= 𝜋𝑝 ∶ 𝐾 → Cat ex∞ be the underlying dia-gram of stable ∞ -categories and let us denote by 𝑞 op ∶ 𝐾 → Cat ex∞ the composite of 𝑞 and the equivalence (−) op ≃ ←←←←←←←→ Cat ex∞ → Cat ex∞ . Since the diagram 𝑝 takes values in Poincaré ∞ -categories and duality preservingfunctors the functors D 𝑘 form the components of a natural transformation 𝑞 op ⇒ 𝑞 , which consequentlyinduce a functor D ∞ ∶ C op∞ → C ∞ , ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 101 where we note that (−) op ∶ Cat ex∞ → Cat ex∞ preserves limits since it is an equivalence. We claim that D ∞ rep-resents the bilinear part of Ϙ . Indeed, since bilinear parts commute with left Kan extensions (Lemma 1.4.3)and colimits (by Lemma 1.1.7) the bilinear part of Ϙ ∞ is given by colim 𝑘 ∈ 𝐾 ( 𝑖 op 𝑘 × 𝑖 op 𝑘 ) ! B Ϙ 𝑘 (− , −) ≃ colim 𝑘 ∈ 𝐾 ( 𝑖 op 𝑘 × 𝑖 op 𝑘 ) ! hom C 𝑘 (− , D 𝑘 (−)) ≃ colim 𝑘 ∈ 𝐾 (id× 𝑖 op 𝑘 ) ! hom C ∞ (− , 𝑖 𝑘 D 𝑘 (−)) ≃≃ colim 𝑘 ∈ 𝐾 (id× 𝑖 op 𝑘 ) ! hom C ∞ (− , D ∞ 𝑖 op 𝑘 (−)) ≃ colim 𝑘 ∈ 𝐾 (id× 𝑖 op 𝑘 ) ! (id× 𝑖 op 𝑘 ) ∗ hom C ∞ (− , D ∞ −) ≃ hom C ∞ (− , D ∞ (−)) , where the last equivalence is by Lemma 6.1.5. We have thus established that the hermitian ∞ -category ( C ∞ , Ϙ ∞ ) is non-degenerate and all the hermitian functors ( 𝑖 𝑘 , 𝜂 𝑘 ) ∶ ( C 𝑘 , Ϙ 𝑘 ) → ( C ∞ , Ϙ ∞ ) are duality pre-serving. To finish the proof it will hence suffice to show that Ϙ ∞ is perfect. But this follows from Re-mark 1.2.9 since D ∞ is an equivalence, being the functor induced on limits by a natural equivalence ofdiagrams 𝑞 op ⇒ 𝑞 . Step 2. If ( D , Φ) is a Poincaré ∞ -category then a hermitian functor ( 𝑓 , 𝜂 ) ∶ ( C ∞ , Ϙ ∞ ) → ( D , Φ) isPoincaré if and only if ( 𝑓 ◦ 𝑖 𝑘 , 𝑖 ∗ 𝑘 𝜂 ) is Poincaré for every 𝑘 ∈ 𝐾 . The hermitian functor ( 𝑓 , 𝜂 ) is a Poincaré functor if and only if the associated natural transformation(130) 𝑓 D ∞ → D ∞ 𝑓 op is an equivalence. Since C ∞ is the colimit of 𝑝 in Cat ex∞ the natural transformation (130) is an equivalenceif and only if the natural transformation 𝑓 D ∞ 𝑖 op 𝑘 → D ∞ 𝑓 op 𝑖 op 𝑘 is an equivalence for all 𝑘 ∈ 𝐾 . But, since the 𝑖 𝑘 are all Poincaré functors, there is a commutative diagram 𝑓 ◦ D ∞ ◦ 𝑖 op 𝑘 D ∞ ◦ 𝑓 op ◦ 𝑖 op 𝑘 𝑓 ◦ 𝑖 𝑘 ◦ D 𝑘 ≃ Thus, (130) is an equivalence if and only if ( 𝑓 ◦ 𝑖 𝑘 , 𝑖 ∗ 𝑘 𝜂 ) is a Poincaré functor for all 𝑘 ∈ 𝐾 .The case of limits is similar and slightly easier. Indeed, as above, the natural equivalence of diagrams 𝑞 op ⇒ 𝑞 induced by the collection of dualities D 𝑘 induces an equivalence D ∞ ∶ C op∞ = lim 𝐾 𝑞 op → lim 𝐾 𝑞 = C ∞ This time, showing that D ∞ represents the bilinear form of Ϙ ∞ = lim 𝑘 ∈ 𝐾 𝑟 ∗ 𝑘 Ϙ 𝑘 is even simpler. Indeed, sincetaking bilinear forms commutes with restriction (Remark 1.1.6), and limits (by Lemma 1.1.7), the bilinearpart of Ϙ ∞ is given by lim 𝑘 ∈ 𝐾 ( 𝑟 𝑘 × 𝑟 𝑘 ) ∗ B Ϙ 𝑘 ≃ lim 𝑘 ∈ 𝐾 hom C 𝑘 ( 𝑟 𝑘 (−) , D 𝑘 (−)) = lim 𝑘 ∈ 𝐾 hom C 𝑘 ( 𝑟 𝑘 (−) , 𝑟 𝑘 ◦ D ∞ ) ≃ hom C (− , D ∞ (−)) , and this concludes the proof of Step 1 in the case of limits. The proof of Step 2 is completely dual to thatof colimits. (cid:3) Remark.
By Proposition 6.1.4 the ∞ -category Cat p∞ has small limits and colimits, and those arepreserved by the inclusion in Cat h∞ , which itself also has small limits and colimits. Since the forgetfulfunctor Cat p∞ → Cat h∞ is a conservative, it consequently also detects limits and colimits. One then says thatlimits and colimits in Cat p∞ are computed in Cat h∞ .6.1.7. Proposition.
The ∞ -categories Cat ex∞ , Cat h∞ and Cat p∞ are all semi-additive, i.e. products and co-products agree.Proof. For
Cat ex∞ , as in the proof of Proposition 6.1.1 we can embed
Cat ex∞ as a reflective full subcategory of
Cat rex∞ . It will then suffice to verify that
Cat rex∞ is semi-additive, which in turn follows from [Lur09a, Lemma7.3.3.4]. In particular, the coproduct of C and C ′ in Cat ex∞ is given by the product C × C ′ , and exhibited bythe two inclusions 𝑖 ∶ C × {0} → C × C ′ ← {0} × C ′ ∶ 𝑖 ′ .
02 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
Turning to the other two cases, we first observe that since limits and colimits in
Cat p∞ are computed in Cat h∞ (Remark 6.1.6) it is in fact sufficient to show that Cat h∞ is semi-additive. We first observe that Cat h∞ is pointed. Indeed, the hermitian ∞ -category ({0} , , whose underlying stable ∞ -category consists of asingle object , equipped with the hermitian structure which sends to the zero spectrum, is both initial andfinal in Cat h∞ .Now let ( C , Ϙ ) and ( C ′ , Ϙ ′ ) be two hermitian ∞ -categories. By the explicit description of Remark 6.1.3we have that the coproduct of ( C , Ϙ ) and ( C ′ , Ϙ ′ ) in Cat h∞ is given by ( C × C ′ , 𝑖 ! Ϙ ⊕ 𝑖 ′! Ϙ ′ ) , while the productis given by ( C × C ′ , 𝑝 ∗ Ϙ ⊕ ( 𝑝 ′ ) ∗ Ϙ ′ ) , where 𝑝, 𝑝 ′ are the projections on C and C ′ , respectively. It will hencesuffice to show that the canonical natural transformation(131) 𝑖 ! Ϙ ⊕ 𝑖 ′! Ϙ ′ ⇒ 𝑝 ∗ Ϙ ⊕ ( 𝑝 ′ ) ∗ Ϙ ′ is an equivalence. But, since 𝑖 is also right adjoint to 𝑝 and 𝑖 ′ is right adjoint of 𝑝 ′ we have 𝑖 ! Ϙ ≅ 𝑝 ∗ Ϙ and 𝑖 ′! Ϙ ′ ≅ ( 𝑝 ′ ) ∗ Ϙ ′ , and equivalence under which the canonical map (131) becomes the identity, since 𝑝 ◦ 𝑖 and 𝑝 ′ ◦ 𝑖 are therespective identities while 𝑝 ◦ 𝑖 ′ and 𝑝 ′ ◦ 𝑖 are the zero functors. (cid:3) We close this section with following result, which we will use in Paper [IV] for proving that
Cat p∞ and Cat h∞ are compactly generated presentable ∞ -categories:6.1.8. Proposition.
The functors
Fm ∶ Cat h∞ → S and Pn ∶ Cat p∞ → S commute with filtered colimits. Inparticular, the object ( S 𝑝 f , Ϙ u ) which corepresents these functors (Proposition 4.1.3) is compact in both Cat h∞ and Cat p∞ .Proof. Let us first prove the statement for Fm . Let E → Cat ∞ be the presentable fibration classified by thefunctor Cat ∞ → CAT ∞ sending C to Psh( C ) , so the objects of E are given by a pair ( C , 𝑆 ) where 𝑆 ∶ C op → S is a presheaf, and morphisms are pairs ( 𝑓 , 𝜂 ) where 𝑓 ∶ C → C ′ is a functor and 𝜂 ∶ 𝑆 → 𝑓 ∗ 𝑆 ′ is a naturaltransformation. There is a functor Φ ∶ Cat h∞ → E lying above the inclusion Cat ex∞ → Cat ∞ sending ( C , Ϙ ) to ( C , Ω ∞ Ϙ ) . Then we can factor Fm(−) as Cat h∞ Φ ←←←←←←←←→ E → S where the second functor is the functor corepresented by the final object ∗∶= (Δ , ∗ ∶ Δ → S ) in E . Notethat Φ is a morphism of cocartesian fibrations by Proposition 1.4.3 and that it preserves filtered colimitsfibrewise. Moreover it lies above the inclusion Cat ex∞ → Cat ∞ , which preserves filtered colimits by [Lur17,Proposition 1.1.4.6]. Hence the functor Φ preserves filtered colimits. It will hence suffice to show that ∗ is compact in E . Now by the naturality of the straightening-unstraightening equivalence as recordedin [GHN17, Corollary A.31], the unstaightening procedure determines an equivalence E ≃ RFib , where RFib ⊆ Ar(Cat ∞ ) is the full subcategory spanned by the right fibrations. It will hence suffice to show thatthe final object in RFib is compact. This object corresponds to the identify right fibration Δ → Δ , whichis compact when considered as an object of Ar(Cat ∞ ) since Δ is compact in Cat ∞ . It will hence suffice toshow that the inclusion RFib ⊆ Ar(Cat ∞ ) is closed under filtered colimits. Indeed, the condition of being aright fibration can be phrased as being local with respect to the maps in Ar(Cat ∞ ) encoded by the squaresof the form Λ 𝑛𝑖 Δ 𝑛 Δ 𝑛 Δ 𝑛 with < 𝑖 ≤ 𝑛 . This is an arrow between two compact objects of Ar(Cat ∞ ) and hence the locality conditionit defines is closed under filtered colimits.Now we want to prove the statement for Pn . We know that the functors Cat ex∞ → S sending C to 𝜄 C and C ↦ 𝜄 TwAr( C ) ≃ 𝜄 Fun(Δ , C ) commute with filtered colimits since Δ and Δ are compact in Cat ∞ andthe inclusion Cat ex∞ ⊆ Cat ∞ preserves filtered colimits by [Lur17, Proposition 1.1.4.6]. We then consider ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 103 the cartesian square (see (44)) Pn( C , Ϙ ) Fm( C , Ϙ )TwAr( 𝜄 C ) 𝜄 TwAr( C ) where we identify TwAr( 𝜄 C ) with the subspace of 𝜄 TwAr( C ) spanned by those arrows that are equivalencesin C . Since all the corners of the square except Pn preserve filtered colimits in Cat h∞ (and so in Cat p∞ ), sodoes Pn . (cid:3) Internal functor categories.
In the present section we will show that the symmetric monoidal struc-tures on
Cat h∞ and Cat p∞ constructed in §5.2 are closed . In particular, for two hermitian ∞ -categories ( C , Ϙ ) and ( C ′ , Ϙ ′ ) , we will promote the stable ∞ -category Fun ex ( C , C ′ ) of exact functors from C to C ′ to a hermitian ∞ -category Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) , characterized by a natural equivalence Map
Cat h∞ (( C ′′ , Ϙ ′′ ) , Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ ))) ≃ Map Cat h∞ (( C ′′ , Ϙ ′′ ) ⊗ ( C , Ϙ ) , ( C ′ , Ϙ ′ )) for ( C ′′ , Ϙ ′′ ) ∈ Cat h∞ . Furthermore, if ( C , Ϙ ) and ( C ′ , Ϙ ′ ) are Poincaré ∞ -categories, then the hermitian ∞ -category Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) is Poincaré as well and satisfies Map
Cat p∞ (( C ′′ , Ϙ ′′ ) , Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ ))) ≃ Map Cat p∞ (( C ′′ , Ϙ ′′ ) ⊗ ( C , Ϙ ) , ( C ′ , Ϙ ′ )) . The internal functor category
Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) enjoys the following useful property: its hermitian ob-jects correspond exactly the hermitian functors ( C , Ϙ ) → ( C ′ , Ϙ ′ ) , and when ( C , Ϙ ) and ( C ′ , Ϙ ′ ) are Poincaréthe Poincaré objects in Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) correspond to Poincaré functors ( C , Ϙ ) → ( C ′ , Ϙ ′ ) . In partic-ular, this allows one to view the notions of hermitian functors and hermitian objects in a unified setting,and describe the relation between hermitian and Poincaré functors in terms of that which holds betweenhermitian and Poincaré objects.6.2.1. Definition.
Let ( C , Ϙ ) and ( C ′ , Ϙ ′ ) be two hermitian ∞ -categories. We set nat Ϙ ′ Ϙ ∶ Fun ex ( C , C ′ ) op → S 𝑝, 𝑓 ↦ nat( Ϙ , 𝑓 ∗ Ϙ ′ ) , where nat denotes the spectrum of natural transformations between two spectrum valued functors.6.2.2. Proposition.
The functor nat Ϙ ′ Ϙ ∶ Fun ex ( C , C ′ ) op → S 𝑝 is quadratic. Its bilinear part is given by B nat Ϙ ′ Ϙ ( 𝑓 , 𝑔 ) ∶= nat(B Ϙ , ( 𝑓 × 𝑔 ) ∗ B Ϙ ′ ) and its linear part makes the diagram L nat Ϙ ′ Ϙ ( 𝑓 ) nat(L Ϙ , 𝑓 ∗ L Ϙ ′ )B nat Ϙ ′ Ϙ ( 𝑓 , 𝑓 ) tC nat( Ϙ , 𝑓 ∗ B Δ Ϙ ′ ) tC nat ( Ϙ , 𝑓 ∗ (B Δ Ϙ ′ ) tC ) , ≃ cartesian, where the left bottom equivalence is by the adjunction of Lemma 1.1.7. Here, as in §1.1, fora bilinear form B we denote by B Δ = Δ ∗ B its pre-composition with the diagonal. If both Ϙ and Ϙ ′ arenon-degenerate, so is nat Ϙ ′ Ϙ with duality given by D nat Ϙ ′ Ϙ ( 𝑓 ) = D Ϙ ′ 𝑓 op D op Ϙ . Finally, if both Ϙ and Ϙ ′ are perfect, then so is nat Ϙ ′ Ϙ .Proof. We begin by computing the cross effect of nat Ϙ ′ Ϙ . It is given by B nat Ϙ ′ Ϙ ( 𝑓 , 𝑔 ) = f ib [ nat( Ϙ , ( 𝑓 ⊕ 𝑔 ) ∗ Ϙ ′ ) ⟶ nat( Ϙ , 𝑓 ∗ Ϙ ′ ) ⊕ nat( Ϙ , 𝑔 ∗ Ϙ ′ ) ] ≃ nat ( Ϙ , f ib [ ( 𝑓 ⊕ 𝑔 ) ∗ Ϙ ′ ⟶ 𝑓 ∗ Ϙ ′ ⊕ 𝑔 ∗ Ϙ ′ ]) ≃ nat ( Ϙ , (( 𝑓 × 𝑔 ) ∗ B Ϙ ′ ) Δ ) ≃ nat ( B Ϙ , ( 𝑓 × 𝑔 ) ∗ B Ϙ ′ )
04 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE where the last equivalence is by the adjunction of Lemma 1.1.7. In particular, the cross effect of nat Ϙ ′ Ϙ isbi-exact. Taking 𝑓 = 𝑔 we similarly get that(132) B Δnat Ϙ ′ Ϙ ( 𝑓 ) ≃ nat ( B Ϙ , ( 𝑓 × 𝑓 ) ∗ B Ϙ ′ ) ≃ nat( Ϙ , 𝑓 ∗ B Δ Ϙ ′ ) , and hence(133) f ib [ nat Ϙ ′ Ϙ ( 𝑓 ) → (B Δnat Ϙ ′ Ϙ ( 𝑓 )) hC ] ≃ nat( Ϙ , 𝑓 ∗ f ib[ Ϙ ′ → (B Δ Ϙ ′ ) hC ]) . Since f ib[ Ϙ ′ → B Δ Ϙ ] is exact by Proposition 1.1.13 it then follows that (133) is exact in 𝑓 , and hence by thesame proposition we have that nat Ϙ ′ Ϙ is quadratic.We now compute the linear part of nat Ϙ ′ Ϙ . Applying nat( Ϙ , 𝑓 ∗ (−)) to the classifying square of Ϙ ′ andusing the equivalence (132) and the equivalence nat(L Ϙ , 𝑓 ∗ L Ϙ ′ ) ≃ ←←←←←←←→ nat( Ϙ , 𝑓 ∗ L Ϙ ′ ) given by Lemma 1.1.24we obtain an exact square nat Ϙ ′ Ϙ ( 𝑓 ) nat(L Ϙ , 𝑓 ∗ L Ϙ ′ )B nat Ϙ ′ Ϙ ( 𝑓 , 𝑓 ) hC nat( Ϙ , 𝑓 ∗ (B Δ Ϙ ′ ) tC ) in which the terms on the left hand side are quadratic in 𝑓 and the terms on the right hand side are exact in 𝑓 . The desired formula for L nat Ϙ ′ Ϙ is now obtained by taking linear parts.Now suppose that Ϙ and Ϙ ′ are non-degenerate. Applying Lemma 1.2.4 and the adjunction between D Ϙ and D op Ϙ we obtain natural equivalences B nat Ϙ ′ Ϙ ( 𝑓 , 𝑓 ) ≃ nat(B Ϙ , ( 𝑓 × 𝑔 ) ∗ B Ϙ ′ ) ≃ nat( 𝑓 D Ϙ , D Ϙ ′ 𝑔 op ) ≃ nat( 𝑓 , D Ϙ 𝑔 op D op Ϙ ) , which shows that nat Ϙ ′ Ϙ is non-degenerate with duality D nat Ϙ ′ Ϙ ( 𝑓 ) ≃ D Ϙ 𝑓 op D op Ϙ . If Ϙ and Ϙ ′ are in additionperfect then the dualities D Ϙ and D Ϙ ′ are equivalences and hence so is D nat Ϙ ′ Ϙ by the above formula. Thehermitian structure nat Ϙ ′ Ϙ is then also perfect, as desired. (cid:3) Definition.
For hermitian ∞ -categories ( C , Ϙ ) , ( C ′ , Ϙ ′ ) we will denote by Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) ∶= (Fun ex ( C , C ′ ) , nat Ϙ ′ Ϙ ) the hermitian ∞ -category given by Proposition 6.2.2. We will refer to Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) as the internalfunctor category from ( C , Ϙ ) to ( C ′ , Ϙ ′ ) .6.2.4. Remark. If ( C , Ϙ ) and ( C ′ , Ϙ ′ ) are Poincaré then Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) is Poincaré. This follows fromthe last part of Proposition 6.2.2.6.2.5. Example.
Let ( C , Ϙ ) be a hermitian ∞ -category. Then under the natural equivalence Fun ex ( S 𝑝 f , C ) ≃ C given by evaluation at the sphere spectrum, the functor nat ϘϘ u corresponds to Ϙ by virtue of Lemma 4.1.1.We consequently obtain a natural equivalence Fun ex (( S 𝑝 f , Ϙ u ) , ( C , Ϙ )) ≃ ( C , Ϙ ) of hermitian ∞ -categories.6.2.6. Construction.
For two hermitian ∞ -categories ( C , Ϙ ) , ( C ′ , Ϙ ′ ) we construct a hermitian functor ev ∶ ( C , Ϙ ) ⊗ Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) → ( C ′ , Ϙ ′ ) as follows. We first observe that by definition of the monoidal structure on Cat h∞ , specifying such a hermitianfunctor is equivalent to specifying a functor 𝑓 ∶ C × Fun ex ( C , C ′ ) → C ′ which is exact in each variableseparately, together with a natural transformation 𝜂 ∶ Ϙ ⊗ nat Ϙ ′ Ϙ ⇒ 𝑓 ∗ Ϙ ′ ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 105 of functors C op × Fun ex ( C , C ′ ) op → S 𝑝 . In the case at hand we then take as 𝑓 the usual evaluation functor ev ∶ C × Fun ex ( C , C ′ ) → C ′ . To construct 𝜂 we use the curry-equivalence Fun( C op × Fun ex ( C , C ′ ) op , S 𝑝 ) ≃ Fun(Fun ex ( C , C ′ ) op , Fun( C op , S 𝑝 )) , and pull the evaluation transformation Ϙ ⊗ nat( Ϙ , −) ⇒ id Fun( C op , S 𝑝 ) back along the postcomposition functor Ϙ ′ ◦ (−) ∶ Fun ex ( C , C ′ ) op → Fun( C op , S 𝑝 ) , which is the curry of ev ∗ Ϙ ′ .6.2.7. Proposition.
Let ( C , Ϙ ) and ( C ′ , Ϙ ′ ) be two hermitian ∞ -categories. Then the evaluation functor (ev , 𝜂 ) of Construction 6.2.6 exhibits Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) as the internal mapping object in the symmetricmonoidal ∞ -category Cat h∞ . That is, for any ( C ′′ , Ϙ ′′ ) hermitian ∞ -category the composite hom Cat h∞ (( C ′′ , Ϙ ′′ ) , Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ ))) ⟶ hom Cat h∞ (( C , Ϙ ) ⊗ ( C ′′ , Ϙ ′′ ) , ( C , Ϙ ) ⊗ Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ ))) (ev ,𝜂 ) ∗ ←←←←←←←←←←←←←←←←←←←←←←←←←→ hom Cat h∞ (( C , Ϙ ) ⊗ ( C ′′ , Ϙ ′′ ) , ( C ′ , Ϙ ′ )) is an equivalence of spaces. By the Yoneda lemma we immediately find:6.2.8.
Corollary.
The association ( C , Ϙ ) , ( C ′ , Ϙ ′ ) ↦ Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) canonically extends to a functor (Cat h∞ ) op × Cat h∞ → Cat h∞ in a way that renders the evaluation map of Construction 6.2.6 a natural trans-formation. Corollary.
The symmetric monoidal structures on
Cat h∞ constructed in §5.2 is closed. Corollary.
The symmetric monoidal product on
Cat h∞ preserves small colimits in each variable. Remark.
Since the hermitian evaluation functor of Construction 6.2.6 refines by definition the usualevaluation functor of
Cat ex∞ , it follows that the forgetful functor
U ∶ Cat h∞ → Cat ex∞ is not only symmetricmonoidal (Theorem 5.2.7ii)), but also closed symmetric monoidal, that is, for ( C , Ϙ ) , ( C ′ , Ϙ ′ ) ∈ Cat h∞ thecomposed map U( C , Ϙ ) ⊗ U Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) ≃ ←←←←←←←→ U(( C , Ϙ ) ⊗ Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) U(ev) ←←←←←←←←←←←←←←←←←←←←→ U( C ′ , Ϙ ′ ) exhibits U Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) as the internal mapping object in Cat ex∞ from U( C , Ϙ ) = C to U( C ′ , Ϙ ′ ) = C ′ .In other words, the transposed map U Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) → Fun ex (U( C , Ϙ ) , U( C , Ϙ )) is an equivalence. Proof of Proposition 6.2.7.
The forgetful functor
Cat h∞ → Cat ex∞ induces a commutative diagram(134) hom
Cat h∞ (( C ′′ , Ϙ ′′ ) , Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ ))) hom Cat h∞ (( C , Ϙ ) ⊗ ( C ′′ , Ϙ ′′ ) , ( C ′ , Ϙ ′ ))hom Cat ex∞ ( C ′′ , Fun ex ( C , C ′ )) hom Cat ex∞ ( C ⊗ C ′′ , C ′ ) ≃ in which the bottom horizontal arrow is an equivalence since on the level of stable ∞ -categories, the bilinearfunctor ev ∶ C × Fun ex ( C , C ′ ) → C ′ already exhibits Fun ex ( C , C ′ ) ∈ Cat ex∞ as the internal mapping objectfrom C to C ′ in Cat ex∞ . It will hence suffice to show that the map induced by (134) on vertical homotopyfibres is an equivalence. Let us thus fix a linear functor 𝑔 ∶ C ′′ → Fun ex ( C , C ′ ) , and write 𝑔 ∶ C ⊗ C ′′ → C ′
06 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE for its image in the bottom right corner of (134). Since
Cat h∞ → Cat ex∞ is a cartesian fibration the inducedmap on vertical fibres in (134) can be identified with the composed map of spaces(135)
Nat( Ϙ ′′ , 𝑔 ∗ nat Ϙ ′ Ϙ ) → Nat( Ϙ ⊗ Ϙ ′′ , Ϙ ⊗ 𝑔 ∗ nat Ϙ ′ Ϙ ) → Nat( Ϙ ⊗ Ϙ ′′ , 𝑔 ∗ Ϙ ′ ) , where the second map is induced by the natural transformation 𝜂 ∶ Ϙ ⊗ nat Ϙ ′ Ϙ ⇒ ev ∗ Ϙ ′ of Construction 6.2.6,restricted along (id , 𝑔 ) ∶ C × C ′′ → C × Fun ex ( C , C ′ ) . Let us now identify functors ( C × C ′′ ) op → S 𝑝 withfunctors C ′′op → Fun( C ′′op , S 𝑝 ) , and write Ϙ ′ ∶ Fun ex ( C , C ′ ) op → Fun( C op , S 𝑝 ) for the curried functordetermined by ev ∗ Ϙ ′ ∶ ( C × Fun ex ( C , C ′ )) op → S 𝑝 . Unwinding the definitions we may rewrite (135) as hom Fun( C ′′op , S 𝑝 ) ( Ϙ ′′ , nat( Ϙ , 𝑔 ∗ Ϙ ′ )) → hom Fun( C ′′op , Fun( C op , S 𝑝 )) ( Ϙ ⊗ Ϙ ′′ , Ϙ ⊗ nat( Ϙ , 𝑔 ∗ Ϙ ′ ))) → hom Fun( C ′′op , Fun( C op , S 𝑝 )) ( Ϙ ⊗ Ϙ ′′ , 𝑔 ∗ Ϙ ′ ) , where we understand nat( Ϙ , −) ∶ Fun( C ′′op , Fun( C op , S 𝑝 )) → Fun( C ′′op , S 𝑝 ) as the functor obtained by ap-plying nat( Ϙ , −) ∶ Fun( C op , S 𝑝 ) → S 𝑝 levelwise. To finish the proof it will hence suffice to show that theevaluation natural transformation Ϙ ⊗ nat( Ϙ , −) ⇒ id Fun( C op , S 𝑝 ) exhibits Ϙ ⊗ − ∶ Fun( C ′′op , S 𝑝 ) → Fun( C ′′op , Fun( C op , S 𝑝 )) as left adjoint to nat( Ϙ , −) ∶ Fun( C ′′ , Fun( C op , S 𝑝 )) → Fun( C ′′op , S 𝑝 ) . Indeed, this is simply the adjunction induced by the canonical adjunction Ϙ ⊗ (−) ∶ S 𝑝 ⟂ Fun( C op , S 𝑝 ) ∶ nat( Ϙ , −) encoding the structure of Fun( C op , S 𝑝 ) as tensored over S 𝑝 . (cid:3) Specializing Proposition 6.2.7 to the unit Poincaré ∞ -category ( C ′′ , Ϙ ′′ ) = ( S 𝑝 f , Ϙ u ) and using Proposi-tion 4.1.3 we recover a natural equivalence(136) Fm(Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ ))) ≃ ←←←←←←←→ Hom
Cat h∞ (( C , Ϙ ) , ( C ′ , Ϙ ′ )) , which by comparing with the analogous claim for Cat ex∞ we can place in a commutative square
Fm(Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ ))) Map Cat h∞ (( C , Ϙ ) , ( C ′ , Ϙ ′ )) 𝜄 Fun ex ( C , C ′ ) Map Cat ex∞ ( C , C ′ ) ≃≃ In particular, the equivalence (136) is of a somewhat tautological nature: a hermitian object in
Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) consists of a pair ( 𝑓 , 𝜂 ) , where 𝑓 ∶ C → C ′ is an exact functor and 𝜂 ∈ Ω ∞ nat Ϙ ′ Ϙ ( 𝑓 ) is a form for nat Ϙ ′ Ϙ , thatwhich by definition means a natural transformation 𝜂 ∶ Ϙ ⇒ 𝑓 ∗ Ϙ ′ . The equivalence (136) then associatesto this hermitian object the same pair ( 𝑓 , 𝜂 ) , now considered as a hermitian functor from ( C , Ϙ ) to ( C ′ , Ϙ ′ ) .Consulting the proof of Proposition 6.2.2 we observe that when ( C , Ϙ ) and ( C ′ , Ϙ ′ ) are non-degenerate themap 𝜂 ♯ ∶ 𝑓 → D nat Ϙ ′ Ϙ 𝑓 ≅ D Ϙ ′ 𝑓 op D op Ϙ associated to a hermitian object ( 𝑓 , 𝜂 ) , corresponds to the map 𝜏 𝜂 ∶ 𝑓 D Ϙ → D Ϙ ′ 𝑓 op (see Definition 1.2.5) via the adjunction between pre-composition with D Ϙ and pre-composition with D op Ϙ .When D Ϙ is an equivalence we thus have that 𝜂 ♯ is an equivalence if and only if 𝜏 𝜂 is an equivalence. In par-ticular, for Poincaré ∞ -categories ( C , Ϙ ) , ( C ′ , Ϙ ′ ) we have that a hermitian object ( 𝑓 , 𝜂 ) in Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 107 is Poincaré if and only if the corresponding hermitian functor ( 𝑓 , 𝜂 ) ∶ ( C , Ϙ ) → ( C ′ , Ϙ ′ ) is Poincaré. In par-ticular, in this case the equivalence (136) restricts to an equivalence(137) Pn(Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ ))) ≃ ←←←←←←←→ Hom
Cat p∞ (( C , Ϙ ) , ( C ′ , Ϙ ′ )) . We may summarize the situation as follows:6.2.12.
Corollary.
For a hermitian ∞ -category ( C , Ϙ ) , the hermitian object id ( C , Ϙ ) in Fun ex (( C , Ϙ ) , ( C , Ϙ )) corresponding to the identity under the equivalence (136) , exhibits ( C , Ϙ ) as representing the functor ( C ′ , Ϙ ′ ) ↦ Fm(Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ ))) in Cat h∞ . In addition, if ( C , Ϙ ) is Poincaré then id ( C , Ϙ ) is Poincaré and exhibits ( C , Ϙ ) as representing thefunctor ( C ′ , Ϙ ′ ) ↦ Pn(Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ ))) in Cat p∞ . Corollary.
Let ( C , Ϙ ) , ( C ′ , Ϙ ′ ) and ( C ′′ , Ϙ ′′ ) be hermitian ∞ -categories. Then there is an equivalenceof hermitian ∞ -categories (138) Fun ex (( C ′′ , Ϙ ′′ ) , Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ ))) ≃ Fun ex (( C , Ϙ ) ⊗ ( C ′′ , Ϙ ′′ ) , ( C ′ , Ϙ ′ )) . which is natural in ( C , Ϙ ) , ( C ′ , Ϙ ′ ) , ( C ′′ , Ϙ ′′ ) . In addition, for ( C ′′ , Ϙ ′′ ) = Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) this equiva-lence sends the hermitian object on the left hand side corresponding to the identity to the hermitian objecton the right side corresponding to the evaluation functor of Construction 6.2.6.Proof. This is a formal consequence of the Yoneda lemma that holds true in any closed symmetric monoidal ∞ -category. Indeed, embedding both sides in presheaves to spaces we may use Proposition 6.2.7 and Corol-lary 6.2.8 to construct natural equivalences between the resulting presheaves Map
Cat h∞ ((−) , Fun ex (( C ′′ , Ϙ ′′ ) , Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ ))) ≃ Map Cat h∞ (( C ′′ , Ϙ ′′ ) ⊗ (−) , Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ ))) ≃Map Cat h∞ (( C , Ϙ ) ⊗ ( C ′′ , Ϙ ′′ ) ⊗ (−) , ( C ′ , Ϙ ′ )) ≃ Map Cat h∞ (− , Fun ex (( C , Ϙ ) ⊗ ( C ′′ , Ϙ ′′ ) , ( C ′ , Ϙ ′ ))) The additional claim in the case ( C ′′ , Ϙ ′′ ) = Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) can be obtained by taking ( D , Φ) =( S 𝑝 f , Ϙ u ) and tracing through the equivalences on both sides. (cid:3) Taking Poincaré objects in (138) and using the equivalence (137) we thus conclude:6.2.14.
Corollary.
Let ( C , Ϙ ) and ( C ′ , Ϙ ′ ) be two Poincaré ∞ -categories. Then the evaluation functor (ev , 𝜂 ) of Construction 6.2.6 is Poincaré and exhibits (Fun ex ( C , C ′ ) , nat Ϙ ′ Ϙ ) as the internal mapping objects in thesymmetric monoidal category Cat p∞ . In particular, it determines an equivalence of spaces hom Cat p∞ (( C ′′ , Ϙ ′′ ) , Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ ))) ≃ hom Cat p∞ (( C , Ϙ ) ⊗ ( C ′′ , Ϙ ′′ ) , ( C ′ , Ϙ ′ )) for ( C ′′ , Ϙ ′′ ) ∈ Cat p∞ . Corollary.
The symmetric monoidal structure on
Cat p∞ constructed in §5.2 is closed. Corollary.
The association ( C , Ϙ ) , ( C ′ , Ϙ ′ ) ↦ Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) canonically extends to a func-tor (Cat p∞ ) op × Cat p∞ → Cat p∞ in a way that renders the evaluation map of Construction 6.2.6 a naturaltransformation. Corollary.
The symmetric monoidal product on
Cat p∞ preserves small colimits in each variable. Remark.
Since the evaluation functor used in Corollary 6.2.14 to exhibit internal functor categoriesin
Cat p∞ is the same as the one that was used to exhibit internal functor categories in Cat h∞ it follows formallythat the inclusion Cat p∞ ↪ Cat h∞ is not only symmetric monoidal (Theorem 5.2.7iii)), but also closed symmetric monoidal, that is, preserves internal functor categories. It then follows from Remark 6.2.11 thatthe composed functor Cat p∞ → Cat h∞ → Cat ex∞ is closed symmetric monoidal as well.
08 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
Definition.
For hermitian ∞ -categories ( C , Ϙ ) and ( C ′ , Ϙ ′ ) we define the category of hermitian func-tors from ( C , Ϙ ) to ( C ′ , Ϙ ′ ) . Fun h (( C , Ϙ ) , ( C ′ , Ϙ ′ )) ∶= He(Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ ))) . Remark.
Combining Corollary 6.2.13 and Corollary 6.2.12 we deduce that for hermitian ∞ -categories ( C , Ϙ ) , ( C ′ , Ϙ ′ ) and ( C ′′ , Ϙ ′′ ) , the evaluation functor of Construction 6.2.6 determines a natural equivalence Fun h (( C ′′ , Ϙ ′′ ) , Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ ))) ≃ Fun h (( C , Ϙ ) ⊗ ( C ′′ , Ϙ ′′ ) , ( C ′ , Ϙ ′ )) . Remark.
Proposition 6.2.7 tells us that
Cat h∞ is a closed monoidal category, so we can turn it intoan ∞ -category enriched over itself via [GH15, Cor. 7.4.10]. In particular, for three hermitian ∞ -categories ( C , Ϙ ) , ( C ′ , Ϙ ′ ) and ( C ′′ , Ϙ ′′ ) we have natural composition hermitian functors Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) ⊗ Fun ex (( C ′ , Ϙ ′ ) , ( C ′′ , Ϙ ′′ )) → Fun ex (( C , Ϙ ) , ( C ′′ , Ϙ ′′ )) . which one can of course also easily write down without using enriched technology. Applying [GH15,Cor. 5.7.6] to the lax monoidal functor He ∶ Cat h∞ → Cat ∞ we see that Cat h∞ is canonically endowed withan enrichment over Cat ∞ , with composition functors Fun h (( C , Ϙ ) , ( C ′ , Ϙ ′ )) × Fun h (( C ′ , Ϙ ′ ) , ( C ′′ , Ϙ ′′ )) → Fun h (( C , Ϙ ) , ( C ′′ , Ϙ ′′ )) , and identities given by the tautological hermitian objects id ( C , Ϙ ) of Corollary 6.2.12. In particular, oneshould consider Cat h∞ as an (∞ , -category, with mapping categories given by Fun h (− , −) . Though wewill not make explicit use of this point of view, these hermitian functor categories will play a role in §6.3-6.4 when we study the tensor-cotensor constructions, a structure most naturally viewed with the (∞ , -categorical perspective in mind.6.3. Cotensoring of hermitian categories.
Given a hermitian ∞ -category ( C , Ϙ ) and an ∞ -category I ,our goal in this section is to promote the diagram category C I ∶= Fun( I , C ) to a hermitian ∞ -category ( C , Ϙ ) I , which we call the cotensor of ( C , Ϙ ) by I . We will characterize the resulting hermitian ∞ -categoryby a universal property, see Proposition 6.3.10 below, which can be considered as witnessing it being thecotensor structure of Cat h∞ over Cat ∞ with respect to the enrichment of the former in the latter describedin §6.2. This construction will feature prominently in Paper [II] via the hermitian Q -construction.6.3.1. Construction.
Let I be a small ∞ -category and ( C , Ϙ ) a hermitian ∞ -category. We will denote by C I ∶= Fun( I , C ) the stable ∞ -category of functors I → C . Let ev ∶ I × C I → C be the evaluation functor,which, under the exponential equivalence Fun( I × C I , C ) ≃ Fun( I , Fun( C I , C )) corresponds to the functor which associates to 𝑖 ∈ I the evaluation-at- 𝑖 functor ev 𝑖 ∶ C I → C . Define afunctor Ϙ I ∶ ( C I ) op → S 𝑝 by Ϙ I ∶= lim 𝑖 ∈ I ev ∗ 𝑖 Ϙ . On a given diagram 𝜑 ∈ C I the functor Ϙ I is given by the formula Ϙ I ( 𝜑 ) = lim 𝑖 ∈ I op Ϙ ( 𝜑 ( 𝑖 )) . Proposition.
The functor Ϙ I ∶ C I → S 𝑝 is quadratic. Its bilinear part is given by B I ( 𝜑, 𝜓 ) ∶= lim 𝑖 ∈ I op B Ϙ ( 𝜑 ( 𝑖 ) , 𝜓 ( 𝑖 )) , and the linear part L I makes the square (139) L I ( 𝜑 ) lim 𝑖 ∈ I op L Ϙ ( 𝜑 ( 𝑖 ))B I ( 𝜑, 𝜑 ) tC [ lim 𝑖 ∈ I op B Ϙ ( 𝜑 ( 𝑖 ) , 𝜑 ( 𝑖 )) ] tC lim 𝑖 ∈ I op B Ϙ ( 𝜑 ( 𝑖 ) , 𝜑 ( 𝑖 )) tC ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 109 cartesian. If Ϙ is non-degenerate and C admits ( I ∕ 𝑖 ) op -shaped limits for all 𝑖 ∈ I , then Ϙ I is also non-degenerate with duality given by [ D I ( 𝜑 ) ] ( 𝑖 ) = lim [ 𝑖 → 𝑗 ]∈( I 𝑖 ∕ ) op D Ϙ ( 𝜑 ( 𝑗 )) . Remark. If I is a finite ∞ -category then L I ( 𝜑 ) = lim I op L Ϙ 𝜑 op , since the bottom horizontal mapin (139) is then an equivalence. Proof of Proposition 6.3.2.
The functor Ϙ I is defined as a limit of the functors ev ∗ 𝑖 Ϙ , each of which is qua-dratic since Ϙ itself is quadratic and each ev 𝑖 is exact. Since the collection of quadratic functors is closedunder limits (Remark 1.1.15) it follows that Ϙ I is quadratic. Since the formation of bilinear parts is com-patible with restriction (Remark 1.1.6) and commutes with limits (e.g, by Lemma 1.1.7), it follows that B I ≃ lim 𝑖 ∈ I (ev ∗ 𝑖 × ev ∗ 𝑖 )B Ϙ , and in particular B I ( 𝜑, 𝜑 ) ≃ lim 𝑖 ∈ I B Ϙ ( 𝜑 ( 𝑖 ) , 𝜑 ( 𝑖 )) . The formation of linear parts however does not commute with limits. To compute it, we apply lim I op to thesquare classifying Ϙ via Corollary 1.3.12, yielding the square Ϙ I ( 𝜑 ) lim 𝑖 ∈ I L Ϙ ( 𝜑 ( 𝑖 ))B I ( 𝜑, 𝜑 ) hC lim 𝑖 ∈ I B Ϙ ( 𝜑 ( 𝑖 ) , 𝜑 ( 𝑖 )) tC of quadratic functors in 𝜑 whose left hand side consists of exact functors. Taking linear parts we then getthe desired description of L I .We now prove the desired formula for the duality. For this, we henceforth assume that ( C , Ϙ ) is non-degenerate and that C admits ( I ∕ 𝑖 ) op -indexed limits for every 𝑖 ∈ I . Let 𝑠 ∶ TwAr( I ) → I and 𝑡 ∶ TwAr( I ) → I op be the source and target functors, respectively. Define D I ∶ C I → C I by the composite formula C I D Ϙ ◦ (−) ←←←←←←←←←←←←←←←←←←←←←←←←←←←→ C I op 𝑡 ∗ ←←←←←←←←→ C TwAr( I ) 𝑠 ∗ ←←←←←←←←←→ C I where 𝑠 ∗ stands for right Kan extension. This right Kan extension indeed exists: since 𝑠 ∶ TwAr( I ) → I is a cartesian fibration classified by the functor 𝑖 ↦ ( I 𝑖 ∕ ) op this right Kan extension is given by the explicitformula [D I ( 𝜑 )]( 𝑖 ) = lim [ 𝑖 → 𝑗 ]∈( I 𝑖 ∕ ) op D Ϙ ( 𝜑 ( 𝑗 )) where the required limits exist in C by assumption. We now claim that D I represents the bilinear functor B I .To prove this, note first that since the right Kan extension functor 𝑠 ∗ is right adjoint to the correspondingrestriction functor 𝑠 ∗ we get that nat( 𝜑, D I ( 𝜓 )) = nat( 𝜑, 𝑠 ∗ 𝑡 ∗ D Ϙ 𝜓 ) ≃ nat( 𝑠 ∗ 𝜑, 𝑡 ∗ D Ϙ 𝜓 )≃ lim [ 𝜎 ∶ 𝛼 ⇒ 𝛽 ] ∈TwAr(TwAr( I )) op hom C ( 𝜑 ( 𝑠𝛼 ) , D Ϙ 𝜓 ( 𝑡𝛽 )) ≃ lim [ 𝜎 ∶ 𝛼 ⇒ 𝛽 ] ∈TwAr(TwAr( I )) op B Ϙ ( 𝜑 ( 𝑠𝛼 ) , 𝜓 ( 𝑡𝛽 )) , where we have used the standard formula for the spectrum of natural transformations as a limit over thetwisted arrow category. We wish to show that the last limit above is equivalent to lim 𝑖 ∈ I op B Ϙ ( 𝜑 ( 𝑖 ) , 𝜓 ( 𝑖 )) .For this we will make using several cofinality arguments. To facilitate readability in what follows, we invitethe reader to visualize an object [ 𝜎 ∶ 𝛼 ⇒ 𝛽 ] ∈ TwAr(TwAr( I )) op as a diagram of the form 𝑖 𝑙𝑗 𝑘 𝛼𝛽
10 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
To begin, consider the commutative diagram [ 𝑖 → 𝑗 → 𝑘 → 𝑙 ] TwAr(TwAr( I )) TwAr( I × I op ) TwAr( I ) × TwAr( I op ) ([ 𝑖 → 𝑗 ] , [ 𝑙 ← 𝑘 ])[ 𝑗 → 𝑘 ] TwAr( I ) op ( I × I op ) op I op × I ( 𝑗, 𝑘 )∈ TwAr(s×t)t t×t ∋∈ (s×t) op ∋ Since ( 𝑠 × 𝑡 ) ∶ TwAr( C ) → C × C op is a right fibration it induces an equivalence on over categories. It thenfollows that the commutative square on the left is cartesian, and hence the induced map TwAr(TwAr( I )) ≃ ←←←←←←←→ TwAr( I ) op × [ I op × I ] [TwAr( I ) × TwAr( I op )] ≃ TwAr( I ) × I op TwAr( I ) op × I TwAr( I op ) is an equivalence. In particular, the projection(140) TwAr(TwAr( I )) → TwAr( I ) × I op TwAr( I ) op is a cartesian fibration whose fibres have terminal objects, being pulled back from the (target) cartesianfibration TwAr( I op ) → I given by [ 𝑙 ← 𝑘 ] ↦ 𝑘 which has this property. By [Lur09a, Lemma 4.1.3.2] itthen follows that the functor (140) is cofinal. To avoid confusion, we note that in the fibre product in (140),the map TwAr( I ) → I op is the target projection [ 𝑖 → 𝑗 ] ↦ 𝑗 and the map TwAr( I ) op → I op is the oppositeof the source projection [ 𝑗 → 𝑘 ] ↦ 𝑗 . Now the composed functor(141) TwAr( I ) × I op TwAr( I ) op → TwAr( I ) × I op [ I op × I ] = TwAr( I ) × I → I is cocartesian fibration, being a composition of a left fibration and a constant cocartesian fibration. Bycompatibility with base change we see that this cocartesian fibration is classified by the functor 𝑘 ↦ TwAr( I ) × I op ( I ∕ 𝑘 ) op . Since the projection I ∕ 𝑘 → I is a right fibration it follows by the same argumentas above that the map TwAr( I ∕ 𝑘 ) → TwAr( I ) × I op ( I ∕ 𝑘 ) op (induced by the target projection) is an equiv-alence. The cocartesian (141) is hence also classified by the equivalent functor 𝑘 ↦ TwAr( I ∕ 𝑘 ) . Nowconsider the map of cocartesian fibrations (over I )(142) TwAr( I ) × I op TwAr( I ) op ≃ ∫ I TwAr( I ∕ 𝑘 ) → ∫ I I ∕ 𝑘 ≃ Ar( I ) induced by the source projections TwAr( I ∕ 𝑘 ) → I ∕ 𝑘 . Then (142) is a map of cocartesian fibrations whichis fibrewise cofinal by Lemma [Lur09a, Lemma 4.1.3.2] and is hence itself cofinal. Since cofinal maps areclosed under composition we may now conclude that the composed projection TwAr(TwAr( I )) → Ar( I ) is cofinal. On the other hand the canonical inclusion I → Ar( I ) sending 𝑥 to id 𝑥 is also cofinal since it hasa left adjoint (the target functor). We may there conclude that lim [ 𝜎 ∶ 𝛼 ⇒ 𝛽 ] ∈TwAr(TwAr( I )) op B Ϙ ( 𝜑 ( 𝑠𝛼 ) , 𝜓 ( 𝑡𝛽 )) ≃ lim [ 𝑖 → 𝑘 ]∈Ar( I ) op B Ϙ ( 𝜑 ( 𝑖 ) , 𝜓 ( 𝑘 )) ≃ lim 𝑖 ∈ I op B Ϙ ( 𝜑 ( 𝑖 ) , 𝜓 ( 𝑖 )) , and so D I represents B Ϙ , as desired. (cid:3) Definition.
For a hermitian ∞ -category ( C , Ϙ ) and an ∞ -category I we will denote by ( C , Ϙ ) I ∶=( C I , Ϙ I ) the ∞ -category given by Proposition 6.3.2. We will refer to it as the cotensor of ( C , Ϙ ) by I .6.3.5. Remark. If I is a finite poset then the comma ∞ -categories I ∕ 𝑖 are finite for every 𝑖 ∈ I , and henceevery stable ∞ -category admits ( I ∕ 𝑖 ) op indexed colimits. In particular, in this case ( C , Ϙ ) I is non-degenerateas soon as ( C , Ϙ ) is non-degenerate.6.3.6. Example.
For I = TwAr(Δ ) we may identify C I with the ∞ -category of spans 𝑥 ← 𝑤 → 𝑦 in C ,with Ϙ I given by Ϙ I ([ 𝑥 ← 𝑤 → 𝑦 ]) = Ϙ ( 𝑥 ) × Ϙ ( 𝑤 ) Ϙ ( 𝑦 ) , and the duality (when ( C , Ϙ ) is non-degenerate) given by D I ([ 𝑥 ← 𝑤 → 𝑦 ]) = [D Ϙ 𝑥 ← D Ϙ 𝑥 × D Ϙ 𝑤 D Ϙ 𝑦 → D Ϙ 𝑦 ] . It is then straightforward to verify that this duality is perfect whenever D Ϙ is perfect, in which case ( C , Ϙ ) I is Poincaré. This example will feature prominently in subsequent parts of the present paper in the context ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 111 of the cobordism category of a Poincaré ∞ -category, where we will view the above duality as an algebraicincarnation of Lefschetz duality for manifolds.6.3.7. Warning.
For a Poincaré ( C , Ϙ ) and I arbitrary, the hermitian ∞ -category ( C , Ϙ ) I might fail to bePoincaré, even if I is a finite poset. This happens for example if I has a final object but is not itself equivalentto a point; indeed, in this case the image of D I is the full subcategory of C I spanned by the constant diagrams.On the other hand, we will see in §6.4 that cotensor by I does preserve Poincaré ∞ -categories when I is theposet of faces of a finite simplicial complex.We go on to establish the universal property of these hermitian diagram categories, from which we willalso deduce their functoriality. We will require the following lemma:6.3.8. Lemma.
Let I be a small ∞ -category and C , C ′ be two hermitian ∞ -categories. Then the fibre of themap Fun( I , Fun h (( C , Ϙ ) , ( C ′ , Ϙ ′ ))) → Fun( I , Fun ex ( C , C ′ )) over a functor 𝑓 ∶ I × C → C ′ is naturally equivalent to the space Nat( 𝑝 ∗ Ϙ , 𝑓 ∗ Ϙ ′ ) of natural transformations 𝑝 ∗ C Ϙ ⇒ 𝑓 ∗ Ϙ ′ , where 𝑝 C ∶ I × C → C denotes the projection to C .Proof. We want to describe the space of dotted lifts
Fun h ( C , C ′ ) I Fun ex ( C , C ′ ) 𝑓 Recall that the vertical map above is the right fibration classified by functor Ω ∞ nat Ϙ ′ Ϙ ∶ Fun ex ( C , C ′ ) op → S ,and so by [Lur09a, Corollary 3.3.3.2], the space of sections of this right fibration coincides with the limit lim 𝑖 ∈ I op Ω ∞ nat Ϙ ′ Ϙ ( 𝑓 𝑖 ) ≅ lim 𝑖 ∈ I op nat( Ϙ , 𝑓 ∗ 𝑖 Ϙ ′ ) ≅ nat ( Ϙ , lim 𝑖 ∈ I op 𝑓 ∗ 𝑖 Ϙ ′ ) On the other hand, the space of natural transformations nat( 𝑝 ∗ Ϙ , 𝑓 ∗ Ϙ ′ ) ≅ nat( Ϙ , 𝑝 ∗ 𝑓 ∗ Ϙ ′ ) ≅ nat( Ϙ , lim 𝑖 ∈ I op 𝑓 ∗ 𝑖 Ϙ ′ ) where 𝑝 C ∶ I × C → C is the projection on C , since right Kan extensions along 𝑝 C are computed by takingthe limit fibrewise, as can be seen from the pointwise formula for right Kan extensions. Hence the twoconstructions are naturally equivalent. (cid:3) Construction.
For a hermitian ∞ -category ( C , Ϙ ) and an ∞ -category I we define a functor ev ∶ I → Fun h (( C , Ϙ ) I , ( C , Ϙ )) as follows. Let 𝑝 ∶ I op × ( C I ) op → ( C I ) op be the projection on the second factor and ev ∶ I op × ( C I ) op → C op the evaluation. By Lemma 6.3.8 the additional data needed in order to define ev is a natural transformation(143) 𝜏 ∶ 𝑝 ∗ Ϙ I ⇒ ev ∗ Ϙ . We then define 𝜏 by taking the counit transformation const I op lim I op ⇒ id Fun( I op , S 𝑝 ) , currying it into morphism in Fun( I op × Fun( I op , S 𝑝 ) , S 𝑝 ) and finally pre-composing with the functor I op × ( C I ) op → I op × Fun( I op , S 𝑝 ) induced by Ϙ .
12 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
Proposition.
Let ( C , Ϙ ) and ( C ′ , Ϙ ′ ) be two hermitian ∞ -categories and I a small ∞ -category. Thenthe composite map Fun h (( C ′ , Ϙ ′ ) , ( C , Ϙ ) I ) × I id×ev ←←←←←←←←←←←←←←←←←←←←←→ Fun h (( C ′ , Ϙ ′ ) , ( C , Ϙ ) I ) × Fun h (( C , Ϙ ) I , ( C , Ϙ )) ⟶ Fun h (( C ′ , Ϙ ′ ) , ( C , Ϙ )) defined using the functor ev of Construction 6.3.9 and the composition functor of Remark 6.2.21, determinesan equivalence of categories (144) Fun h (( C ′ , Ϙ ′ ) , ( C , Ϙ ) I ) ≃ ←←←←←←←→ Fun( I , Fun h (( C ′ , Ϙ ′ ) , ( C , Ϙ ))) , and in particular an equivalence (145) Hom
Cat h∞ (( C ′ , Ϙ ′ ) , ( C , Ϙ ) I ) ≃ 𝜄 Fun( I , Fun h (( C ′ , Ϙ ′ ) , ( C , Ϙ ))) . We will give the proof of Proposition 6.3.10 at the end of this subsection. Before let us explore someof its consequences. First, as in the case of internal functor categories, the Yoneda lemma immediatelyimplies:6.3.11.
Corollary.
The association ( I , ( C , Ϙ )) ↦ ( C , Ϙ ) I extends canonically to a functor Cat op∞ × Cat h∞ → Cat h∞ that rendered the equivalence from proposition 6.3.10 natural. Remark.
Unwinding the definitions, if 𝛼 ∶ I → J is a functor of small ∞ -categories and ( C , Ϙ ) is ahermitian ∞ -category then the hermitian functor ( 𝛼 ∗ , 𝜂 𝛼 ) ∶ ( C , Ϙ ) J → ( C , Ϙ ) I issued via the functoriality of Corollary 6.3.11 is given by the usual restriction functor 𝛼 ∗ ∶ C J → C I onthe underlying stable ∞ -categories accompanied by the usual restriction-induced map 𝜂 𝛼𝜑 ∶ Ϙ J ( 𝜑 ) = lim 𝑗 ∈ J Ϙ ( 𝜑 ( 𝑗 )) → lim 𝑖 ∈ I Ϙ ( 𝜑 ( 𝛼 ( 𝑖 ))) = Ϙ I ( 𝛼 ∗ 𝜑 ) on limits.6.3.13. Remark.
As pointed out in Warning 6.3.7, the cotensor construction does not restrict to a functor
Cat op∞ × Cat p∞ → Cat p∞ . In particular, while this construction is best understood by considering Cat h∞ asan (∞ , -category, the (∞ , -categorical perspective does not seem to extend to Cat p∞ in a meaningfulmanner.6.3.14. Remark.
It follows from Proposition 6.3.10 that when I is an ∞ -groupoid the cotensor ( C , Ϙ ) I co-incides with the limit in Cat h∞ of the constant I -diagram with value ( C , Ϙ ) . In particular, it follows formProposition 6.1.4 that for such an I the functor ( C , Ϙ ) ↦ ( C , Ϙ ) I does preserve Poincaré ∞ -categories.Taking ( C ′ , Ϙ ′ ) = ( S 𝑝 𝜔 , Ϙ u ) in Proposition 6.3.10 yields:6.3.15. Corollary.
There is a natural equivalence
He(( C , Ϙ ) I ) ≃ Fun( I , He( C , Ϙ )) . Remark.
We know of no analogous formula for the Poincaré objects of ( C , Ϙ ) I when the latterhappens to be Poincaré. It is certainly not true, for example, that the individual objects of a Poincarédiagram are Poincaré objects themselves, as demonstrated by the case I = TwAr(Δ ) , see Example 6.3.6.The following is again a formal consequence:6.3.17. Corollary.
For any hermitian ∞ -categories ( C , Ϙ ) , ( C ′ , Ϙ ′ ) and any ∞ -category I there is a canonicalequivalence Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ ) I ) ≃ Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) I of hermitian ∞ -categories. Proposition.
Let ( C , Ϙ ) be non-degenerate and 𝛼 ∶ I → J a functor between small categories, suchthat ( C , Ϙ ) admits both ( I 𝑖 ∕ ) op - and ( J 𝑗 ∕ ) op -shaped limits for all 𝑖 ∈ I and 𝑗 ∈ J . If the induced maps I 𝑖 ∕ → J 𝛼 ( 𝑖 )∕ are cofinal for every 𝑖 ∈ I then the hermitian functor ( 𝛼 ∗ , 𝜂 𝛼 ) ∶ ( C , Ϙ ) J → ( C , Ϙ ) I is duality preserving. In particular, if ( C , Ϙ ) J and ( C , Ϙ ) I are Poincaré then ( 𝛼 ∗ , 𝜂 𝛼 ) is a Poincaré functor. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 113 Proof.
This follows directly from the explicit description of the duality in Proposition 6.3.2. (cid:3)
Remark.
In the situation of Proposition 6.3.18, if 𝛼 is a map of posets, then the given criterion forpreservation of duality can be rephrased more explicitly as by saying that for every 𝑖 ∈ I and 𝑗 ∈ J with 𝑗 ≥ 𝛼 ( 𝑖 ) , the realization of the poset { 𝑘 ∈ I ∣ 𝑖 ≤ 𝑘, 𝑗 ≤ 𝛼 ( 𝑘 )} is contractible. Proof of Proposition 6.3.10.
We argue similarly to the proof of Proposition 6.2.7. The forgetful functordetermines a commutative diagram of ∞ -categories(146) Fun h (( C ′ , Ϙ ′ ) , ( C , Ϙ ) I ) Fun( I , Fun h (( C ′ , Ϙ ′ ) , ( C , Ϙ )))Fun ex ( C ′ , C I ) Fun( I , Fun ex ( C ′ , C )) ≃ in which the vertical maps are right fibrations and where the bottom arrow is an equivalence since thefunctor ev ′ ∶ I → Fun ex ( C I , C ′ ) underlying ev already exhibits C I as the cotensor of C over I in Cat ex∞ . Itwill hence suffice to show the map induced by (146) on vertical fibres is an equivalence. Let us hence fixan exact functor 𝑔 ∶ C ′ → C I and let 𝑔 I ∶= { 𝑔 𝑖 } ∶ I → Fun ex ( C ′ , C ) be its image in the bottom right cornerof (146). Now the fibre of the right vertical map in (146) over 𝑔 I is the space of sections of the base change(147) Fun h ( C ′ , C ) × Fun ex ( C ′ , C ) I → I , where the fibre product us taken with respect to the map 𝑔 I . By the compatibility of base change andstraightening we see that (147) is the right fibration classified by the functor 𝑖 ↦ Nat( Ϙ ′ , 𝑔 ∗ 𝑖 Ϙ ) . By [Lur09a,Corollary 3.3.3.2] evaluation at the various 𝑖 ∈ I exhibits the space of sections of (147) as the limit lim 𝑖 Nat( Ϙ ′ , 𝑔 ∗ 𝑖 Ϙ ) . We may then identify the map induced by (146) from the fibre over 𝑔 to the fibre of 𝑔 I with the map Nat( Ϙ ′ , 𝑔 ∗ Ϙ I ) → lim 𝑖 ∈ I Nat( Ϙ ′ , 𝑔 ∗ 𝑖 Ϙ ) whose components Nat( Ϙ ′ , 𝑔 ∗ Ϙ I ) → Nat( Ϙ ′ , 𝑔 ∗ 𝑖 Ϙ ) are induced by the components 𝜏 𝑖 ∶ Ϙ I → ev ∗ 𝑖 Ϙ of (143).Since pulling back functors preserve limits the desired result now follows from the fact that the collectionof maps Ϙ I → ev ∗ 𝑖 Ϙ exhibit Ϙ I as the limit of the diagram {ev ∗ 𝑖 Ϙ } , by definition. (cid:3) Tensoring of hermitian categories.
In this section we will consider the dual of the cotensor construc-tion studied in §6.3, which we will refer to as tensoring a Poincaré ∞ -category ( C , Ϙ ) by an ∞ -category I .In general this construction is somewhat less accessible then the cotensor construction, but we will be ableto say more about it when I satisfies certain finiteness conditions, see §6.5 below. We will exploit the tensorconstruction in Paper [II] in order to form the dual Q -construction, which is needed in the proof of theuniversal property of the Grothendieck-Witt spectrum.6.4.1. Construction.
Let ( C , Ϙ ) be a hermitian ∞ -category and I a small ∞ -category. For 𝑖 ∈ I and 𝑥 ∈ C ,let us denote by 𝑅 𝑖,𝑥 ∶ I → Pro( C ) the functor 𝑅 𝑖,𝑥 = ( 𝜄 𝑖 ) ∗ ( 𝑥 ) right Kan extended along the inclusion 𝜄 𝑖 ∶ { 𝑖 } ↪ I of 𝑖 of the functor { 𝑖 } → Pro( C ) with value 𝑥 ∈ C ⊆ Pro( C ) . We then let C I ⊆ Pro( C ) I op bethe smallest full subcategory containing 𝑅 𝑖,𝑥 for 𝑖 ∈ I and 𝑥 ∈ C and closed under finite limits. Then C I isalso closed under suspensions (since the collection 𝑅 𝑖,𝑥 is, as suspension in C commutes with finite limits)and is hence stable. It is also equipped by construction with a functor(148) 𝜄 ∶ C × I → C I ( 𝑥, 𝑖 ) ↦ 𝑅 𝑖,𝑥 . We then promote C I to a hermitian ∞ -category by endowing it with the quadratic functor Ϙ I ∶ C I → S 𝑝 obtained by taking the left Kan extension of 𝑝 ∗ C Ϙ ∶ C op × I op → S 𝑝 along 𝜄 op ∶ C × I op → C I (that whichresults in a reduced functor) and then applying to it the -excisive approximation of Construction 1.1.26,left adjoint to the inclusion Fun q ( C I ) ⊆ Fun ∗ ( C op I , S 𝑝 ) . Here we denotes by 𝑝 C ∶ I × C → C projection to C . We then set ( C , Ϙ ) I ∶= ( C I , Ϙ I ) , and refer to it as the tensor of ( C , Ϙ ) by I . By construction the functor Ϙ I supports a natural transformation 𝑝 ∗ C Ϙ ⇒ 𝜄 ∗ Ϙ I ,
14 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE where 𝑝 C ∶ I × C → C is the projection to C , and by Lemma 6.3.8 this transformations determines a functor(149) coev ∶ I → Fun h ( C , C I ) , which to 𝑖 ∈ I associates the exact functor 𝑥 ↦ 𝑅 𝑖,𝑥 , equipped with the natural transformation Ϙ ( 𝑥 ) ⇒ Ϙ I ( 𝑅 𝑖,𝑥 ) given by the construction of Ϙ I .6.4.2. Remark.
The functor coev ∶ C × I → C I exhibits C I as universal among stable ∞ -categories equippedwith a functor from C × I which is exact in the first entry. To see this, we may replace the term “exact” by“finite limit preserving”. In other words, it will suffice to show that (148) is universal among maps from C × I to a finitely complete ∞ -category which preserve finite limits in the first variable. Such universalconstructions are explicitly described in [Lur09a, §5.3.6]. In particular, it will suffice to show that C I coincides with the construction appearing in the proof of [Lur09a, Proposition 5.3.6.2]. Indeed: Pro( C ) I op identifies with the full subcategory of Fun( C × I , S ) op spanned by those functors which preserve finite limitsin the second variable, and C I ⊆ Pro( C ) I op identifies with the further full subcategory spanned by the imageof I × C under finite limits.To describe Ϙ I more explicitly, let ̃ Ϙ ∶ Pro( C ) op → S 𝑝 be the left Kan extention of Ϙ along the Yonedaembedding C op ↪ Pro( C ) op . Then ̃ Ϙ is quadratic by [Lur17, Proposition 6.1.5.4]. Its bilinear part thencoincides with the essentially unique bilinear functor ̃ B Ϙ ∶ Pro( C ) op × Pro( C ) op → S 𝑝 which extends B andpreserve colimits in each variable separately, and its linear part is the essentially unique colimit preservingfunctor ̃ L Ϙ ∶ Pro( C ) op → S 𝑝 extending L Ϙ .6.4.3. Proposition.
Let ( C , Ϙ ) be a hermitian ∞ -category and I a small ∞ -category. Then the quadraticfunctor Ϙ I of Construction 6.4.1 is given explicitly by the formula Ϙ I ( 𝜑 ) = colim 𝑖 ∈ I ̃ Ϙ ( 𝜑 ( 𝑖 )) . Its bilinear and linear parts are given by B I ( 𝜑, 𝜑 ) ∶= colim 𝑖 ∈ I ̃ B Ϙ ( 𝜑 ( 𝑖 ) , 𝜑 ( 𝑖 )) and L I ( 𝜑 ) ∶= colim 𝑖 ∈ I ̃ L Ϙ ( 𝜑 ) , respectively.Proof. To establish the formula for Ϙ I we first note that colim 𝑖 ∈ I ̃ Ϙ ( 𝜑 ( 𝑖 )) is quadratic, being a colimit of thequadratic functors ev ∗ 𝑖 ̃ Ϙ for 𝑖 ∈ I op , see Remark 1.1.15. Its linear and bilinear parts are then given by theindicated formulas since taking linear and bilinear parts commutes with colimits. It will hence suffice toidentify colim 𝑖 ∈ I ̃ Ϙ ( 𝜑 ( 𝑖 )) with the left Kan extension of 𝑝 ∗ C Ϙ along 𝜄 op . For this, consider the commutativesquare ( C × I ) op C op I Fun( C × I , S ) Fun( I op , Pro( C )) op 𝜄 op 𝑗 where the left vertical map 𝑗 is the Yoneda embedding and the bottom horizontal map is the left adjoint tothe inclusion Fun( I op , Pro( C )) op ↪ Fun( C × I , S ) induced by the inclusion Pro( C ) ↪ Fun( I , S ) op as the fullsubcategory spanned by left exact functors. Since the right vertical map is fully-faithful we may compute 𝜄 ! 𝑝 ∗ C Ϙ by further Kan extending to Fun( I op , Pro( C )) op and then restricting back to C I . By the commutativityof the above square the left Kan extension to Fun( I op , Pro( C )) op can be performed by first left Kan extendingto Fun( C × I , S ) and then left Kan extending to Fun( I op , Pro( C )) op , the latter given by restriction along theright adjoint Fun( I op , Pro( C )) op ↪ Fun( C × I , S ) . Now the left Kan extension of 𝑝 ∗ C Ϙ along the Yonedaembedding results in the coend construction Fun( C × I , S ) ∋ 𝜌 ↦ ∫ C × I 𝜌 ⊗ 𝑝 ∗ C Ϙ ≃ ∫ C ( 𝑝 C ) ! 𝜌 ⊗ Ϙ ≃ ∫ Pro( C ) 𝑞 ! 𝜌 ⊗ ̃ Ϙ , where ⊗ denotes the tensor of spectra over spaces, and 𝑞 is the composed functor C × I 𝑝 C ←←←←←←←←←←←→ C → Pro( C ) ,which can also be written as the composite C × I → Pro( C ) × I → Pro( C ) . Now, in the case where 𝜌 ∶ C × I → S is of the form ( 𝑥, 𝑖 ) ↦ hom C ( 𝜑 ( 𝑖 ) , 𝑥 ) for some Pro( C ) -valued presheaf 𝜑 ∶ I op → Pro( C ) , ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 115 its left Kan extension to Pro( C ) × I → S is given by the formula ( 𝑥, 𝑖 ) ↦ hom Pro( C ) ( 𝜑 ( 𝑖 ) , 𝑥 ) , and so 𝑞 ! 𝜌 isgiven by 𝑞 ! 𝜌 ( 𝑥 ) = colim 𝑖 ∈ I hom Pro( C ) ( 𝜑 ( 𝑖 ) , 𝑥 ) . We may then conclude that in this case Ϙ I ( 𝜑 ) = 𝜄 ! 𝑝 ∗ C Ϙ ( 𝜌 ) = ∫ Pro( C ) 𝑞 ! 𝜌 ⊗ ̃ Ϙ ≃ colim 𝑖 ∈ I ∫ Pro( C ) hom Pro( C ) ( 𝜑 ( 𝑖 ) , 𝑥 ) ⊗ ̃ Ϙ ≃ colim 𝑖 ∈ I ̃ Ϙ ( 𝜑 ( 𝑖 )) , as desired. To obtain the description of the bilinear and linear part we may consider the above formula asexpressing Ϙ I as a colimit of an I -indexed diagram of quadratic functors. Since the formation of linear andbilinear parts commutes with colimits and restrictions their respective formulas readily follow. (cid:3) We shall now address the universal property of the tensor construction.6.4.4.
Proposition.
Let ( C , Ϙ ) and ( C ′ , Ϙ ′ ) be hermitian ∞ -categories and I a small ∞ -category. Then thecomposed map I × Fun h (( C , Ϙ ) I , ( C ′ , Ϙ ′ )) coev×id ←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ Fun h (( C , Ϙ ) , ( C , Ϙ ) I ) × Fun h (( C , Ϙ ) I , ( C ′ , Ϙ ′ )) → Fun h (( C , Ϙ ) , ( C ′ , Ϙ ′ )) induces an equivalence (150) Fun h (( C , Ϙ ) I , C ′ ) ≃ Fun( I , Fun h (( C , Ϙ ) , ( C ′ , Ϙ ′ )) . and in particular an equivalence (151) Hom
Cat h∞ (( C ′ , Ϙ ′ ) I , ( C , Ϙ )) ≃ 𝜄 Fun( I , Fun h (( C ′ , Ϙ ′ ) , ( C , Ϙ ))) . As in the case of the cotensor construction the universal characterization implies functoriality:6.4.5.
Corollary.
The association ( I , ( C , Ϙ )) ↦ ( C , Ϙ ) I extends canonically to a functor Cat op∞ × Cat h∞ → Cat h∞ that rendered the equivalence from proposition 6.4.4 natural. Remark.
Comparing universal properties, we see that there are canonical equivalences of hermitian ∞ -categories Fun ex (( C , Ϙ ) I , ( C ′ , Ϙ ′ )) ≃ Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) I ≃ Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ ) I ) . Remark.
Comparing universal properties we see that there are canonical equivalences of hermitian ∞ -categories ( C , Ϙ ) I ⊗ ( C ′ , Ϙ ′ ) ≃ ( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ ) I ≃ (( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ )) I . Remark.
It follows from Remarks 6.4.6 6.4.7 and 6.2.4 that for a given small ∞ -category I theconditionsi) the functor ( C , Ϙ ) ↦ ( C , Ϙ ) I preserves Poincaré ∞ -categories;ii) the hermitian ∞ -category ( S 𝑝 f , Ϙ u ) I is Poincaré;are equivalent, and that when these equivalent conditions hold the functor ( C , Ϙ ) ↦ ( C , Ϙ ) I preservesPoincaré ∞ -categories as well.6.4.9. Remark. If 𝛼 ∶ I → J is a map between small ∞ -categories then the hermitian functor C I → C J resulting from the functoriality of Corollary 6.4.5 must induce the associated restriction functor 𝛼 ∗ ∶ Fun( J , Fun h ( C , C ′ )) → Fun( I , Fun h ( C , C ′ )) under the equivalence of Proposition 6.4.4, upon mapping into any ( C ′ , Ϙ ′ ) . The underlying exact functor isconsequently the essentially unique one (see Remark 6.4.2) making the diagram I × C J × CC I C J coev 𝛼 ×id coev
16 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE commute, and must therefore coincide with the restriction to C I of the right Kan extension functor 𝛼 ∗ ∶ Fun( I op , Pro( C )) → Fun( J op , Pro( C )) . By a slight abuse of notation we will denote this restriction by 𝛼 ∗ ∶ C I → C J as well. Using the formula ofProposition 6.4.3 the hermitian structure on 𝛼 ∗ is the given by the natural map Ϙ I ( 𝜑 ) = colim 𝑖 ∈ I ̃ Ϙ ( 𝜑 ( 𝑖 )) → colim 𝑖 ∈ I ̃ Ϙ ( 𝛼 ∗ 𝛼 ∗ 𝜑 ( 𝑖 )) → colim 𝑗 ∈ J ̃ Ϙ ( 𝛼 ∗ 𝜑 ) = Ϙ I ( 𝛼 ∗ 𝜑 ) for 𝜑 ∈ C I .6.4.10. Remark.
The natural equivalences (151) and (151) exhibit (−) I as left adjoint to (−) I . Furthermore,if 𝛼 ∶ I → J is a map of finite posets then this adjunction intertwines the restriction functor 𝛼 ∗ ∶ ( C , Ϙ ) J → ( C , Ϙ ) I with the (restricted) right Kan extension functor 𝛼 ∗ ∶ ( C , Ϙ ) I → ( C , Ϙ ) J . Proof of Proposition 6.4.4.
The forgetful functor
Cat h∞ → Cat ex∞ determines a commutative square of ∞ -categories(152) Fun h (( C , Ϙ ) I , ( C ′ , Ϙ ′ )) Fun( I , Fun h (( C , Ϙ ) , ( C ′ , Ϙ ′ )))Fun ex ( C I , C ′ ) Fun( I , Fun ex ( C , C ′ )) in which the vertical maps are right fibrations. By Remark 6.4.2 the bottom horizontal map is an equivalence.It will hence suffice to show that (152) induces an equivalence on vertical fibres. Let 𝑔 ∶ C I → C ′ be anexact functor and let 𝑔 I = { 𝑔 𝑖 } ∶ I → Fun ex ( C , C ′ ) be its image in the bottom right corner of (152). Toidentify the fibre on the left side, let 𝑔 ∶ I × C → C I 𝑔 → C ′ be the composed functor, so that 𝑔 correspondsto 𝑔 I under the identification of functors I → Fun( C , C ′ ) and functors I × C → C ′ . In light of the definitionof Ϙ I via left Kan extensions and -excisive approximations we may identify the fibre of the left verticalarrow in (152) over 𝑔 with Nat I × C ( 𝑝 ∗ Ϙ , 𝑔 ∗0 Ϙ ′ ) , where 𝑝 ∶ I × C → C is the projection. The map between thevertical fibres in (152) can then be identified with the map(153) Nat I × C ( 𝑝 ∗ Ϙ , 𝑔 ∗0 Ϙ ′ ) → lim 𝑖 ∈ I Nat C ( Ϙ , 𝑔 ∗ 𝑖 Ϙ ′ ) , whose 𝑖 ’th component Nat I × C ( 𝑝 ∗ Ϙ , 𝑔 ∗0 Ϙ ′ ) → Nat C ( Ϙ , 𝑔 ∗ 𝑖 Ϙ ′ ) is given by restricting to { 𝑖 } × C ⊆ C . This mapis an equivalence by Lemma 6.3.8, and so the proof is complete. (cid:3) Finite tensors and cotensors.
In this section we will consider the tensor and cotensor constructionsin the case where the ∞ -category I satisfies strong finiteness conditions, e.g., when I is a finite poset. In thiscase the tensor construction admits a more accessible description, and sends non-degenerate Poincaré ∞ -categories to non-degenerate ones, with explicit induced duality, see Proposition 6.5.8. In addition, we willshow that under these conditions the functor ( C , Ϙ ) ↦ ( C , Ϙ ) I is not only right adjoint to ( C , Ϙ ) ↦ ( C , Ϙ ) I ,but also left adjoint to it, and exact some useful consequences. Finally, in this case both the tensor andcotensor constructions are functorial not only in maps 𝛼 ∶ I → J , but also in cofinal maps 𝛽 ∶ J → I goingin the other direction, a phenomenon we refer to as exceptional functoriality , see Construction 6.5.14.To begin, recall that an ∞ -category I is said to be finite if it is categorically equivalent to a simplicialset with only finitely many non-degenerate simplices. If I is a space then the condition that I is finite asan ∞ -category is equivalent to the condition that I is finite as space, that is, that it is weakly equivalent toa simplicial set with finitely many non-degenerate simplices. We will use the term finite (co)limits to referto (co)limits indexed by finite ∞ -categories. We recall that any stable ∞ -category admits finite limits andcolimits, and that these are preserved by any exact functor. In particular, in any stable ∞ -category whichadmits small (co)limits, the latter automatically commute with finite (co)limits.6.5.1. Definition.
We will say that an ∞ -category I is strongly finite if it is finite, and in addition for every 𝑖, 𝑗 ∈ I the mapping space Map I ( 𝑖, 𝑗 ) is finite.6.5.2. Example.
Any finite poset is strongly finite.6.5.3.
Example.
Any Reedy category with finitely many objects and finitely many morphisms is stronglyfinite. This follows by induction from [Lur09a, Proposition A.2.9.14]. For example, the full subcategory Δ ≤ 𝑛 ⊆ Δ spanned by the ordinals [ 𝑘 ] for 𝑘 ≤ 𝑛 is strongly finite. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 117 Remark. If I → J is a cartesian or cocartesian fibration such that J is finite and the fibres I 𝑗 arefinite for every 𝑗 ∈ J then I is finite. This follows from the explicit description of cartesian fibrations overthe 𝑛 -simplex via generalized mapping cones, see [Lur09a, §3.2.3]. It then follows that for a strongly finite ∞ -category I the twisted arrow category TwAr( I ) is finite. Similarly, if 𝛼 ∶ I → J is a functor betweenstrongly finite ∞ -categories then all the comma ∞ -categories of 𝛼 are finite.6.5.5. Remark.
Any localisation of a finite ∞ -category by a finite set of arrows is finite, since it can bewritten as a pushout of finite ∞ -categories. In particular, if I is an ∞ -category such that TwAr( I ) is finitethen I is finite, since I can be written as a localisation of TwAr( I ) by a collection of arrows of the form [ 𝑓 ∶ 𝑥 → 𝑦 ] → [id ∶ 𝑥 → 𝑥 ] where 𝑓 runs over a set of representatives of equivalence types in TwAr( I ) .Combining this with Remark 6.5.4 it follows that the condition that I is strongly finite is equivalent to thecondition that TwAr( I ) is finite and all mapping space in I are finite.6.5.6. Lemma.
Let I be a small ∞ -category and C a stable ∞ -category. Then the following holds:i) If the mapping spaces of I are finite then C I is contained in Fun( I op , C ) ⊆ Fun( I op , Pro( C )) .ii) If the twisted arrow category TwAr( I op ) is finite then Fun( I op , C ) is contained in C I .In particular, if I is a strongly finite ∞ -category then C I = Fun( I op , C ) as full subcategories of Fun( I op , Pro( C )) . Remark.
The objects 𝑅 𝑥,𝑖 are cocompact in Fun( I op , Pro( C )) and generate it under limits. Since C I is by definition the closure of 𝑅 𝑥,𝑖 under finite limits it follows that the inclusion C I ⊆ Fun( I op , Pro( C )) induces an equivalence Pro( C I ) ≃ Fun( I op , Pro( C )) . When I is strongly finite Lemma 6.5.6 then gives anequivalence Pro(Fun( I op , C )) ≃ Fun( I op , Pro( C )) . This generalizes [Lur09a, Proposition 5.3.5.15] (in the case of 𝜅 = 𝜔 ) from finite posets to all strongly finite ∞ -categories. Proof of Lemma 6.5.6.
To prove i) it will suffice to show that C I op , which is closed under finite limits in Pro( C ) I op , contains the objects 𝑅 𝑖,𝑥 for every ( 𝑖, 𝑥 ) ∈ I × C . Indeed 𝑅 𝑖,𝑥 ( 𝑗 ) = 𝑥 hom I ( 𝑖,𝑗 ) is contained in C ⊆ Pro( C ) since 𝑥 is in C , hom I ( 𝑖, 𝑗 ) is a finite space, and C is closed inside Pro( C ) under finite limits.Let us now prove ii). We need to show that if TwAr( I ) op is finite then any C -valued presheaf 𝜑 ∶ I op → C is a finite limit of cofree presheaves of the form 𝑅 𝑖,𝑥 . But it is a standard fact that any presheaf 𝜑 iscanonically the limit of the composed TwAr( I ) op -indexed diagram TwAr( I ) op I op × I C × I Pro( C ) I op [ 𝛼 ∶ 𝑖 → 𝑗 ] ( 𝑖, 𝑗 ) ( 𝜑 ( 𝑖 ) , 𝑗 ) 𝑅 𝑗,𝜑 ( 𝑖 )id× 𝜑 which takes values in cofree presheaves. To see this note that the TwAr( I ) op -indexed family of maps 𝑐 [ 𝑖 → 𝑗 ] ∶ 𝜑 ( 𝑗 ) → 𝜑 ( 𝑖 ) determines a TwAr( I ) op -indexed family of maps 𝜑 ⇒ 𝑅 𝑗,𝜑 ( 𝑖 ) , and hence a map 𝜑 ⇒ lim [ 𝑖 → 𝑗 ]∈TwAr( I ) op 𝑅 𝑗,𝜑 ( 𝑖 ) . Evaluating at 𝑘 ∈ I op , the resulting map 𝜑 ( 𝑘 ) ⇒ lim [ 𝑖 → 𝑗 ]∈TwAr( I ) op 𝜑 ( 𝑖 ) hom I ( 𝑗,𝑘 ) ≃ lim [ 𝑖 → 𝑗 → 𝑘 ]∈TwAr( I ∕ 𝑘 ) op 𝜑 ( 𝑖 ) is then seen to be an equivalence by the cofinality of the functors TwAr( I ∕ 𝑘 ) dom ←←←←←←←←←←←←←←←←→ I ∕ 𝑘 ← {id 𝑘 } . (cid:3) Proposition.
Let I be a strongly finite ∞ -category (e.g., any finite poset). Then under the identification C I = Fun( I op , C ) ⊆ Fun( I op , Pro( C )) of Lemma 6.5.6, the quadratic functor Ϙ I corresponds to the functor Ϙ I ( 𝜑 ) = colim 𝑖 ∈ I Ϙ ( 𝜑 ( 𝑖 )) . Its bilinear and linear parts are then given by B I ( 𝜑 ) = colim 𝑖 ∈ I B Ϙ ( 𝜑 ( 𝑖 ) , 𝜑 ( 𝑖 )) and L I ( 𝜑 ) = colim 𝑖 ∈ I L Ϙ ( 𝜑 ( 𝑖 )) ,
18 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE respectively. In addition, if ( C , Ϙ ) is non-degenerate then ( C , Ϙ ) I is non-degenerate with duality (154) [D I 𝜑 ]( 𝑗 ) = colim 𝑖 ∈ I D Ϙ ( 𝜑 ( 𝑖 )) Map I ( 𝑖,𝑗 ) . Proof of Proposition 6.5.8.
The identification of Ϙ I together with its linear and bilinear parts follows di-rectly from Proposition 6.4.3 and Lemma 6.5.6. Now assume that ( C , Ϙ ) is non-degenerate. To prove theformula for the duality, we need to show that for diagrams 𝜑, 𝜓 ∶ I op → C there is an equivalence B I ( 𝜑, 𝜓 ) ≅ nat( 𝜑, D I 𝜓 ) natural in 𝜑, 𝜓 , where D I is given by (154). Expanding the right hand side and using the standard formulafor natural transformations we obtain nat( 𝜑, D I 𝜓 ) ≅ lim [ 𝑖 → 𝑗 ]∈TwAr( I ) op hom C ( 𝜑 ( 𝑗 ) , (D I 𝜓 )( 𝑖 ))≅ lim [ 𝑖 → 𝑗 ]∈TwAr( I ) op hom C ( 𝜑 ( 𝑗 ) , colim 𝑘 ∈ I (D Ϙ 𝜓 ( 𝑘 )) Map I ( 𝑘,𝑖 ) ) ≅ colim 𝑘 ∈ I lim [ 𝑖 → 𝑗 ]∈TwAr( I ) op hom C ( 𝜑 ( 𝑗 ) , D Ϙ 𝜓 ( 𝑘 )) Map I ( 𝑘,𝑖 ) ≅ colim 𝑘 ∈ I lim [ 𝑖 → 𝑗 ]∈TwAr( I 𝑘 ∕ ) op hom C ( 𝜑 ( 𝑗 ) , D Ϙ 𝜓 ( 𝑘 ))≅ colim 𝑘 ∈ I hom C ( 𝜑 ( 𝑘 ) , D Ϙ 𝜓 ( 𝑘 )) ≅ B I ( 𝜑, 𝜓 ) , where we have used the finiteness of TwAr( I ) and Map I (− , −) to commute limits and colimits and thecofinality of the maps TwAr( I 𝑘 ∕ ) cod ←←←←←←←←←←←←←→ I op 𝑘 ∕ ← {id 𝑘 } . (cid:3) We now turn our attention to some structural properties of the tensor and cotensor constructions whichare special to the strongly finite case. Recall from Remark 6.4.10 that for a fixed ∞ -category I , the functor ( C , Ϙ ) ↦ ( C , Ϙ ) I is right adjoint to the functor ( C , Ϙ ) ↦ ( C , Ϙ ) I . Our next goal is to show that when I isstrongly finite the functor ( C , Ϙ ) ↦ ( C , Ϙ ) I is also left adjoint to the functor ( C , Ϙ ) ↦ ( C , Ϙ ) I . To exhibit this,consider for hermitian ∞ -categories ( C , Ϙ ) , ( C ′ , Ϙ ′ ) the evaluation hermitian functor(155) ( C , Ϙ ) ⊗ Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) → ( C ′ , Ϙ ′ ) from Construction 6.2.6. By the universal property of internal functor categories this transposes to a her-mitian functor ( C , Ϙ ) → Fun ex (Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) , ( C ′ , Ϙ ′ )) , and consequently induces for I ∈ Cat ∞ a hermitian functor ( C , Ϙ ) I → Fun ex (Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) , ( C ′ , Ϙ ′ )) I ≃ Fun ex (Fun ex (( C , Ϙ ) , ( C , Ϙ ′ )) I , ( C ′ , Ϙ ′ )) , where we have used the equivalence of Remark 6.4.6. The resulting functor then transposes twice to give ahermitian functor Fun ex (( C , Ϙ ) , ( C , Ϙ ′ )) I → Fun ex (( C , Ϙ ) I , ( C ′ , Ϙ ′ )) . On the other hand, using the equivalence of Remark 6.4.7 the evaluation functor (155) induces a hermitianfunctor ( C , Ϙ ) ⊗ Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) I ≃≃ ( C , Ϙ ) ⊗ Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) ⊗ ( S 𝑝 f , Ϙ u ) I → ( C ′ , Ϙ ′ ) ⊗ ( S 𝑝 f , Ϙ u ) I ≃ ( C ′ , Ϙ ′ ) I which transposes to give a hermitian functor Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) I → Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ ) I ) . Combining the above constructions we hence obtain a pair hermitian functors(156)
Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ ) I ) ⟵ Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) I ⟶ Fun ex (( C , Ϙ ) I , ( C ′ , Ϙ ′ )) . natural in ( C , Ϙ ) , ( C ′ , Ϙ ′ ) and I (indeed, all the operations used above have already been proven natural in§6.2, §6.3 and §6.4 through the various universal properties they encode). ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 119 Proposition. If I is strongly finite then the hermitian functors in (156) are equivalences of hermitian ∞ -categories. Passing to hermitian objects (see (136) ) they then determine a natural equivalence Map
Cat h∞ (( C , Ϙ ) , ( C ′ , Ϙ ′ ) I ) ≃ Map Cat h∞ (( C , Ϙ ) I , ( C ′ , Ϙ ′ )) exhibiting (−) I as left adjoint to (−) I .Proof. To begin, we note that on the level of underlying stable ∞ -categories both functors in (156) areequivalences by Lemma 6.5.6. Indeed, replacing I with I op these identify with the equivalences of stable ∞ -categories(157) Fun ex ( C , C ′ I op ) ≃ ←←←←←←←← Fun ex ( C , C ′ ) I op ≃ ←←←←←←←→ Fun ex ( C I op , C ′ ) underlying those of Remark 6.4.6. Explicitly, the equivalence on the left hand side of (157) associates toa diagram 𝜑 ∶ I op → Fun ex ( C , C ′ ) the exact functor 𝑔 𝜑 ∶ C → C ′ I op = C ′ I given by [ 𝑔 𝜑 ( 𝑥 )]( 𝑖 ) = 𝜑 𝑖 ( 𝑥 ) .Unwinding the definitions, the hermitian structure of the left hand side functor in (156) is given by the map colim 𝑖 ∈ I nat( Ϙ , 𝜑 ∗ 𝑖 Ϙ ′ ) → nat ( Ϙ , colim 𝑖 ∈ I 𝜑 ∗ 𝑖 Ϙ ′ ) = nat ( Ϙ , 𝑔 ∗ 𝜑 Ϙ I ) , which is indeed an equivalence since I is finite and nat( Ϙ , −) is an exact functor.Similarly, the equivalence on the right hand side of (157) associates to a diagram 𝜑 ∶ I op → Fun ex ( C , C ′ ) an exact functor ℎ 𝜑 ∶ C I op = C I → C ′ such that 𝜑 can be recovered from ℎ 𝜑 as 𝜑 𝑖 ( 𝑥 ) = ℎ 𝜑 ( 𝑅 𝑖,𝑥 ) . Let usdenote by ev 𝑖 ∶ C I → C the evaluation at 𝑖 ∈ I functor and by ran 𝑖 ∶ C → C I its right adjoint, given byright Kan extension. In particular, we have ran 𝑖 ( 𝑥 ) = 𝑅 𝑖,𝑥 by definition. Unwinding the definitions, thehermitian structure of the right hand side functor in (156) is then given by the map colim 𝑖 ∈ I nat( Ϙ , 𝜑 ∗ 𝑖 Ϙ ′ ) = colim 𝑖 ∈ I nat( Ϙ , ran ∗ 𝑖 ℎ ∗ 𝜑 Ϙ ′ ) = colim 𝑖 ∈ I nat(ev ∗ 𝑖 Ϙ , ℎ ∗ 𝜑 Ϙ ′ ) → nat(lim 𝑖 ev ∗ 𝑖 Ϙ , ℎ ∗ 𝜑 Ϙ ′ ) = nat( Ϙ I , ℎ ∗ 𝜑 Ϙ ′ ) , which is indeed an equivalence since I is finite and nat(− , ℎ ∗ 𝜑 Ϙ ′ ) is an exact functor. (cid:3) Corollary.
Let I be a strongly finite ∞ -category. Then the functor ( C , Ϙ ) ↦ ( C , Ϙ ) I from Cat h∞ toitself preserves all limits and the functor ( C , Ϙ ) ↦ ( C , Ϙ ) I preserves all colimits. Corollary.
For a fixed strongly finite ∞ -category I , the functor ( C , Ϙ ) ↦ ( C , Ϙ ) I is internally corep-resented by ( S 𝑝 f , Ϙ u ) I . More precisely, there is an equivalence of hermitian ∞ -categories ( C , Ϙ ) I ≃ Fun ex (( S 𝑝 f , Ϙ u ) I , ( C , Ϙ )) natural in ( C , Ϙ ) and I . Corollary.
For a strongly finite ∞ -category I the following conditions are equivalent:i) the operation ( C , Ϙ ) ↦ ( C , Ϙ ) I preserves Poincaré ∞ -categories and Poincaré functors.ii) the hermitian ∞ -category ( S 𝑝 f , Ϙ u ) I is Poincaré;iii) the operation ( C , Ϙ ) ↦ ( C , Ϙ ) I preserves Poincaré ∞ -categories and Poincaré functors.iv) the hermitian ∞ -category ( S 𝑝 f , Ϙ u ) I is Poincaré; In a similar spirit, we may deduce that the criterion for duality preservation of Proposition 6.3.18 holdsfor tensors as well in the strongly finite case:6.5.13.
Corollary.
Let 𝛼 ∶ I → J be a map of strongly finite ∞ -categories and ( C , Ϙ ) a non-degeneratehermitian ∞ -category. If the induced map I 𝑖 ∕ → J 𝛼 ( 𝑗 )∕ is cofinal for every 𝑖 ∈ I then the induced hermitianfunctor ( 𝛼 ∗ , 𝜂 𝛼 ) ∶ ( C , Ϙ ) I → ( C , Ϙ ) J is duality preserving. In particular, if ( C , Ϙ ) I and ( C , Ϙ ) J are Poincaréthen ( 𝛼 ∗ , 𝜂 𝛼 ) is a Poincaré functor.Proof. Identify ( C , Ϙ ) I with Fun ex (( S 𝑝 f , Ϙ u ) I , ( C , Ϙ )) as a functor of I using Corollary 6.5.11 and applyProposition 6.3.18. (cid:3) We now describe some additional functoriality exhibited by the tensor and cotensor constructions in thestrongly finite case.
20 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
Construction (Exceptional functoriality) . Let 𝛽 ∶ J → I be a functor between strongly finite ∞ -categories. Then by Lemma 6.5.6 the full subcategory C I ⊆ Fun( I op , Pro( C )) is sent into C J ⊆ Fun( J op , Pro( C )) by restriction along any functor 𝛽 ∶ J → I . On the cotensor side, the comma categories of 𝛽 are all finite byRemark 6.5.4 and hence the operation of right Kan extension 𝛽 ∗ ∶ C J → C I exists for any stable C . Nowsuppose that 𝛽 is cofinal . Then we can refine 𝛽 ∗ ∶ C I → C J and 𝛽 ∗ ∶ C J → C I to hermitian functors asfollows. In the tensor case we simply note that the cofinality of 𝛽 yields a natural equivalence Ϙ J ( 𝛽 ∗ 𝜑 ) = colim 𝑗 ∈ J Ϙ ( 𝜑 ( 𝛽 ( 𝑗 )) ≃ colim 𝑖 ∈ I Ϙ ( 𝜑 ( 𝑖 )) = Ϙ I ( 𝜑 ) and so we obtain a heritian functor ( 𝛽 ∗ , 𝜗 𝛽 ) ∶ ( C , Ϙ ) I → ( C , Ϙ ) J in which 𝜗 𝛽 ∶ Ϙ I ⇒ Ϙ J ◦ 𝛽 ∗ is an equivalence.For cotesors we consider the counit 𝛽 ∗ 𝛽 ∗ 𝜓 ⇒ 𝜓 and use the coinitiality of 𝛽 op to obtain a map Ϙ J ( 𝜓 ) ⇒ Ϙ J ( 𝛽 ∗ 𝛽 ∗ 𝜓 ) = lim 𝑗 ∈ J op Ϙ ( 𝛽 ∗ 𝜓 ( 𝛽 ( 𝑗 )) ≃ lim 𝑖 ∈ I op Ϙ ( 𝛽 ∗ 𝜓 ( 𝑖 )) = Ϙ I ( 𝛽 ∗ 𝜓 ) giving a hermitian refinement ( 𝛽 ∗ , 𝜗 𝛽 ) ∶ ( C , Ϙ ) J → ( C , Ϙ ) I .6.5.15. Example.
A common source of cofinal maps are maps 𝛽 ∶ J → I which admit a left adjoint 𝛼 ∶ I → J . Unwinding the definitions and using Remarks 6.3.12 and 6.4.9, we see that in this case the exceptionalhermitian functors ( 𝛽 ∗ , 𝜗 𝛽 ) and ( 𝛽 ∗ , 𝜗 𝛽 ) of Construction 6.5.14 coincide with the direct hermitian functors ( 𝛼 ∗ , 𝜂 𝛼 ) and ( 𝛼 ∗ , 𝜂 𝛼 ) , respectively.6.5.16. Remark.
Let I , J be two strongly finite ∞ -categories and ( C , Ϙ ) a Poincaré ∞ -category. The identi-fication(158) ( C , Ϙ ) I ≃ Fun ex (( S 𝑝 f , Ϙ u ) I , ( C , Ϙ )) of Remark 6.4.6, being natural in I , identifies, for every map 𝛼 ∶ I → J , the associated hermitian functor ( 𝛼 ∗ , 𝜂 𝛼 ) ∶ ( C , Ϙ ) J → ( C , Ϙ ) I with the one induced from ( 𝛼 ∗ , 𝜂 𝛼 ) ∶ ( S 𝑝 f , Ϙ u ) I → ( S 𝑝 f , Ϙ u ) J upon taking upontaking internal functor categories to ( C , Ϙ ) . Similarly, the identification(159) ( C , Ϙ ) I ≃ Fun ex (( S 𝑝 f , Ϙ u ) I , ( C , Ϙ )) of Corollary 6.5.11 identifies the hermitian functor ( 𝛼 ∗ , 𝜂 𝛼 ) ∶ ( C , Ϙ ) I → ( C , Ϙ ) J with the one induced from ( 𝛼 ∗ , 𝜂 𝛼 ) ∶ ( S 𝑝 f , Ϙ u ) I → ( S 𝑝 f , Ϙ u ) J . Unravelling all definitions, and observing the similarity between theformulas for the direct and exceptional functorialities, we see that the equivalence (158) also identifies forevery 𝛽 ∶ J → I the exceptional hermitian functor ( 𝛽 ∗ , 𝜗 𝛽 ) ∶ ( C , Ϙ ) J → ( C , Ϙ ) I with the one induced from theexceptional functor ( 𝛽 ∗ , 𝜗 𝛽 ) ∶ ( S 𝑝 f , Ϙ u ) I → ( S 𝑝 f , Ϙ u ) J upon taking internal functor categories to ( C , Ϙ ) , andsimilarly the equivalence (159) also identifies the exceptional hermitian functor ( 𝛽 ∗ , 𝜗 𝛽 ) ∶ ( C , Ϙ ) I → ( C , Ϙ ) J with the one induced from ( 𝛽 ∗ , 𝜗 𝛽 ) ∶ ( S 𝑝 f , Ϙ u ) J → ( S 𝑝 f , Ϙ u ) I .6.5.17. Remark.
Like the ordinary functoriality of the tensor and cotensor constructions, the exceptionalfunctorialities are compatible with composition. More precisely, if K 𝛽 ′ ←←←←←←←←←→ J 𝛽 ←←←←←←→ I are a pair of composablecofinal maps then the hermitian functor (( 𝛽 ◦ 𝛽 ′ ) ∗ , 𝜗 𝛽 ◦ 𝛽 ′ ) ∶ ( C , Ϙ ) I → ( C , Ϙ ) K is naturally equivalent to thecomposite of ( 𝛽 ∗ , 𝜗 𝛽 ) and (( 𝛽 ′ ) ∗ , 𝜗 𝛽 ′ ) and the hermitian functor (( 𝛽 ◦ 𝛽 ′ ) ∗ , 𝜗 𝛽 ◦ 𝛽 ′ ) ∶ ( C , Ϙ ) K → ( C , Ϙ ) I isnaturally equivalent to the composite of ( 𝛽 ∗ , 𝜗 𝛽 ) and ( 𝛽 ′∗ , 𝜗 𝛽 ′ ) . Indeed, by Remark 6.5.16 it will sufficeto check this for the tensor construction, where it amounts to the fact the Beck-Chevalley transformationrelating restriction and colimits is compatible with composition of restriction maps.6.5.18. Remark.
Comparing the explicit formulas of the direct and exceptional functorialities we see thatif 𝛽 ∶ J → I is a cofinal map then the resulting hermitian structure on the exceptional-direct composite ( 𝛽 ∗ , 𝜂 𝛽 ) ◦ ( 𝛽 ∗ , 𝜗 𝛽 ) ∶ ( C , Ϙ ) J → ( C , Ϙ ) J is given by the map Ϙ J ( 𝜓 ) ⇒ Ϙ J ( 𝛽 ∗ 𝛽 ∗ 𝜓 ) induced by the counit 𝛽 ∗ 𝛽 ∗ 𝜓 ⇒ 𝜓 . Similarly, the resulting hermitian structure on the direct-exceptionalcomposite ( 𝛽 ∗ , 𝜗 𝛽 ) ◦ ( 𝛽 ∗ , 𝜂 𝛽 ) ∶ ( C , Ϙ ) J → ( C , Ϙ ) J is induced by the counit of C I ⟂ C J in a similar manner. Inparticular, if 𝛽 is fully-faithful then these counits are equivalences, in which case we get that the exceptionalfunctoriality of 𝛽 gives a one sided inverse to its direct functoriality. More generally, it can be checked thatfor any cofinal 𝛽 the exceptional functoriality ( 𝛽 ∗ , 𝜗 𝛽 ) is right adjoint to the direct functoriality ( 𝛽 ∗ , 𝜂 𝛽 ) , ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 121 and the exceptional functoriality ( 𝛽 ∗ , 𝜗 𝛽 ) is left adjoint to the direct functoriality ( 𝛽 ∗ , 𝜂 𝛽 ) , when these termsare understood with respect to the (∞ , -categorical structure of Cat h∞ determined by the internal functorcategories of §6.2.6.6. Finite complexes and Verdier duality.
In this final section we explore the particular case when I isthe poset of simplices of a finite simplicial complex. In this context the tensor construction plays a centralrole in [Lur11], where its duality functor is identified as a form of Verdier duality . Filling in the details insome of the arguments of loc. cit., we will show that tensoring and cotensoring by the poset of faces of a finitesimplicial complex preserves Poincaré ∞ -categories, and that the hermitian functors associated to maps ofsimplicial complexes (direct functoriality) and refinements of triangulations (exceptional functoriality) arePoincaré.Recall that a finite simplicial complex 𝐾 consists of a finite set of vertices 𝐾 and a collection I 𝐾 ofnon-empty subsets 𝑆 ⊆ 𝐾 , called faces, which contain all singletons and are downwards closed in thesense that if 𝑆 is a face of 𝐾 and 𝑆 ′ ⊆ 𝑆 then 𝑆 ′ is a face of 𝐾 as well. The dimension of a face 𝑆 is bydefinition dim( 𝑆 ) ∶= | 𝑆 | − 1 . We may realize a simplicial complex 𝐾 geometrically as the subspace | 𝐾 | ofthe full simplex on 𝐾 obtained as the union of the given faces. For a finite simplicial complex 𝐾 we willconsider I 𝐾 as a poset, and consequently a category, by inverse inclusion, that is, there is a unique morphism 𝑆 → 𝑆 ′ if 𝑆 ′ ⊆ 𝑆 . A map of simplicial complexes 𝐾 → 𝐿 is by definition a map of sets 𝐾 → 𝐿 whichsends every face of 𝐾 to a face of 𝐿 . In particular, such a map determines a map of posets 𝛼 ∶ I 𝐾 → I 𝐿 .If 𝐾 is a simplicial complex then a refinement of 𝐾 consists of a simplicial complex 𝐿 together with ahomeomorphism | 𝐿 | ≅ ←←←←←←←→ | 𝐾 | which carries the realization of every face of 𝐿 into the realization of a faceof 𝐾 , and such that the realization of every face of 𝐾 in | 𝐾 | is a union of faces of 𝐿 . We note that such ahomeomorphism determines in particular a map of posets 𝛽 ∶ I 𝐿 → I 𝐾 which sends every face of 𝐿 to theface of 𝐾 containing it under the homoemorphism | 𝐿 | ≅ | 𝐾 | .6.6.1. Proposition ([Lur11, Lecture 19, Proposition 3]) . Let ( C , Ϙ ) be a Poincaré ∞ -category and 𝐾 a finitesimplicial complex with poset of faces I 𝐾 . Then the hermitian ∞ -categories ( C , Ϙ ) I 𝐾 and ( C , Ϙ ) I 𝐾 arePoincaré.Proof. By Corollary 6.5.12 it will suffice to prove the claim for ( C , Ϙ ) I 𝐾 . We need to show that for every 𝜑 ∈ C I 𝐾 the map(160) 𝜑 → D I 𝐾 D I 𝐾 𝜑 is an equivalence. To do so it suffices to show it for a system of objects that generate under colimits. Wechoose the set 𝜑 = 𝑅 𝑥,𝑆 for 𝑆 ∈ I 𝐾 and 𝑥 ∈ C . Now the face 𝑆 corresponds to an injective map of simplicialcomplexes Δ 𝑛 → 𝐾 which in turn determines an inclusion of posets 𝛼 ∶ I Δ 𝑛 ⊆ I 𝐾 such that 𝛼 ([ 𝑛 ]) = 𝑆 and 𝑅 𝑥,𝑆 is the right Kan extension along 𝛼 of the constant diagram 𝜑 𝑥 ∶ I Δ 𝑛 → C with value 𝑥 . Now the map 𝛼 satisfies the hypothesis of Corollary 6.5.13 (in fact, the map the needs to be cofinal is an isomorphismof posets, see also Proposition 6.6.2 below), and so the hermitian functor ( 𝛼 ∗ , 𝜂 𝛼 ) ∶ C I Δ 𝑛 → C I 𝐾 is dualitypreserving. We can hence reduce to the case of 𝐾 = Δ 𝑛 and 𝜑 = 𝜑 𝑥 . Using Proposition 6.5.8 we now have(161) D I Δ 𝑛 𝜑 𝑥 ( 𝑆 ) = colim ∅ ≠ 𝑇 ⊆ [ 𝑛 ] { D 𝑥 𝑆 ⊆ 𝑇 otherwise . To calculate this colimit let us denote by 𝜄 ∶ I 𝑆 Δ 𝑛 ⊆ I Δ 𝑛 the subposet spanned by those 𝑇 ⊆ [ 𝑛 ] such that 𝑆 ⊈ 𝑇 . Then the functor whose colimit is calculated in (161) can be identified with the cofibre of the map 𝜄 ∗ 𝜄 ! 𝜑 𝑥 → 𝜑 𝑥 , and so we get that [D I Δ 𝑛 𝜑 𝑥 ]( 𝑆 ) = cof [ colim I 𝑆 Δ 𝑛 D 𝑥 → colim I Δ 𝑛 D 𝑥 ] . Now since I 𝑆 Δ 𝑛 is closed under subfaces in I Δ 𝑛 it corresponds to some subcomplex Δ 𝑛 , which we readilyidentify as the join 𝜕 Δ 𝑆 ∗ Δ 𝑆 ′ , where 𝑆 ′ = [ 𝑛 ] − 𝑆 is the complement of 𝑆 ′ , and we have used the notation Δ 𝑆 and Δ 𝑆 ′ to denote the corresponding faces, considered as subcomplexes. The poset I 𝑆 Δ 𝑛 is hence weaklycontractible if 𝑆 ′ ≠ ∅ , that is, if 𝑆 ≠ [ 𝑛 ] , and is weakly equivalent to 𝜕 Δ 𝑛 of 𝑆 = [ 𝑛 ] . We thus concludethat [D I Δ 𝑛 𝜑 𝑥 ]( 𝑆 ) = { Σ 𝑛 D 𝑥 𝑆 = [ 𝑛 ]0 otherwise .
22 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
By the same argument we then have [D I Δ 𝑛 D I Δ 𝑛 𝜑 𝑥 ]( 𝑆 ) = colim ∅ ≠ 𝑇 ⊆ [ 𝑛 ] { DΣ 𝑛 D 𝑥 𝑇 = [ 𝑛 ]0 otherwise , so that D I Δ 𝑛 D I Δ 𝑛 𝜑 𝑥 is constant with value Σ 𝑛 DΣ 𝑛 D 𝑥 ≃ 𝑥 , and is in particular equivalent to 𝜑 𝑥 . We needhowever to make sure that specifically the evaluation map is an equivalence. Since both D I Δ 𝑛 D I Δ 𝑛 𝜑 𝑥 and 𝜑 𝑥 are constant it will suffice to show that the component of the evaluation map at 𝑆 = [ 𝑛 ] is an equivalence.Unwinding the definitions, this is the composed map 𝑥 ≃ ←←←←←←←→ DD 𝑥 → colim 𝑇 ∈ I Δ 𝑛 lim 𝑇 ′ ∈ I opΔ 𝑛 { DD 𝑥 if 𝑇 = 𝑇 ′ = [ 𝑛 ]0 otherwise. = lim 𝑇 ′ ∈ I opΔ 𝑛 colim 𝑇 ∈ I Δ 𝑛 { DD 𝑥 if 𝑇 = 𝑇 ′ = [ 𝑛 ]0 otherwise.whose component at level 𝑇 ′ is for 𝑇 ′ ≠ [ 𝑛 ] and is the inclusion of the 𝑇 = [ 𝑛 ] component in the colimitotherwise. The invertibility of this map then reduces to the fact that in a stable ∞ -category an 𝑛 -cube iscartesian if and only if it is cocartesian. (cid:3) The perfectness of hermitian structures asserted in Proposition 6.6.1 is accompanied by the followingduality preservation statement:6.6.2.
Proposition. [Lur11, Lecture 19]
Let ( C , Ϙ ) be a Poincaré ∞ -category.i) If 𝐾 → 𝐿 is a map of finite simplicial complexes with 𝛼 ∶ I 𝐾 → I 𝐿 the induced map of posets of facesthen the induced hermitian functors ( 𝛼 ∗ , 𝜂 𝛼 ) ∶ ( C , Ϙ ) I 𝐾 → ( C , Ϙ ) I 𝐿 and ( 𝛼 ∗ , 𝜂 𝛼 ) ∶ ( C , Ϙ ) I 𝐿 → ( C , Ϙ ) I 𝐾 are Poincaré.ii) If 𝐿 is a refinement of a simplicial complex 𝐾 and 𝛽 ∶ I 𝐿 → I 𝐾 is the associated map of posetsof faces then 𝛽 is cofinal and the exceptional hermitian functors ( 𝛽 ∗ , 𝜗 𝛽 ) ∶ ( C , Ϙ ) I 𝐾 → ( C , Ϙ ) I 𝐿 and ( 𝛽 ∗ , 𝜗 𝛽 ) ∶ ( C , Ϙ ) I 𝐿 → ( C , Ϙ ) I 𝐾 are Poincaré.Proof. By Remark 6.5.16 it will suffice to prove the tensor case. For the first statement we observe thatthe functor 𝛼 ∶ I 𝐾 → I 𝐿 satisfies the criterion of Corollary 6.5.13, since for every face 𝑆 ∈ I 𝐾 the functor ( I 𝐾 ) 𝑆 ∕ → ( I 𝐿 ) 𝛼 ( 𝑆 )∕ admits a left adjoint sending 𝑇 ⊆ 𝛼 ( 𝑆 ) to its inverse image in 𝑆 .To prove the second statement, we begin by showing that 𝛽 is cofinal. Indeed for any simplex 𝑆 ∈ I 𝐾 theposet I 𝐿 × I 𝐾 ( I 𝐾 ) 𝑆 ∕ has geometric realization homeomorphic to a simplex and so it is weakly contractible.To prove that ( 𝛽 ∗ , 𝜗 𝛽 ) is Poincaré, it will suffice to show that for every generator 𝑅 𝑥,𝑆 ∈ C I 𝐾 the associatedmap 𝛽 ∗ D I 𝐾 𝑅 𝑥,𝑆 → D I 𝐿 𝛽 ∗ 𝑅 𝑆,𝑥 is an equivalence. Now the face 𝑆 corresponds to an injective map of simplicial complexes Δ 𝑆 → 𝐾 (where Δ 𝑆 denotes the full simplex with vertex set 𝑆 ) , which in turn determines an inclusion of posets 𝛼 ∶ I Δ 𝑆 ⊆ I 𝐾 such that 𝑅 𝑥,𝑆 is the right Kan extension along 𝛼 of the constant diagram 𝜑 𝑥 ∶ I Δ 𝑆 → C with value 𝑥 . Theinverse image of I Δ 𝑆 in I 𝐿 then determines a subcomplex 𝐿 𝑆 ⊆ 𝐿 which is a refinement of Δ 𝑆 .Let us denote by 𝛽 𝑆 ∶ I 𝐿 𝑆 → I Δ 𝑆 the induced refinement map and by ̃𝛼 ∶ I 𝐿 𝑆 ↪ I 𝐿 the inclusion. Since I Δ 𝑆 and J are downward closed the pointwise formula for right Kan extensions implies that the square ( C , Ϙ ) I Δ 𝑆 ( C , Ϙ ) I 𝐿𝑆 ( C , Ϙ ) I 𝐾 ( C , Ϙ ) I 𝐿 ( 𝛽 ∗ 𝑆 ,𝜗 𝛽𝑆 )( 𝛼 ∗ ,𝜂 𝛼 ) ( ̃𝛼 ∗ ,̃𝜂 𝛼 )( 𝛽 ∗ ,𝜗 𝛽 ) commutes. Since the vertical hermitian functor are Poincaré by the first part we may reduce to the casewhere 𝐾 is the 𝑛 -simplex Δ 𝑛 , 𝐿 is some refinement of Δ 𝑛 , and 𝜑 = 𝜑 𝑥 . Now for 𝑇 ∈ I 𝐿 let us denoteby I 𝑇𝐿 ⊆ I 𝐿 the subposet spanned by those faces which do not contain 𝑇 , and by J 𝑇𝐿 ⊆ I 𝑇𝐿 the subposetspanned by those faces whose image in I 𝐾 does not contain 𝛽 ( 𝑇 ) . We note that both of these subposets aredownward closed and correspond to subcomplexes of 𝐿 . In particular, I 𝑇𝐿 corresponds to the subcomplex ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 123 𝐿 𝑇 ⊆ 𝐿 obtained by removing all faces which contain 𝑇 , and J 𝑇𝐿 corresponds to the subcomplex 𝐿 𝑇 ⊆ 𝐿 𝑇 obtained by removing all faces whose image in 𝐾 contains 𝑇 . We also note that 𝐿 𝑇 is a refinement of thesubcomplex 𝜕 Δ 𝛽 ( 𝑇 ) ∗ Δ [ 𝑛 ]− 𝛽 ( 𝑇 ) ⊆ Δ 𝑛 obtained from Δ 𝑛 by removing all the faces which contain 𝛽 ( 𝑇 ) .Calculating as in the proof of Proposition 6.6.1 and using that refinement maps are cofinal as establishedabove we may identify the cofibre of the map [ 𝛽 ∗ D I Δ 𝑛 𝜑 𝑥 ]( 𝑇 ) → [D I 𝐿 𝛽 ∗ 𝜑 𝑥 ]( 𝑇 ) for 𝑇 ∈ I 𝐿 with the total cofibre of the square colim J 𝑇𝐿 D 𝑥 colim I 𝐿 D 𝑥 colim I 𝑇𝐿 D 𝑥 colim I 𝐿 D 𝑥 . To finish the proof it will hence suffice to show that | 𝐿 𝑇 | → | 𝐿 𝑇 | is a weak homotopy equivalence. Let 𝑝 ∈ | Δ 𝑛 | be a point in the interior of the face 𝑇 (and hence also in the interior of the face 𝛽 ( 𝑇 ) ). Then wehave a sequence of inclusions | 𝐿 𝑇 | ⊆ | 𝐿 𝑇 | ⊆ 𝑈 where 𝑈 ⊆ | Δ 𝑛 | is the complement of 𝑝 . The desiredresult now follows from the fact that both | 𝐿 𝑇 ⊆ 𝑈 and | 𝐿 𝑇 | ⊆ 𝑈 are deformation retracts; this is a generalproperty of simplicial complexes: if one removes a point from the realization of a simplicial complex thenthe result deformation retracts to the subcomplex spanned by all the simplices which do not contain thatpoint. (cid:3)
7. H
YPERBOLIC AND METABOLIC P OINCARÉ CATEGORIES
From a conceptual view point, it is arguably tempting to regard hermitian structures on stable ∞ -categoriesas categorified versions of hermitian forms on modules. Similarly, Poincaré ∞ -categories correspond tomodules equipped with a unimodular hermitian form. Inspired by this informal perspective, in this sectionwe will identify a surprisingly comprehensive variety of such categorified counterparts, including the cat-egorified analogues of bilinear forms, perfect bilinear forms, hyperbolic objects, metabolic objects and thealgebraic Thom construction. Beyond its conceptually pleasing effect, it turns out that many of the con-structions encountered via this perspective give explicit left and right adjoints to various natural functors,a feature which we will repeatedly exploit in subsequent instalments of this project. In particular, the mainpractical outcomes of our categorified stroll will include the following:i) After exploring the categorified counterparts of bilinear forms in §7.2, we will deduce that the associa-tion C ↦ Hyp( C ) described in §2.2 is both left and right adjoint to the forgetful functor Cat p∞ → Cat ex∞ ,with unit and counit given on the side of
Cat p∞ by the Poincaré functors hyp∶ Hyp( C ) → ( C , Ϙ ) and f gt ∶ ( C , Ϙ ) → Hyp( C ) described in §2.2. In addition, we will show that Hyp is C -equivariant withrespect to the op action on Cat ex∞ and the trivial action on
Cat p∞ , and that hyp and f gt are equivari-ant natural transformations. All this information is best organized in the setting of C -categories andMackey functors, which we will explore in §7.4. This will also be the basis for our organization inPaper [II] of algebraic K -theory, Grothendieck-Witt theory and L -theory into a single functor takingvalues in genuine C -spectra, which we call the real K -theory spectrum .ii) While forming the categorified analogue of metabolic objects, Lagrangians, and the algebraic Thomconstruction in §7.3, we will deduce an explicit formula for the left and right adjoints to the inclusion Cat p∞ → Cat h∞ . This will be exploited in Paper [II] when setting up the framework of algebraic surgery in the context of Poincaré ∞ -categories, and in analysing the effect of additive and bordism invariantfunctors applied to the Q -construction. We will also use it Paper [IV] for constructing the localisinganalogue of the Grothendieck-Witt spectrum and for proving that Cat p∞ is compactly generated.The present section is organized as follows. We begin in §7.1 with some preliminary material on bifibra-tions , a notion used for encoding space valued bifunctors which are covariant in one entry and contravariantin the other. By translating results from [Lur17, §5.2.1] to the context of bifibrations we deduce in particularthat the ∞ -category of perfect symmetric bifibrations is equivalent to Cat hC ∞ . In §7.2 we specialize to thesetting of stable ∞ -categories and replace space valued bifunctors by spectrum valued ones. This leads tothe notion of bilinear ∞ -categories as analogous of pairs of modules equipped with a bilinear form. We
24 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE also consider the variants of requiring the form to be perfect and/or symmetric, and identify, using §7.1,the notion of a perfect symmetric ∞ -category with that of a stable ∞ -category equipped with a perfectduality. All these notions accept natural forgetful functors from either Cat p∞ (in the perfect case) or Cat h∞ .Studying all of them on equal footing allows one to efficiently identify left and right adjoints to these for-getful functors, which constitutes the main content of §7.2. In particular, we will see that the association C ↦ Hyp( C ) described in §2.2 gives a two-sided adjoint to the forgetful functor Cat p∞ → Cat ex∞ . In §7.3we will discuss the categorified analogous of metabolic objects and Lagrangians, and will show that thecategorified analogue of the Thom construction enables one to produce both a left and a right adjoint to theinclusion
Cat p∞ → Cat h∞ . We will also prove a generalization of the non-categorified Thom construction,thus providing in particular a proof for Proposition 2.3.17 which was stated in §2.3. Finally, in §7.4 we willdiscuss C -categories and Mackey functors, and show how to view various players in the present paper inthat context. In particular, we will show that the relations between quadratic and bilinear functors, betweenhermitian and bilinear ∞ -categories, and between Poincaré and stable ∞ -categories, can all be understoodon the same footing, giving another take on the categorified perspective. The results of §7.4 will be primar-ily used in Paper [II] in order to define the real K -theory spectrum, but the C -equivariant properties of thehyperbolic construction resulting from it will be useful for a variety of other purposes as well.7.1. Preliminaries: pairings and bifibrations.
Let A , B be two ∞ -categories. By a correspondence on the pair ( A , B ) we will simply mean a functor 𝑏 ∶ A op × B → S . In particular, we will think of acorrespondence as a space valued functor on pairs ( 𝑥, 𝑦 ) with 𝑥 ∈ A and 𝑦 ∈ B , which is contravariant in 𝑥 and covariant in 𝑦 . The prototypical example to have in mind is taking A = B = C for some ∞ -category C , with 𝑏 ( 𝑥, 𝑦 ) = Map C ( 𝑥, 𝑦 ) . Given a correspondence 𝑏 ∶ A op × B → S and 𝑦 ∈ B , evaluation at 𝑦 yieldsa presheaf of spaces 𝑏 (− , 𝑦 ) on A , which we can unstraighten to obtain a right fibration(162) ∫ 𝑥 ∈ A 𝑏 ( 𝑥, 𝑦 ) → A . Since the presheaf 𝑏 (− , 𝑦 ) ∈ Fun( A op , S ) depends functorially 𝑦 , so does the domain of (162). In fact,we can identify the arrow (162) with the map ∫ 𝑥 ∈ A 𝑏 ( 𝑥, 𝑦 ) → ∫ 𝑥 ∈ A ∗ associated to the terminal map ofcorrespondences 𝑏 → ∗ , and so the entire arrow (162) depends functorially in 𝑦 . Equivalently, we may viewit as a natural transformation between two Cat ∞ -valued functors on B , the second of which is constant withvalue A . We then define(163) Pair( A , B , 𝑏 ) ∶= ∫ 𝑦 ∈ B ∫ 𝑥 ∈ A 𝑏 ( 𝑥, 𝑦 ) → A × B , to be the ∞ -category over A × B obtained by unstraightening (162) over B . The ∞ -category Pair( A , B , 𝑏 ) can informally be described as having objects triples ( 𝑥, 𝑦, 𝛽 ) where 𝑥 ∈ A and 𝑦 ∈ B are objects and 𝛽 ∈ 𝑏 ( 𝑥, 𝑦 ) is a 𝑏 -valued pairing on 𝑥 and 𝑦 . A map from ( 𝑥, 𝑦, 𝛽 ) to ( 𝑥 ′ , 𝑦 ′ , 𝛽 ′ ) is then given by maps 𝑓 ∶ 𝑥 → 𝑥 ′ , 𝑔 ∶ 𝑦 → 𝑦 ′ and a homotopy 𝜂 ∶ 𝑔 ∗ 𝛽 ∼ 𝑓 ∗ 𝛽 ′ ∈ 𝑏 ( 𝑥, 𝑦 ′ ) , where 𝑓 ∗ and 𝑔 ∗ encode the contravariant andcovariant dependence of 𝑏 on 𝑥 and 𝑦 , respectively.We point out that since the above construction involves both the cartesian unstraightening ∫ 𝑥 ∈ A and thecocartesian straightening ∫ 𝑦 ∈ B , the resulting arrow in (163) is neither a cartesian nor a cocartesian fibration.We can nonetheless describe it as follows: recall that a bifibration (see [Lur09a, Definition 2.4.7.2]) is apair of maps A 𝑞 ←←←←←←← X 𝑝 ←←←←←←→ B consisting of a cartesian fibration 𝑞 ∶ X → A and a cocartesian fibration 𝑝 ∶ X → B , such that the 𝑞 -cartesian edges are exactly those projecting to equivalences in B and the 𝑝 -cocartesian edges are exactlythose projecting to equivalences in A . Equivalently, the pair of maps 𝑝, 𝑞 forms a bifibration if and onlyif ( 𝑞, 𝑝 ) ∶ X → A × B is a map of cocartesian fibrations over B whose fibres are right fibrations, and ifand only if ( 𝑞, 𝑝 ) ∶ X → A × B is map of cartesian fibrations over A whose fibres are left fibrations. Inparticular, one readily verifies that for a correspondence 𝑏 ∶ A op × B → S , the pair of projections A ← Pair( A , B , 𝑏 ) → B constitutes a bifibration. In fact, this association determines an equivalence between correspondences 𝑏 ∶ A op × B → S and bifibrations A ← X → B (see [Ste18] and [AF20, §4]), and can be considered ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 125 as a bivariant form of the space-valued straightening-unstraightening equivalence. In particular, using thecoherent compatibility of the straightening-unstraightening equivalence with base change as establishedin [GHN17, Corollary A.31], this equivalence integrates to an equivalence(164) (−) ← Pair(− , − , −) → (−) ∶ ∫ ( A , B )∈Cat ∞ ×Cat ∞ Fun( A op × B , S ) ≃ ←←←←←←←→ BiFib , where BiFib ⊆ Fun(Λ , Cat ∞ ) is the full subcategory spanned by the bifibrations.7.1.1. Example.
In the case of B = A = C and the canonical correspondence 𝑚 C ∶= Map C (− , −) ∶ C op × C → S , the ∞ -category Pair( C , C , 𝑚 C ) canonically identifies with the arrow category Ar( C ) ∶= Fun(Δ , C ) .7.1.2. Remark.
The notion of a correspondence A op × B → S can equivalently be encoded by a rightfibration M → A × B op . Right fibrations of this form were studied in [Lur17, §5.2.1] under the name pairings . In particular, the ∞ -category CPair of [Lur17, Construction 5.2.1.14] of pairings is naturallyequivalent to
BiFib , since both are equivalent to ∫ A , B∈Cat ∞ Fun( A op × B , S ) . Under this equivalence, thecanonical bifibration C ← Ar( C ) → C of Example 7.1.1 encoding the mapping space correspondencecorresponds to the right fibration TwAr( C ) → C × C op .7.1.3. Definition.
We will say that a right biexact correspondence 𝑏 ∶ A op × B → S is right-representable if the associated map B → Psh( A ) factors through the image of the Yoneda embedding 𝜄 ∶ A ↪ Psh( A ) . Inthis case resulting the functor 𝑑 ∶ B → A is characterized by a natural equivalence Map C ( 𝑥, 𝑑 ( 𝑦 )) ≃ 𝑏 ( 𝑥, 𝑦 ) .7.1.4. Remark.
If a correspondence 𝑏 ∶ A op × B → S is right-representable then Pair( A , B , 𝑏 ) naturallyidentifies with the fibre product Ar( A ) × A B along the target projection Ar( A ) → A and 𝑑 ∶ B → A .More generally, if we Yoneda embed A in Psh( A ) = Fun( A op , S ) , then 𝑏 becomes tautologically rightrepresentable by the functor ̃𝑑 ∶ B → Psh( A ) sending 𝑦 ∈ B to 𝑏 (− , 𝑦 ) ∈ Psh( A ) . We may then write Pair( A , B , 𝑏 ) ≃ A × Psh( A ) Ar(Psh( A )) × Psh( A ) B where the fibre product is taken along the Yoneda embedding A → Psh( A ) and ̃𝑑 ∶ B → Psh( B ) , and Ar(Psh( A )) projects to the domain on the left hand side and to the target on the right hand side.7.1.5. Definition.
We will say that a correspondence 𝑏 ∶ A op × B → S is perfect if 𝑏 is right-representableand the associated representing functor 𝑑 ∶ B → A is an equivalence. In this case, we will say that a pairing ( 𝑥, 𝑦, 𝛽 ) ∈ Pair( A , B , 𝑏 ) is perfect if the map 𝑥 → 𝑑 ( 𝑦 ) determined by 𝛽 is an equivalence. Similarly, wewill say that a bifibration A ← X → B is perfect if its classifying correspondence A op × B → S is. In thiscase we will say that an object in X is perfect if it corresponds to a perfect pairing under the identificationof X with Pair( A , B , 𝑏 ) . We will then say that a map of bifibration [ A ← X → B ] → [ A ′ ← X ′ → B ′ ] is perfect if it sends perfect objects of X to perfect objects of X ′ . We will denote by BiFib p ⊆ BiFib the(non-full) subcategory spanned by the perfect bifibrations and the perfect maps between them.While
BiFib p ⊆ BiFib is not a full subcategory, it does satisfy the following weaker property [Lur16,Definition 20.1.1.2]:7.1.6.
Definition.
Let C be an ∞ -category and C ′ ⊆ C a subcategory. We will say that C ′ is replete if forevery 𝑥 ∈ C ′ and 𝑦 ∈ C such that there is an equivalence 𝛼 ∶ 𝑥 ≃ ←←←←←←←→ 𝑦 in C , then 𝑦 ∈ C ′ and there exists anequivalence 𝛽 ∶ 𝑥 ≃ ←←←←←←←→ 𝑦 ′ in C ′ whose image in C is homotopic to 𝛼 .7.1.7. Remark.
The condition of being a replete subcategory C ′ ⊆ C is detected on the level of the homotopycategories: it is equivalent to saying that Ho C ′ is closed under isomorphisms and for every 𝑥, 𝑦 ∈ C ′ thesubset Hom Ho C ′ ( 𝑥, 𝑦 ) ⊆ Hom Ho C ( 𝑥, 𝑦 ) contains all isomorphisms from 𝑥 to 𝑦 in C .7.1.8. Example.
The subcategory
BiFib p ⊆ BiFib is replete. This follows from the observation that anyequivalence between perfect bifibrations is necessarily a perfect map.7.1.9.
Example.
The subcategory
Cat ex∞ ⊆ Cat ∞ is replete.
26 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
Remark.
For any subcategory C ′ ⊆ C the induced map Map C ′ ( 𝑥, 𝑦 ) → Map C ( 𝑥, 𝑦 ) is a (−1) -truncated map of spaces for every 𝑥, 𝑦 ∈ C ′ , that is, its fibres are either empty or contractible. If thesubcategory C ′ is replete then the induced map 𝜄 C ′ → 𝜄 C on core groupoids is also (−1) -truncated. Thisimplies that every replete subcategory inclusion C ′ ↪ C is (−1) -truncated as a map in Cat ∞ , that is, forevery test ∞ -category D the induced map Map
Cat ∞ ( D , C ′ ) → Map
Cat ∞ ( D , C ) is (−1) -truncated. In many contexts, this property is what makes replete subcategories behave more like“subobjects” than general subcategories.The following proposition records the content of [Lur17, Remark 5.2.1.20] in the context of bifibrations:7.1.11. Proposition.
The composed functor (165)
BiFib p BiFib Cat ∞ × Cat ∞ Cat ∞ ( A , B , 𝑏 ) ( A , B ) A is an equivalence of ∞ -categories. An inverse is given by C ↦ [ C ← Ar( C ) → C ] .Proof. Since the association C ↦ [ C ← Ar( C ) → C ] is visibly a one-sided inverse to (165) we see that thelatter is in particular essentially surjective, and it will hence suffice to show that it is also fully-faithful, Onthe other hand, if A ← X → B is a perfect bifibration then it is equivalent in BiFib to A ← Ar A → A via the associated functor 𝑑 ∶ B → A and the natural equivalence 𝑏 ( 𝑥, 𝑦 ) ≃ hom A ( 𝑥, 𝑑 ( 𝑦 )) . It will hencesuffice to show that for every C ∈ Cat ∞ and every perfect bifibration A ← X → B the induced map Map
BiFib p ([ A ← X → B ] , [ C ← Ar( C ) → C ]) → Map
Cat ∞ ( A , C ) is an equivalence. But this now follows from [Lur17, Proposition 5.2.1.18] under the equivalence betweenbifibrations as above and pairings in the sense of [Lur17, Definition 5.2.1.5], see Remark 7.1.2. (cid:3) The ∞ -category BiFib ⊆ Fun(Λ , Cat ∞ ) of bifibrations carries a natural action of C induced by theaction of C on Λ switching the vertices Δ {1} and Δ {2} and post-composing with the op action on Cat ∞ .Explicitly, this action sends a bifibration A ← X → B to the bifibration B op ← X op → A op (indeed, thelatter is again a bifibration since taking opposites switches cartesian and cocartesian fibrations). A C -fixedstructure on a given bifibration A ← X → B can then be described as a duality D ∶ X ≃ ←←←←←←←→ X op on X , anequivalence A ≃ B op , and a a duality-preserving refinement of X → A × B ≃ A × A op . We will refer tosuch a structure as a Λ -duality on A ← X → B , and will call a bifibration equipped with a Λ -duality a symmetric bifibrations .7.1.12. Proposition.
The C -fixed ∞ -category BiFib hC participates in a cartesian fibration BiFib hC → Cat ∞ classified by the functor C ↦ Fun( C op × C op , S ) hC .Proof. Equipping Δ {1} ∐ Δ {2} with the swap action and restricting along the C -equivariant inclusion Δ {1} ∐ Δ {2} ⊆ Λ we get that the cocartesian fibration(166) BiFib → Cat ∞ × Cat ∞ , naturally refines to a C -equivariant functor, where C acts on the target by flipping the factors and takingopposites. Since cartesian fibration are preserved under limits, taking C -fixed points results in a cartesianfibration(167) BiFib hC → (Cat ∞ × Cat ∞ ) hC . We now observe that the equivalence (id , (−) op ) ∶ Cat ∞ × Cat ∞ ≃ ←←←←←←←→ Cat ∞ × Cat ∞ ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 127 intertwines the flip-op action on the left hand side with the flip action on the right hand side, and so thetarget of (166) is a coinduced C -object. We may consequently identify the target of (167) with Cat ∞ andwrite is a cocartesian fibration(168) BiFib hC → Cat ∞ . Since taking fibres commutes with fixed points we may identify the fibres of (168) with the C -fixed points ofthe corresponding fibres of (166). Now C ∈ Cat ∞ corresponds to the fixed object ( C , C op ) ∈ Cat ∞ × Cat ∞ ,and so the fibre of (168) over C is the C -fixed points of Fun( C op × C op , S ) , as claimed. (cid:3) Construction.
By Proposition 7.1.12 we may identify the notion of a symmetric bifibration withthat of a pair ( C , 𝑏 ) where C is an ∞ -category and 𝑏 ∈ Fun( C op × C op , S ) hC is a symmetric functor. Given ( C , 𝑏 ) , the associated symmetric bifibration is C op ← Pair( C , C op , 𝑏 ) → C equipped with its Λ -duality which we will denote by (D pair , 𝜏 ) . It is given explicitly by the duality D pair ( 𝑥, 𝑦, 𝛽 ) = ( 𝑦, 𝑥, 𝜎 𝑥,𝑦 ( 𝛽 ) ) , on Pair( C , C op , 𝑏 ) , where 𝜎 𝑥,𝑦 ∶ 𝑏 ( 𝑥, 𝑦 ) ≃ ←←←←←←←→ 𝑏 ( 𝑦, 𝑥 ) is given by the symmetric structure of 𝑏 , equipped withthe tautological duality-preserving structure of the projection Pair( C , C op , 𝑏 ) → C × C op .7.1.14. Proposition.
The C -action on BiFib restricts to a C -action on BiFib p . Under the equivalence BiFib p ≃ Cat ∞ of Proposition 7.1.11, this action corresponds to the op -action on Cat ∞ . In particular,we may identify the notion of a perfect symmetric bifibration with that of an ∞ -category equipped with aperfect duality.Proof. We claim that the functor
Cat ∞ → BiFib C ↦ [ C ← Ar( C ) → C ] admits a natural C -equivariant structure, where C acts on Cat ∞ via the op -action. To see this, it willsuffice to construct a C -equivariant structure for the composed map Cat ∞ → BiFib ↪ Fun(Λ , Cat ∞ ) .This composed functor is by definition given by mapping out of the diagram 𝑒 ∶= [Δ {0} → Δ ← Δ {1} ] ,and so it will suffice to put a C -equivariant structure on 𝑒 ∶ (Λ ) op → Cat ∞ . Such an equivariant structureis then given by the canonical duality on D Δ ∶ Δ → (Δ ) op which switches {0} and {1} (since Δ is thenerve of an ordinary category not much coherence is needed in order to verify this fact).By Proposition 7.1.11 it now follows in particular that the C -action on BiFib preserves the subcategory
BiFib p . This subcategory is replete by Example 7.1.8, and so by Remark 7.1.10 the C -action on BiFib restricts to an essentially unique C -action on BiFib p making the inclusion BiFib p ↪ BiFib equivariant.By the above this action must then coincide with the op -action via the equivalence BiFib p ≃ Cat ∞ , asdesired. (cid:3) Example.
In the situation of Construction 7.1.13, if the symmetric correspondence 𝑏 ∶ C op × C op → S is perfect with duality D ∶ C ≃ ←←←←←←←→ C then the associated symmetric bifibration identifies by Proposition 7.1.11with C ← Ar( C ) → C and by Proposition 7.1.14 (and its proof) the associated Λ -duality D pair corresponds to the arrow duality [ 𝑥 → 𝑦 ] ↦ [D 𝑦 → D 𝑥 ] induced on the functor category Ar( C ) = Fun(Δ , C ) from the dualities of Δ and C . The remainder of this section is devoted to producing an explicit formula expressing the mapping spacesin Pair( A , B , 𝑏 ) in terms of 𝑏 and the mapping spaces in A × B . This will be useful for us in §7.3 when wewill need to upgrade the pairings construction to the hermitian setting. To begin, consider the following
28 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE diagram in
Fun(Λ , Cat ∞ ) : [ 𝜕 Δ {0 , ← 𝜕 Δ {0 , ⨿ 𝜕 Δ {1 , → 𝜕 Δ {1 , ] [ 𝜕 Δ {0 , ← Δ {0} ⨿ Δ {1} ⨿ Δ {2} → 𝜕 Δ {1 , ][ Δ {0 , ← Δ {0 , ∐ Δ {1 , → Δ {1 , ] [ Δ {0 , ← Δ → Δ {1 , ][ Δ {0 , ← Δ {0} ∐ Δ {2} → Δ {1 , ] [ Δ {0 , ← Δ {0 , → Δ {1 , ] . Here, left going internal arrows are all given on vertices by [0 ↦ , [1 ↦ , [2 ↦ and all the right goinginternal arrows are given by [0 ↦ , [1 ↦ , [2 ↦ , while the external arrows always preserve the vertexlabels. The upper square is cocartesian, as can be tested levelwise using the standard categorical equivalence Δ {0 , ∐ Δ {1} Δ {1 ,
2} ≃ ←←←←←←←→ Δ . In addition, all entries in this square carry compatible Λ -dualities, inducedby the canonical duality Δ ←←←←←←←→ (Δ ) op which switches between and and between Δ {0 , and (Δ {1 , ) op .Mapping out of the above square now yields a C -equivariant functor Fun(Λ , Cat ∞ ) → Fun(Δ ×Δ , Cat ∞ ) sending [ A 𝑞 ←←←←←←← X 𝑝 ←←←←←←→ B ] to the diagram(169) X Δ {0} × B Δ{1} X Δ {1} × A Δ{1} X Δ {2} X Δ {0} × B Δ{1} X Δ {1} × ( A × B ) Δ{1} X Δ {1} × A Δ{1} X Δ {2} X Δ × ( A × B ) Δ2 [ A Δ {0 , × B Δ {1 , ] X Δ {0 , ⨿ Δ {1 , × ( A × B ) Δ{0 , ⨿ Δ{1 , [ A Δ {0 , × B Δ {1 , ] X Δ {0 , × ( A × B ) Δ{0 , [ A Δ {0 , × B Δ {1 , ] X Δ {0} ⨿ Δ {2} × ( A × B ) Δ{0} ⨿ Δ{2} [ A Δ {0 , × B Δ {1 , ] in which the top square is cartesian. In addition, using that all entries in this diagram compatibly project to X Δ {0} and X Δ {2} we will view this as a diagram in (Cat ∞ ) ∕ X × X .7.1.16. Proposition.
When [ A ← X → B ] is a bifibration the bottom vertical arrows in (169) are equiv-alences. In particular, inverting these and taking the external rectangle yields a C -equivariant functor BiFib → Fun(Δ × Δ , Fun(Λ , Cat ∞ )) which sends [ A ← X → B ] to a cartesian square (170) of the form (170) Ar( X ) X × A X × B XX × A × B [Ar( A × B )] × A × B X X × A X × A × B X × B X . in (Cat ∞ ) ∕ X × X . Here, the two projections to X are given by the domain and codomain projections in the caseof Ar( X ) and by the projection to the two extremal factors in the three other cases. In addition, this functortakes values in Fun(Δ ×Δ , BiFib) after post-composing with the inclusion (Cat ∞ ) ∕ X × X → Fun(Λ , Cat ∞ ) . Writing X = Pair( A , B , 𝑏 ) for some correspondence 𝑏 ∶ A op × B → S the square of bifibrations (170)corresponds to a cartesian square of correspondences X op × X → S of the form(171) Map X (( 𝑥, 𝑦, 𝛽 ) , ( 𝑥 ′ , 𝑦 ′ , 𝛽 ′ )) 𝑏 ( 𝑥, 𝑦 ′ )Map A ( 𝑥, 𝑥 ′ ) × Map B ( 𝑦, 𝑦 ′ ) 𝑏 ( 𝑥, 𝑦 ′ ) × 𝑏 ( 𝑥, 𝑦 ′ ) , giving, in particular, an explicit pullback formula for the mapping spaces in Pair( A , B , 𝑏 ) .7.1.17. Remark.
In the situation of Proposition 7.1.16, the C -equivariance of the functor in question meansin particular that if a bifibration A ← X → B carries a Λ -duality then all the entries in the square (170)inherit such a duality and all arrows in the square are duality preserving. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 129 Proof of Proposition 7.1.16.
By definition the arrows in X which map to equivalences in A are exactly the 𝑝 -cocartesian arrows, and the arrows which map to equivalences in B are exactly the 𝑞 -cartesian arrows. Itthen follows that the projections X Δ {0 , × B Δ{0 , B → X Δ {0} × A Δ{0} A Δ {0 , and X Δ {1 , × A Δ{1 , A → X Δ {2} × B Δ{2} B Δ {1 , are equivalences, and so the right bottom vertical arrow in (169) (which is the fibre product of these twomaps over the identity on A × B ) is an equivalence. Similarly, the projections X Δ × B Δ{0 , B → A Δ × A Λ20 X Λ × B Δ{0 , B → A Δ × A Δ{0 , X Δ {0 , are both equivalences, from which it follows that the left bottom vertical arrow in (169), which is a basechange of the above composite, is an equivalence. (cid:3) Bilinear and symmetric ∞ -categories. In the present section we define and study the notion of bi-linear and perfect bilinear ∞ -categories. We then show that the notion of an ∞ -category equipped with asymmetric bilinear form, which we call a symmetric ∞ -category, can be identified with a C -fixed bilinear ∞ -category, and similarly in the perfect case. Using the results of §7.1 we then identify the ∞ -categoryof perfect bilinear ∞ -categories with Cat ex∞ itself, through which we also deduce an equivalence between ∞ -categories with perfect dualities and C -fixed objects of Cat ex∞ with respect to the op -action. We then con-struct left and right adjoints to the forgetful functors from hermitian to symmetric to bilinear ∞ -categories,and similarly in the Poincaré/perfect case, where we finally recover the hyperbolic construction C ↦ Hyp( C ) acting as both left and right adjoint to the forgetful functor Cat p∞ → Cat ex∞ .Let A , B be stable ∞ -categories. We will say that a correspondence 𝑏 ∶ A op × B → S is right biexact if it preserves finite limits in each variable separately. Such a correspondence lifts in an essentially uniquemanner to a bilinear functor A × B → S 𝑝 . More precisely, post-composition with the infinite loop spacefunctor induces an equivalence between bilinear functors A op × B → S 𝑝 and right biexact correspondences A × B op → S .7.2.1. Definition.
We will denote by
Fun b ( A , B ) ⊆ Fun( A op × B , S 𝑝 ) the full subcategory spanned by thebilinear functors and write Cat b ∶= ∫ ( A , B )∈Cat ex∞ ×Cat ex∞ Fun b ( A , B ) for the ∞ -category of triples ( A , B , B) where A , B are stable ∞ -categories and B ∶ A op × B → S is abilinear functor. We will refer to the object ( A , B , B) of Cat b as bilinear categories .7.2.2. Example.
For a stable ∞ -category C the mapping correspondence 𝑚 C ∶ C op × C → S of Example 7.1.1is right biexact. We may hence consider the triple ( C , C , 𝑚 C ) as a bilinear ∞ -category.7.2.3. Example.
Let A , B be stable ∞ -categories. If a correspondence 𝑏 ∶ A × B op → S is right repre-sentable by an exact functor 𝑑 ∶ A → B then 𝑏 is right biexact.7.2.4. Example.
Let C be a stable ∞ -category. Consider the ∞ -category Seq( C ) of exact sequences in C (see Notation 2.3.18). The pair of projections C Seq( C ) C 𝑥 𝑦 then constitute a bifibration. Indeed, a map of exact sequences 𝑦 𝑧 𝑥𝑦 ′ 𝑧 ′ 𝑥 ′ is a cocartesian lift of 𝑦 → 𝑦 ′ if and only if the left square is exact, or, equivalently, if the map 𝑥 → 𝑥 ′ isan equivalence, and similarly forms a cartesian lift of 𝑥 → 𝑥 ′ if and only if its component 𝑦 → 𝑦 ′ is an
30 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE equivalence. To identify the correspondence seq C ∶ C op × C → S associated to this bifibration we use thefact that every exact square as in (52) extends in an essentially unique manner to a diagram 𝑥 ′ 𝑦 𝑧 𝑥 𝑦 ′ in which all squares except the top right one are exact. We note that such a diagram determines in particularequivalences 𝑥 ′ ≃ Ω 𝑥 and 𝑦 ′ ≃ Σ 𝑦 . At the same time, the forgetful functor sending a diagram as a above toits external square(172) 𝑥 ′ 𝑦 ′ is an equivalence as well. In particular, if we denote by Ar Ω ( C ) ⊆ Fun(Δ × Δ , C ) the full subcategoryspanned by the exact squares of the form (172) then we obtain an equivalence of bifibrations C Seq( C ) CC Ar Ω ( C ) C Ω ≃ [− 𝜎 ] The correspondence associated to Ar Ω ( C ) can then be identified with ( 𝑥 ′ , 𝑦 ′ ) ↦ Ω Map C ( 𝑥 ′ , 𝑦 ′ ) , and so thecorrespondence associated to Seq can be written as seq C ( 𝑥, 𝑦 ) = Ω Map(Ω 𝑥, Σ 𝑦 ) . In particular, it is rightbiexact.7.2.5. Remark.
It follows from Lemma 1.4.1 that the defining cartesian fibration
Cat b → Cat ex∞ × Cat ex∞ isalso a cocartesian fibration. Applying the precise same argument as in the proof of Proposition 6.1.2 wemay consequently conclude that
Cat b has all small limits and colimits, and that those are preserved by theprojection to Cat ex∞ × Cat ex∞ .Given a bilinear category ( A , B , B) we may consider the pairings ∞ -category associated to the underly-ing right biexact correspondence Ω ∞ B . To simplify notation we will denote Pair( A , B , B) ∶= Pair( A , B , Ω ∞ B) . We note that since Ω ∞ B is right biexact the functor 𝑦 ↦ ̃𝑑 ( 𝑦 ) = Ω ∞ B(− , 𝑦 ) from B to Psh( A ) takes valuesin the full subcategory Ind( A ) ⊆ Psh( A ) spanned by the right exact presheaves. In addition, in this case Ind( A ) is also stable, the Yoneda embedding A → Ind( A ) is exact, and the functor ̃𝑑 ∶ B → Ind( A ) givenby Ω ∞ B is exact as well. As in Remark 7.1.4 we may then identify(173) Pair( A , B , B) ≃ A × Ind( A ) Ar(Ind( A )) × Ind( A ) B , a description from which we see that Pair( A , B , B) is stable and that exact squares in Pair( A , B , B) aredetected in A × B . This means in particular that any map of bifibrations [ A ← Pair( A , B , B) → B ] → [ A ′ ← Pair( A ′ , B ′ , B ′ ) → B ′ ] whose component A → A ′ , B → B ′ are exact, is also exact on Pair(− , − , −) . Invoking again the coherentcompatibility of the straightening-unstraightening equivalence with base change as established in [GHN17,Corollary A.31], we may consequently assemble the association ( A , B , B) ↦ Pair( A , B , B) to a functor Pair(− , − , −) ∶ Cat b → Cat ex∞ , taking values in stable ∞ -categories and exact functors between them. It then follows that the bivariantstraightening equivalence (164) restricts to a (non-full) subcategory inclusion(174) Cat b → BiFib ( A , B , B) ↦ [ A ← Pair( A , B , B) → B ] ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 131 whose image is spanned by those bifibrations A ← X → B for which A , B are stable and the associatedcorrespondence is right biexact and by those maps of bifibrations whose components are exact functors.7.2.6. Remark.
The subcategory inclusion (174) is replete (see Definition 7.1.6).7.2.7.
Remark.
The association in (174) can also be viewed as a functor from
Cat b to Fun(Λ , Cat ex∞ ) . Assuch, it is fully-faithful and its image is spanned by those diagrams A ← X → B in Cat ex∞ which arebifibrations with associated correspondence right biexact. From this description we see that
Cat b is closedunder finite products in Fun(Λ , Cat ex∞ ) , and hence inherits from its the property of being semiadditive, seeProposition 6.1.7.7.2.8. Definition.
We will say that a bilinear category ( A , B , B) is perfect if its corresponding bifibrationis perfect in the sense of Definition 7.1.5, and say that a map of perfect bilinear categories is perfect ifthe corresponding map of bifibrations is so. We will denote by Cat pb ⊆ Cat b the (non-full) subcategoryspanned by the perfect bilinear categories and perfect maps between them.7.2.9. Remark.
As in Example 7.1.8, the subcategory inclusion
Cat pb ⊆ Cat b is replete.By construction we have a commutative diagram Cat pb Cat b Cat ex∞ × Cat ex∞
BiFib p BiFib Cat ∞ × Cat ∞ in which both squares are cartesian (the right one because (bi)exact functors to spectra are determined bytheir underlying space-valued functors) and all vertical arrows, as well as the horizontal arrows in the leftsquare, are replete subcategory inclusions.7.2.10. Proposition.
The composed functor (175)
Cat pb Cat b Cat ex∞ × Cat ex∞
Cat ex∞ ( A , B , 𝑏 ) ( A , B ) A is an equivalence of ∞ -categories. An inverse is given by A ↦ ( A , A , 𝑚 A ) .Proof. This follows directly from Proposition 7.1.11 since a perfect bifibration A ← X → B belongs to theimage of Cat pb if and only if A ≃ B is stable (in which case the associated correspondence is automaticallyright biexact by Examle 7.2.3). (cid:3) Recall from §7.1 that the ∞ -category BiFib ⊆ Fun(Λ , Cat ∞ ) of bifibrations carries a natural actionof C induced by the action of C on Λ switching the vertices Δ {1} and Δ {2} and post-composing withthe op action on Cat ∞ . Explicitly, this action which sends a bifibration A ← X → B to the bifibration B op ← X op → A op . Since taking opposites also preserves stable ∞ -categories and exact functors, theabove action induces a C -action on the replete subcategory Cat b ↪ BiFib . Explicitly, the resulting C -action sends a triple ( A , B , B) to the triple ( B op , A , B swap ) , where B swap ∶ B × A op → S is B pre-composedwith the swap equivalence B × A op ≃ A op × B .7.2.11. Proposition.
The C -fixed ∞ -category (Cat b ) hC participates in a cartesian fibration (Cat b ) hC → Cat ex∞ which classifies the functor C ↦ Fun s ( C ) .Proof. The claim follows from its unstable counterpart Proposition 7.1.12 since the restriction along exactfunctors preserves right exact correspondences. (cid:3)
Definition.
We will denote by
Cat sb∞ ∶= (Cat b ) hC the C -fixed points of Cat b .
32 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
By Proposition 7.2.11 we may identify the objects of
Cat sb∞ with pairs ( C , B) where C is a small stable ∞ -category and B ∶ C op × C op → S 𝑝 is a symmetric bilinear functor, that is an object of the ∞ -category Fun s ( C ) = Fun b ( C ) hC . We will refer to such a pair as a symmetric ∞ -category. We will refer to morphisms ( C , B) → ( C ′ , B ′ ) in Cat sb∞ as symmetric functors . Using again Proposition 7.2.11 we may identify thesewith pairs ( 𝑓 , 𝛽 ) where 𝑓 ∶ C → C ′ is an exact functor and 𝛽 ∶ B → ( 𝑓 × 𝑓 ) ∗ B ′ is a natural transformation.7.2.13. Definition.
We will say that a symmetric bilinear ∞ -category ( C , B) is non-degenerate if B is non-degenerate in the sense of Definition 1.2.2. In this case B is induced by a (possibly imperfect) duality D B ∶ C → C op , and every symmetric functor ( 𝑓 , 𝛽 ) ∶ ( C , B) → ( C ′ , B ′ ) induces a natural transformation 𝜏 𝛽 ∶ 𝑓 D B ⇒ D B ′ 𝑓 op via Lemma 1.2.4. We will say that such a symmetric functor ( 𝑓 , 𝛽 ) is duality pre-serving if 𝜏 𝛽 is an equivalence. We will say that ( C , B) is perfect if D is an equivalence. We then denoteby Cat ps∞ ⊆ Cat sb∞ the (non-full) subcategory spanned by the perfect symmetric ∞ -categories and dualitypreserving functors between them.A version of the following result was already proven in [HLAS16] and [HSV19].7.2.14. Lemma.
The C -action on Cat b preserves the replete subcategory Cat pb of perfect bilinear ∞ -categories. In addition, under the equivalence Cat sb∞ ≃ (Cat b ) hC of Proposition 7.2.10 the subcategory Cat ps∞ corresponds to the subcategory (Cat pb ) hC ⊆ (Cat b ) hC .Proof. We need to verify two things:i) If ( C , B) is a symmetric ∞ -category then B is perfect in the sense of Definition 7.2.13 if and only ifthe right exact correspondence Ω ∞ B is perfect in the sense of Definition 7.1.5.ii) A symmetric ( 𝑓 , 𝛽 ) ∶ ( C , B) → ( C ′ , B ′ ) between perfect symmetric ∞ -categories is duality preservingif and only if the induced functor Pair( C , C op , B) → Pair( C ′ , C ′op , B ′ ) preserves perfect pairings.To prove i) we need to show that B ∶ C op × C op → S 𝑝 can be written as B( 𝑥, 𝑦 ) ≃ hom C ( 𝑥, D 𝑦 ) for someequivalence D ∶ C op → C if and only if Ω ∞ can be written as Map C ( 𝑥, D 𝑦 ) for some equivalence D ∶ C op → C . Clearly the former implies the latter, but the latter also implies the former since post-composition with Ω ∞ induces an equivalence between bilinear functors C op × C op → S 𝑝 and right exact correspondences C op × C op → S . To verify ii), consider the commutative diagram Ω ∞ B( 𝑥, 𝑦 ) Ω ∞ B ′ ( 𝑓 ( 𝑥 ) , 𝑓 ( 𝑦 ))Map C ( 𝑥, D B ( 𝑦 )) Map C ( 𝑓 ( 𝑥 ) , 𝑓 D B ( 𝑦 )) Map C ( 𝑓 ( 𝑥 ) , D B ′ 𝑓 ( 𝑦 )) Ω ∞ 𝛽 ≃ ( 𝜏 𝛽 ) ∗ ≃ furnished by Remark 1.2.6. We then see that the map 𝑓 D B ( 𝑦 ) → D B ′ 𝑓 ( 𝑦 ) is an equivalence if and onlyif 𝑓 sends the tautological perfect pairing (D B ( 𝑦 ) , 𝑦, 𝜄 ) ∈ Pair( C , C op , B) to a perfect pairing in C ′ . Sinceevery perfect pairing is equivalent to a tautological perfect pairing (D B ( 𝑦 ) , 𝑦, 𝜄 ) for some 𝑦 we may thusconclude that 𝜏 𝛽 ∶ 𝑓 D B ⇒ D B ′ 𝑓 op is an equivalence if and only if the induced functor Pair( C , C op , B) → Pair( C ′ , C ′op , B ′ ) preserves perfect pairings, as desired. (cid:3) Combining Lemma 7.2.14 and Proposition 7.1.14 we now conclude:7.2.15.
Corollary.
The forgetful functor
Cat ps∞ → Cat ex∞ lifts to an equivalence
Cat ps∞ ≃ (Cat ex∞ ) hC where C acts on Cat ex∞ by the op -action. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 133 The constructions we have made so far can now be summarized by the following commutative diagram(176)
Cat p∞ Cat ps∞ (Cat ex∞ ) hC Cat ex∞
Cat h∞ Cat sb∞ (Cat b ) hC Cat b Cat ex∞
Cat ex∞ (Cat ex∞ × Cat ex∞ ) hC Cat ex∞ × Cat ex∞ ≃=≃ in which the vertical arrows in the top row are replete subcategory inclusions, the top squares are cartesian,the vertical arrows in the bottom row are cartesian fibrations, and the horizontal arrows in the middle rowpreserve cartesian edges. In fact, all the vertical maps in the bottom row are also cocartesian fibrations andthe horizontal arrows in the middle row also preserve cocartesian edges, see Corollary 1.4.2 and Propo-sition 1.4.3. We also recall that the C -action on Cat ex∞ × Cat ex∞ is given by ( A , B ) ↦ ( B op , A op ) and thecomposed maps Cat ex∞ → Cat ex∞ × Cat ex∞ along the bottom and right sides both equivalent to the diagonalmap.A useful feature of the diagram (176) is that all arrows in it admit both left and right adjoints, and inparticular all arrows preserve all limits and colimits. For the vertical functors on the bottom row, these areall cartesian and cocartesian fibrations and their fibres admit zero objects, thus the zero section gives a twosided adjoint in all three cases. The left and right adjoints of the horizontal functors in (176) will be studiedin this section, with special interest given to the resulting adjoints of the composed arrow
Cat p∞ → Cat ex∞ onthe top tow, which are both given by the construction C ↦ Hyp( C ) of §2.2. The left and right adjoint to thetop left vertical inclusion Cat p∞ ↪ Cat h∞ will be produced in §7.3 below using the pairings construction.Left and right adjoints to the top middle vertical inclusion then follow by formal considerations, while leftand right adjoints to the top right inclusion can be constructed in a similar manner, see Remarks 7.3.19and 7.3.24.The remainder of this section is dedicated to the construction of left and right adjoints to the horizontalfunctors in (176). We begin with the top left square:7.2.16. Proposition.
The functor
Cat h∞ → Cat sb∞ admits fully-faithful left and a right adjoints, given by sending ( C , B) to ( C , Ϙ qB ) and ( C , Ϙ sB ) , respectively.Furthermore, the units and counits of these adjunctions project to equivalences in Cat ex∞ .Proof.
We have a diagram
Cat h∞ Cat sb∞
Cat ex∞ 𝑓𝑝 𝑞 where 𝑝 and 𝑞 are cartesian fibrations and the forgetful functor 𝑓 preserves cartesian edges. The functor 𝑓 has fibrewise left and right adjoints by Corollary 1.3.6, and so by [Lur17, Proposition 7.3.2.6] 𝑓 admitsa left adjoint whose unit transformation is sent to an equivalence in Cat ex∞ . The counit of the adjunction isgiven by the fibrewise counit and thus is an equivalence by Corollary 1.3.6, according to which the fibrewiseleft adjoint is fully faithful. Similarly, using the dual of [Lur17, Proposition 7.3.2.6] we see that 𝑓 admits aright adjoint whose associated unit is mapped to an equivalence in Cat ex∞ and whose counit is an equivalencein
Cat sb∞ . (cid:3) Proposition.
The fully-faithful adjoints of Proposition 7.2.16 map
Cat ps∞ to Cat p∞ , and yield fully-faithful left and right adjoints to the forgetful functor Cat p∞ → Cat ps∞ .Proof.
Since the left and right adjoints constructed in Proposition 7.2.16 are fully-faithful and since thecondition for a hermitian ∞ -category of being non-degenerate is defined via the non-degeneracy of theunderlying symmetric bilinear form we see that these adjoints send non-degenerate symmetric ∞ -categories
34 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE to non-degenerate hermitian ∞ -categories and similarly perfect symmetric ∞ -categories to Poincaré ∞ -categories. For the same reason these adjoints also send duality-preserving symmetric funtors to dualitypreserving functors. To see that these now yield left and right adjoints to Cat p∞ → Cat ps∞ it is enough tocheck that all the units and counits are contained in the respective subcategories. Now on the side of
Cat ps∞ these units and counits are equivalences, and so by triangle identities these unit and counit in
Cat p∞ map toequivalences in Cat ps∞ . This implies that they are duality preserving, as desired. (cid:3)
Definition.
Given a perfect bilinear functor
B ∈ Fun s ( C ) , the images of ( C , B) under the fully-faithful left adjoint and right adjoint of Proposition 7.2.17 are, as in the case of Proposition 7.2.16, givenby ( C , Ϙ qB ) and ( C , Ϙ sB ) , respectively. We will refer to Poincaré ∞ -categories of this form as quadratic and symmetric Poincaré ∞ -categories, respectively.Taking now the top external rectangle in (176) we obtain a square(177) Cat p∞ Cat ex∞
Cat h∞ Cat b in which the right vertical arrow is given by(178) Cat ex∞
Cat pb Cat b A ( A , A , 𝑚 A ) . ≃ ∈ ∈ Proposition.
In the square (177) both horizontal arrows admit left and right adjoints, both compat-ible with the vertical subcategory inclusions. In addition, both adjoints of the bottom horizontal arrow areequivalent and given by the formula ( A , B , B) ↦ ( A × B op , B) . Pre-composing the formula of Proposition 7.2.19 with the functor (178) we conclude7.2.20.
Corollary.
The forgetful functor
U ∶ Cat p∞ → Cat ex∞ admits both a left and a right adjoint. The twoare equivalent and given by the association A ↦ ( A × A op , 𝑚 A ) = Hyp( A ) . Remark.
The unit exhibiting
Hyp as right adjoint to U and the counit exhibiting Hyp as left adjointto U are given respectively by the Poincaré functors Hyp( C ) hyp ←←←←←←←←←←←←←←→ ( C , Ϙ ) fgt ←←←←←←←←←←←←→ Hyp( C ) of (42). The counit U Hyp( C ) = C ⊕ C op → C exhibiting Hyp as right adjoint to U is then given by theprojection on the first fact and the unit C → U Hyp( C ) = C ⊕ C op exhibiting Hyp as left adjoint to U isgiven by the inclusion into the first summand. Here we are using the direct sum notation keeping in mindthat Cat ex∞ is semi-additive, see Proposition 6.1.7.7.2.22.
Remark.
By [Lur17, Corollary 7.3.2.7] the functor
Hyp inherits a lax symmetric monoidal structureby virtue of being right adjoint to the symmetric monoidal functor
U ∶ Cat p∞ → Cat ex∞ (see Theorem 5.2.7).In particular, U ⊣ Hyp is a symmetric monoidal adjunction. Applying the same argument to the symmetricmonoidal functor U op ∶ (Cat p∞ ) op → (Cat ex∞ ) op we get that the adjunction U op ⊣ Hyp op (opposite to Hyp ⊣ U ) is symmetric monoidal and Hyp also carries an oplax symmetric monoidal structure.7.2.23.
Remark.
The symmetric monoidal structure on the forgetful functor
U ∶ Cat p∞ → Cat ex∞ yields forevery ( C , Ϙ ) ∈ Cat p∞ a commutative square Cat p∞ Cat ex∞
Cat p∞ Cat ex∞ . U( C , Ϙ ) ⊗ (−) C ⊗ (−)U ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 135 Passing to right adjoints using Corollary 7.2.20 and Corollary 6.2.15 we obtain a commutative square
Cat p∞ Cat p∞ Cat p∞ Cat ex∞ . Fun ex (( C , Ϙ ) , −) Fun ex ( C , −)HypHyp In particular, we have a natural equivalence
Fun ex (( C , Ϙ ) , Hyp( C ′ , Ϙ ′ )) ≃ Hyp Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) . Proof of Proposition 7.2.19.
Propositions 7.2.16 and 7.2.17 provide left and right adjoints to the horizontalarrows in the left square of (176), which are furthermore compatible with the vertical subcategory inclu-sions. In the same diagram, the horizontal arrows in the middle square are equivalences, and the horizontalarrows in the right square have left and right adjoints as well. Indeed, in both cases these are the naturalmaps from C -fixed points to underlying objects, which admit left and right adjoint since Cat ex∞ and
Cat b admits products and coproducts (see Proposition 6.1.1 and Remark 7.2.5). In addition, since Cat ex∞ is semi-additive (Proposition 6.1.7) these products and coproducts coincide in that case, and we get that the leftand right adjoints of (Cat ex∞ ) hC → Cat ex∞ are both give by the formula C ↦ C × C op . This implies that theleft and right adjoints for the horizontal maps in the right square in (176) are compatible with the verticalreplete subcategory inclusions, and hence we may conclude that the horizontal arrows in (177) admit leftand right adjoints, both compatible with the vertical subcategory inclusions.It is left to obtain to desired explicit formula. First by semi-additivity, for the left and right adjoints of (Cat b ) hC → Cat b , they are both given by the formula ( A , B , B) ↦ ( A , B , B) ⊕ ( B op , A op , B swap ) = ( A × B op , B × A op , B ⊕ B swap ) . Under the identification (Cat b ) hC ≃ Cat sb∞ of Proposition 7.1.12 the C -fixed object on the right hand sidecorresponds to the symmetric ∞ -category ( A × B op , B) where B ∈ Fun s ( A × B op ) is given by B(( 𝑎, 𝑏 ) , ( 𝑎 ′ , 𝑏 ′ )) = B( 𝑎, 𝑏 ′ ) ⊕ B( 𝑎 ′ , 𝑏 ) . We note that this symmetric bilinear form is both induced and coinduced from B . In particular, its C -fixedpoints and C -orbits are both canonically identify with B . In particular, the left and right adjoints of thebottom horizontal map in (177) are both given by the formula ( A , B , B) ↦ ( A × B op , B) . (cid:3) The categorical Thom isomorphism.
In §2.3 we described the algebraic Thom construction , an op-eration introduced by Ranicki which allows one to identify the notion of a metabolic Poincaré object ( 𝑥, 𝑞 ) in ( C , Ϙ ) equipped with a prescribed Lagrangian ( 𝑤 → 𝑥, 𝜂 ) , with the data of a hermitian object in ( 𝑧, 𝑟 ) in C with respect to the shifted Poincaré structure Ϙ [−1] . In this section we will see that this procedure natu-rally fits in a more general perspective. We will begin by refining the construction of pairing ∞ -categoriesdescribed in §7.2 above to the context of hermitian ∞ -categories. This will result in a construction whichtakes a hermitian ∞ -category ( C , Ϙ ) and produces a Poincaré ∞ -category Pair( C , Ϙ ) . When ( C , Ϙ ) is Poincaréthis construction reproduces that of the arrow category Ar( C , Ϙ ) described in §2.3. We will then prove thatPoincaré objects in Pair( C , Ϙ ) correspond to hermitian objects in ( C , Ϙ ) , yielding in particular a proof ofProposition 2.3.17 upon taking ( C , Ϙ ) to be Poincaré.The Poincaré ∞ -categories of the form Pair( C , Ϙ ) can be considered as a categorified form of the notionof a metabolic Poincaré object: they contain a stable full subcategory on which the Poincaré structurevanishes and which is equivalent to its own orthogonal complement, a property which can be consideredas a categorical analogue of the notion of a Lagrangian. From this point of view, we may consider thethe association ( C , Ϙ ) ↦ Pair( C , Ϙ ) as a categorified form of the Thom construction, taking a hermitian ∞ -category and producing a Poincaré ∞ -category with a canonical choice of Lagrangian. We will showthat this process is reversible: given a Poincaré ∞ -category ( D , Φ) with a Lagrangian, one can reconstructa hermitian ∞ -category ( C , Ϙ ) such that Pair( C , Ϙ ) ≃ ( D , Φ) . We consider this as a categorical form of theThom isomorphism. Relying on these results we will then use the pairing construction in order to produceboth a left and a right adjoint to the forgetful functor Cat p∞ → Cat h∞ . This adjunction ties together all theabove results in a conceptual manner, and at the same time is quite useful in practice. In particular, the
36 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE results of this section we will be used in subsequent installments of this project, for example in setting upthe theory of algebraic surgery in Paper [II], and in proving that
Cat p∞ is compactly generated in Paper [IV].We now proceed to introduce the main construction of the current section:7.3.1. Construction.
Given a stable ∞ -category C , any bilinear functor B ∶ C op × C op → S determines aright biexact correspondence on the pair ( C , C op ) , given by ( 𝑥, 𝑦 ) ↦ Ω ∞ B( 𝑥, 𝑦 ) . We will then denote by Pair( C , B) ∶= Pair( C , C op , B) ∈ Cat ex∞ the associated pairings ∞ -category. Given a hermitian structure Ϙ ∶ C op → S 𝑝 on C , we define an associatedhermitian structure Ϙ pair on Pair( C , B Ϙ ) via the pullback square(179) Ϙ pair ( 𝑥, 𝑦, 𝛽 ) Ϙ ( 𝑥 )hom C ( 𝑥, 𝑦 ) B( 𝑥, 𝑥 ) where the bottom horizontal map is given by the association [ 𝑓 ∶ 𝑥 → 𝑦 ] ↦ 𝑓 ∗ 𝛽 ∈ B( 𝑥, 𝑥 ) , canonicallyextended from mapping spaces to mapping spectra.7.3.2. Lemma.
The hermitian structure Ϙ pair is Poincaré. Its bilinear part sits in the pullback square ofsymmetric bilinear forms (180) B pair (( 𝑥, 𝑦, 𝛽 ) , ( 𝑥 ′ , 𝑦 ′ , 𝛽 ′ )) B( 𝑥, 𝑥 ′ )hom C ( 𝑥, 𝑦 ′ ) ⊕ hom C ( 𝑥 ′ , 𝑦 ) B( 𝑥 ′ , 𝑥 ) ⊕ B( 𝑥, 𝑥 ′ ) . ( 𝜏 𝑥,𝑥 ′ , id) and its associated duality coincides with the duality D pair ( 𝑥, 𝑦, 𝛽 ) = ( 𝑦, 𝑥, 𝜏 𝑥,𝑦 ( 𝛽 ) ) , of Construction 7.1.13, which encodes the symmetric structure of B .Proof. Substituting in (179) the direct sum of ( 𝑥, 𝑦, 𝛽 ) , ( 𝑥 ′ , 𝑦 ′ , 𝛽 ′ ) ∈ Pair( C , B Ϙ ) we obtain the square Ϙ pair ( 𝑥 ⊕ 𝑥 ′ , 𝑦 ⊕ 𝑦 ′ , 𝛽 ⊕ 𝛽 ′ ) Ϙ ( 𝑥 ⊕ 𝑥 ′ )hom C ( 𝑥 ⊕ 𝑥 ′ , 𝑦 ⊕ 𝑦 ′ ) B( 𝑥 ⊕ 𝑥 ′ , 𝑥 ⊕ 𝑥 ′ ) , which yields the square (180) upon passing to bireduced replacements. On the other hand, working back-wards from the required duality, we note that D pair is part of a Λ -duality of the associated bifibration C ← Pair( C , B Ϙ ) → C op . Applying Proposition 7.1.16 and Remark 7.1.17 to this bifibration we obtain a cartesian square of symmetricbilinear forms hom
Pair( C , B Ϙ ) (( 𝑥, 𝑦, 𝛽 ) , D pair ( 𝑥 ′ , 𝑦 ′ , 𝛽 ′ )) B( 𝑥, 𝑥 ′ )hom C ( 𝑥, 𝑦 ′ ) ⊕ hom C ( 𝑥 ′ , 𝑦 ) B( 𝑥 ′ , 𝑥 ) ⊕ B( 𝑥, 𝑥 ′ ) , ( 𝜏 𝑥,𝑥 ′ , id) where we are again silently identifying exact functors valued in spectra with right-exact functors valued inspaces. Comparing this square with (180) we thus get that B pair is perfect with duality D pair . (cid:3) Examples. i) For a stable ∞ -category C equipped with the zero hermitian structure, the Poincaré ∞ -category Pair( C , is naturally equivalent to Hyp( C ) . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 137 ii) If the hermitian ∞ -category ( C , Ϙ ) is Poincaré with associated duality D , then the correspondence Ω ∞ B Ϙ on the pair ( C , C op ) is equivalent to the correspondence hom C (− , −) on the pair ( C , C ) via the (id , D) ∶ C op × C op ≃ ←←←←←←←→ C op × C , and so Pair( C , B Ϙ ) is naturally equivalent to the arrow category Ar( C ) of Definition 2.3.15. Under thisequivalence, the Poincaré structure Ϙ pair directly translates to the Poincaré structure Ϙ ar ( 𝑥 → 𝑦 ) = Ϙ ( 𝑥 ) × B Ϙ ( 𝑥,𝑥 ) B Ϙ ( 𝑥, 𝑦 ) , and so we obtain a natural equivalence Pair( C , Ϙ ) ≃ Ar( C , Ϙ ) .iii) Combining the previous example with Lemma 2.3.19 we obtain a natural identification of Poincaré ∞ -categories Pair( C , Ϙ [−1] ) ≃ Ar( C , Ϙ [−1] ) ≃ Met( C , Ϙ ) whenever C , Ϙ is Poincaré.7.3.4. Remark.
Combining Example 7.3.3(ii), Example 7.1.15 and Lemma 7.3.2, we obtain that for ( C , Ϙ ) Poincaré, the underlying duality D ar of Ar( C , Ϙ ) is equivalent to the one induced on Ar( C ) ≃ Fun(Δ , C ) bythe duality D Ϙ on C and the canonical duality of Δ .By construction, the underlying stable ∞ -category of Pair( C , Ϙ ) sits in a bifibration(181) 𝑥 ( 𝑥, 𝑦, 𝛽 ) 𝑦 C Pair( C , B Ϙ ) C op ∈ ∈ ∈ 𝑞 𝑝 so that 𝑞 is a cartesian fibration and 𝑝 is a cocartesian fibration. The fact that C is pointed and B Ϙ is bireducedimplies that these fibrations have fully-faithful adjoints 𝑥 ( 𝑥, , C Pair( C , B Ϙ ) C op (0 , 𝑦, 𝑦 ∈ ∈ 𝑗 𝑖 ∈ ∈ More precisely, 𝑞 has a fully-faithful right adjoint 𝑗 ∶ C → Pair( C , Ϙ ) sending 𝑥 to ( 𝑥, , . Indeed, thecanonical arrows ( 𝑥, 𝑦, 𝛽 ) → ( 𝑥, , in Pair( C , Ϙ ) induce an equivalence on mapping spaces into any tripleof the form ( 𝑥 ′ , , , and thus assemble to form a unit exhibiting 𝑗 as right adjoint to 𝑞 . Similarly, the col-lection of arrows (0 , 𝑦, → ( 𝑥, 𝑦, 𝛽 ) assemble to form a counit exhibiting the functor 𝑖 ∶ C op → Pair( C , Ϙ ) sending 𝑦 to (0 , 𝑦, as left adjoint to 𝑝 . and we observe that the image of 𝑖 coincides with the kernel of 𝑞 and the image of 𝑗 with the kernel of 𝑝 .We now observe that 𝑞 ∶ Pair( C , Ϙ ) → C naturally extends to a hermitian functor(182) ( 𝑞, 𝜂 ) ∶ Pair( C , Ϙ ) → ( C , Ϙ ) 𝑞 ( 𝑥, 𝑦, 𝛽 ) = 𝑥 , 𝜂 ∶ Ϙ pair ( 𝑥, 𝑦, 𝛽 ) → Ϙ ( 𝑥 ) with 𝜂 ∶ Ϙ pair ⇒ 𝑞 ∗ Ϙ given by the natural projection furnished directly from the definition of Ϙ pair . We thenhave the follows:7.3.5. Proposition (The generalized algebraic Thom isomorphism) . For every hermitian ∞ -category ( C , Ϙ ) the composed map Pn(Pair( C , Ϙ )) → Fm(Pair( C , Ϙ )) ( 𝑞,𝜂 ) ∗ ←←←←←←←←←←←←←←←←←←←←←→ Fm( C , Ϙ ) is an equivalence. In particular, Poincaré objects in Pair( C , Ϙ ) classify hermitian objects in ( C , Ϙ ) . Remark.
Applied in the case where ( C , Ϙ ) is Poincaré, Proposition 7.3.5 reduces to the statement ofProposition 2.3.17 via the identification Pair( C , Ϙ ) ≃ Ar( C , Ϙ ) of Examples 7.3.3.Proposition 7.3.5 is a direct consequence of the following lemma:7.3.7. Lemma.
Let ( C , Ϙ ) be a hermitian ∞ -category. Then the functor ( 𝑞, 𝜂 ) ∗ ∶ He(Pair( C , Ϙ )) → He( C , Ϙ ) induced by (182) is a cartesian fibration whose fibres admit final objects. In addition, a hermitian object ( 𝑠 ( 𝑥, 𝑦, 𝛽 ) , 𝑞 ) in Pair( C , Ϙ ) is final in its fibre over He( C , Ϙ ) if and only if it is Poincaré. Given Lemma 7.3.7, the proof of Proposition 7.3.5 is immediate:
38 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
Proof of Proposition 7.3.5.
By Lemma 7.3.7 the homotopy fibres of
Pn(Pair( C , Ϙ )) → Fm( C , Ϙ ) are contractible, and so the desired result follows. (cid:3) Proof of Lemma 7.3.7.
Consider the commutative square
He(Pair( C , Ϙ )) He( C , Ϙ )Pair( C , Ϙ ) C where the vertical arrows are the defining right fibrations of He(−) . Since the bottom horizontal map is acartesian fibration (by the construction of
Pair( C , Ϙ ) as a bifibration) we obtain that the composed dottedmap is a cartesian fibration, and hence we can view the top horizontal map is a map of cartesian fibrationsover C whose target is a right fibration. Such a map is automatically a cartesian fibration (up to equivalence),which gives us the first claim of the lemma. We now verify that its fibres contain final objects. Let X ∶= ∫ ( 𝑥,𝑦,𝛼 )∈Pair( C , B Ϙ ) Map C ( 𝑥, 𝑦 ) → Pair( C , B Ϙ ) be the right fibration classifying the contravariant functor ( 𝑥, 𝑦, 𝛽 ) ↦ Map C ( 𝑥, 𝑦 ) . The defining fibresquare (179) then determines commutative square(183) He(Pair( C , Ϙ )) He( C , Ϙ ) X ∫ 𝑥 ∈ C B Ϙ ( 𝑥, 𝑥 )Pair( C , B Ϙ ) C of ∞ -categories in which (using the same argument as above) the vertical maps are right fibrations andthe horizontal maps are cartesian fibrations. In addition, the top square in (183) is cartesian; indeed, basechanging the right fibrations on the right from C to Pair( C , B Ϙ ) yields a fibre square of right fibrations whichis the straightening of the defining square (179) (after taking Ω ∞ ). It will hence suffice to show that thefibres of the middle horizontal cartesian fibration have final objects. Fix an object ( 𝑥, 𝛽 ) ∈ ∫ 𝑥 ∈ C B Ϙ ( 𝑥, 𝑥 ) .Then the fibre X ( 𝑥,𝛽 ) of the middle horizontal map sits in a right fibration(184) X ( 𝑥,𝛽 ) → Pair( C , B Ϙ ) 𝑥 = ∫ 𝑦 ∈ C op B Ϙ ( 𝑥, 𝑦 ) , where the middle term stands for the fibre of the bottom horizontal map over 𝑥 ∈ C . Using the pullbackformula (171) for the mapping spaces in Pair( C , B Ϙ ) we now calculate X ( 𝑥,𝛽 ) = ∫ ( 𝑦,𝛼 )∈Pair( C , B Ϙ ) 𝑥 Map C ( 𝑥, 𝑦 ) × B Ϙ ( 𝑥,𝑥 ) { 𝛽 } ≃ ∫ ( 𝑦,𝛼 )∈Pair( C , B Ϙ ) 𝑥 Map C op ( 𝑦, 𝑥 ) × B Ϙ ( 𝑥,𝑥 ) { 𝛽 }≃ ∫ ( 𝑦,𝛼 )∈Pair( C , B Ϙ ) 𝑥 Map
Pair( C , B Ϙ ) 𝑥 (( 𝑥, 𝑦, 𝛼 ) , ( 𝑥, 𝑥, 𝛽 )) ≃ (Pair( C , B Ϙ ) 𝑥 ) ∕( 𝑥,𝑥,𝛽 ) , from which we see that the right fibration (184) is visibly represented by ( 𝑥, 𝑥, 𝛽 ) , so that X ( 𝑥,𝛽 ) has a finalobject.To finish the proof we now need to verify that a hermitian object (( 𝑥, 𝑦, 𝛽 ) , 𝑞 ) in Pair( A , B , 𝑏 ) is Poincaréif and only if it is final in the fibre. The top square in (183) being cartesian it will suffice to show that (( 𝑥, 𝑦, 𝛽 ) , 𝑞 ) is Poincaré if and only if its image in X is final in its fibre over ∫ 𝑥 ∈ C B Ϙ ( 𝑥, 𝑥 ) . Indeed, ahermitian form 𝑞 on ( 𝑥, 𝑦, 𝛽 ) determines a self dual map 𝑞 ♯ ∶ ( 𝑥, 𝑦, 𝛽 ) → D pair ( 𝑥, 𝑦, 𝛽 ) = ( 𝑦, 𝑥, 𝛽 ) withcomponents 𝑓 ∶ 𝑥 → 𝑦 and 𝑦 ← 𝑥 ∶ 𝑔 (the latter considered as a map from 𝑦 to 𝑥 in C op ). The form 𝑞 isthen Poincaré if and only if 𝑓 and 𝑔 are equivalences. But since 𝑞 ♯ is self-dual the components 𝑓 and 𝑔 arehomotopic to each other. We then get that 𝑞 is Poincaré if and only if the map 𝑔 is an equivalence. But 𝑔 (or ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 139 𝑓 ) is exactly the image of 𝑞 in Map( 𝑥, 𝑦 ) via the vertical map in the defining square (179), and so the imageof (( 𝑥, 𝑦, 𝛽 ) , 𝑞 ) in X is (( 𝑥, 𝑦, 𝛽 ) , 𝑔 ) . The latter object lies over ( 𝑥, 𝑔 ∗ 𝛽 ) ∈ ∫ 𝑥 ∈ C B Ϙ ( 𝑥, 𝑥 ) and corresponds tothe object(185) [( 𝑥, 𝑦, 𝛽 ) (id ,𝑔 ) ←←←←←←←←←←←←←←←←←←←←→ ( 𝑥, 𝑥, 𝑔 ∗ 𝛽 )] ∈ X ( 𝑥,𝑔 ∗ 𝛽 ) ≃ (Pair( C , B Ϙ ) 𝑥 ) ∕( 𝑥,𝑥,𝑔 ∗ 𝛽 ) We may then conclude that (( 𝑥, 𝑦, 𝛽 ) , 𝑞 ) is Poincaré if and only if 𝑔 is an equivalence, and so if and onlyif (185) is final, as desired. (cid:3) The following almost immediate corollary of Lemma 7.3.7 will also be useful for us later:7.3.8.
Corollary.
The natural transformation 𝜂 ∶ Ϙ pair ⇒ 𝑞 ∗ Ϙ exhibits Ϙ ∶ C op → S 𝑝 as the left Kan extension of Ϙ pair along 𝑞 op ∶ Pair( C , B Ϙ ) op → C op .Proof. By Proposition 1.4.3 and Lemma 1.1.25 it will suffice to show that Ω ∞ 𝜂 exhibits Ω ∞ Ϙ as the leftKan extension of Ω ∞ Ϙ pair . Since 𝑞 op is a cocartesian fibration left Kan extensions along 𝑞 op are calculatedby colimits along the fibres. In particular, we need to show that for every 𝑥 ∈ C the map colim ( 𝑥,𝑦,𝛼 )∈Pair( C , B Ϙ ) op 𝑥 Ω ∞ Ϙ pair ( 𝑥, 𝑦, 𝛼 ) → Ω ∞ Ϙ ( 𝑥 ) , induced by 𝜂 , is an equivalence of spaces. Since colimits in spaces are universal it will suffice to show thatfor every point 𝛽 ∈ Ω ∞ Ϙ ( 𝑥 ) the space colim ( 𝑥,𝑦,𝛼 )∈Pair( C , B Ϙ ) op 𝑥 Ω ∞ Ϙ pair ( 𝑥, 𝑦, 𝛼 ) × Ϙ ( 𝑥 ) { 𝛽 } is contractible. Indeed, this space can in turn be identified with the geometric realization of the fibre of He(Pair( C , Ϙ )) op → He( C , Ϙ ) op over ( 𝑥, 𝛽 ) , and the latter has an initial object by Lemma 7.3.7, so its real-ization is contractible. (cid:3) We now take a closer look at the Poincaré ∞ -categories of the form Pair( C , Ϙ ) . We wish to make theargument that they constitute a categorical analogue of the notion of a metabolic Poincaré object, makingthe passage from ( C , Ϙ ) to Pair( C , Ϙ ) an analogue of the algebraic Thom construction. To identify furtherkey properties we introduce the following piece of notation:7.3.9. Definition.
Let ( D , Φ) be a Poincaré ∞ -category and L ⊆ D a full subcategory. We will denote by L ⟂ ⊆ D the full subcategory spanned by the objects 𝑦 ∈ D such that B Φ ( 𝑥, 𝑦 ) = 0 for every 𝑥 ∈ L . Wewill refer to L ⟂ as the orthogonal complement of L .Using the notion of orthogonal complements, we may identify the following additional properties heldby the full subcategory inclusion 𝑖 ∶ C op ↪ Pair( C , B Ϙ ) :i) The restriction of the quadratic functor Ϙ pair to C op vanishes.ii) The inclusion 𝑖 C op ⊆ ( 𝑖 C op ) ⟂ , furnished by i) above, is an equivalence.iii) 𝑖 admits a right adjoint 𝑝 ∶ Pair( C , B Ϙ ) → C op given by ( 𝑥, 𝑦, 𝛽 ) ↦ 𝑦 (see (181) and the discussionbelow it).The validity of (i)) above is evident from the fibre square (179) defining Ϙ pair . To see that (ii)) holds, notethat B Ϙ ( 𝑖 ( 𝑧 ) , ( 𝑥, 𝑦, 𝛽 )) = hom((0 , 𝑧, , ( 𝑦, 𝑥, 𝛽 )) = hom C op ( 𝑧, 𝑥 ) = hom C ( 𝑥, 𝑧 ) , and hence ( 𝑥, 𝑦, 𝛽 ) ∈ ( 𝑖 C op ) ⟂ if and only if 𝑥 = 0 , i.e., if and only if ( 𝑥, 𝑦, 𝛽 ) ∈ 𝑖 C op . This motivates thefollowing definition:7.3.10. Definition.
Let ( D , Φ) be a Poincaré ∞ -category and L ⊆ D a full subcategory. We will say that L is a Lagrangian in D if it satisfies Properties (i)), (ii)) and (iii)) above. In other words, if Ϙ vanishes whenrestricted to L , the orthogonal complement L ⟂ ⊆ D coincides with L itself and the inclusion L ⊆ D admitsa right adjoint. We will say that a Poincaré ∞ -category is metabolic if it admits a Lagrangian.
40 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
In particular, the Poincaré ∞ -category Pair( C , Ϙ ) contains C op as a Lagrangian. We now claim that thisproperty completely characterizes Poincaré ∞ -categories of the form Pair( C , Ϙ ) . To see this, let ( D , Φ) bea Poincaré ∞ -category with underlying duality D , and let 𝑖 ∶ L ↪ D be a Lagrangian with right adjoint 𝑝 ∶ D → L . Let 𝑗 ∶ L op ⊆ D be the inclusion sending 𝑧 to D( 𝑖 ( 𝑧 )) . Then im( 𝑗 ) = D( 𝑖 L ) = D(( 𝑖 L ) ⟂ ) = ker( 𝑝 ) , and 𝑗 admits a left adjoint 𝑞 ∶ D → L op given by the formula 𝑞 ( 𝑥 ) = 𝑝 (D 𝑥 ) .We then have the following:7.3.11. Proposition (Recognition principle for pairing Poincaré categories) . Let ( D , Φ) be a Poincaré ∞ -category admitting a Lagrangian 𝑖 ∶ L ↪ D with right adjoint 𝑝 ∶ D → L , and let 𝑞 ∶ D ⟂ L op ∶ 𝑗 and 𝜂 be as above. Let Π = 𝑞 ! Φ ∈ Fun q ( L op ) be the left Kan extension of Φ along 𝑞 op ∶ D op → L . Then thereexists a canonical diagram of hermitian ∞ -categories ( L op , Π) Pair( L op , Π) ( L , L op , Π) ( D , Φ) ( L , ≃( 𝑞,𝜂 ) ( 𝑝, in which the middle vertical arrow is an equivalence of Poincaré ∞ -categories. Here the projections onthe top row are the underlying bifibration (181) of Pair( L op , B Π ) promoted to hermitian functors triviallyon the right and as in (182) on the left. Remark.
The recognition principle of Proposition 7.3.11 could also be formulated in the bilinearsetting of §7.2. In particular, given a stable ∞ -category D , equivalences of the form D ≃ Pair( A , B , 𝑏 ) for ( A , B , 𝑏 ) ∈ Cat b correspond to fully-faithful embeddings 𝑖 ∶ B ↪ D which admit a right adjoint 𝑝 ∶ D → B , in which case A is recovered as the kernel of 𝑝 , and 𝑏 is recovered as the restriction of the correspondence sec D ∶ D op × D → S of Example 7.2.4 to A op × B . The proof of this claim essentially amounts to the firsthalf of the proof of Proposition 7.3.11 below. Proof of Proposition 7.3.11.
Let
Seq( D , L ) ⊆ Seq( D ) denote the full subcategory spanned by those exactsequences [ 𝑦 → 𝑧 → 𝑥 ] such that 𝑦 ∈ 𝑖 ( L ) and 𝑥 ∈ 𝑗 ( L op ) . We claim that the projection Seq( D , L ) → D [ 𝑦 → 𝑧 → 𝑥 ] ↦ 𝑧 is an equivalence. To see this, consider first the map Seq( D , L ) → Ar( D ) × D 𝑗 ( L op ) sending [ 𝑦 → 𝑧 → 𝑥 ] to [ 𝑧 → 𝑥 ] . By [Lur09a, Proposition 4.3.2.15] this map is fully-faithful, with essential image the fullsubcategory of Ar( D ) × D 𝑗 ( L op ) spanned by those arrows 𝑧 → 𝑥 with 𝑥 ∈ 𝑗 ( L op ) whose fibre lies in 𝑖 ( L ) .Now since 𝑖 ( L ) = ker( 𝑞 ) the condition that the fibre of 𝑧 → 𝑥 lies in 𝑖 ( L ) is equivalent to the conditionthat 𝑞 ( 𝑧 ) → 𝑞 ( 𝑥 ) ≃ 𝑥 is an equivalence. Now the projection Ar( D ) × D 𝑗 ( L op ) → D sending 𝑧 → 𝑥 is acartesian fibration whose fibre over 𝑧 ∈ D is equivalent to the comma category D 𝑧 ∕ × C 𝑗 ( L op ) . This commacategory is equivalent by adjunction to L op 𝑞 ( 𝑧 )∕ , and the above condition shows that under this equivalencethe full subcategory Seq( D , L ) ⊆ Ar( D ) × D 𝑗 ( L op ) consists of exactly those objects which are initial intheir fibres. The projection Seq( D , L ) → D is consequently an equivalence.We note that under the equivalence between bifibrations and correspondences, base changes on the carte-sian side correspond the restriction along the first entry, while base changes on the cocartesian side corre-spond to base changes in the second entry. In particular, we may identify Seq( D , L ) with Pair( L op , L , (seq D ) | L × L ) as full subcategories of Pair( D , D , seq D ) , and so the projections L op Seq( D , L ) L 𝑥 [ 𝑦 → 𝑧 → 𝑥 ] 𝑦 form a bifibration classified by the restricted correspondence (seq D ) | L × L ∶ L × L → S . Under this equiv-alence Seq( D , L ) ≃ D these projections correspond to adjoints 𝑝 ∶ D → L and 𝑞 ∶ D → L op to theinclusions 𝑖 ∶ L ↪ D and 𝑗 ∶ L op ↪ D . We now observe that for 𝑥 ∈ L op and 𝑦 ∈ L we have seq D ( 𝑥, 𝑦 ) ≃ Ω hom D (Ω 𝑗 ( 𝑥 ) , Σ 𝑖 ( 𝑦 )) ≃ Ω hom D (Ω 𝑗 ( 𝑥 ) , ΣD 𝑗 ( 𝑦 )) ≃ ΩB Φ (Ω 𝑗 ( 𝑥 ) , Ω 𝑗 ( 𝑦 )) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 141 and so if we let Π ∶ L → S 𝑝 be the quadratic functor given by the formula Π( 𝑧 ) = ΩΦ(Ω 𝑗 ( 𝑧 )) then weobtain an equivalence D ≃ Pair( L op , B Π ) on the level of stable ∞ -categories, which is compatible with the inclusions from and projections to L and L op on both sides.We now address the comparison of the Poincaré structures. For every 𝑥 ∈ D the unit of 𝑖 ⊣ 𝑝 and counitof 𝑞 ⊣ 𝑗 yield a sequence(186) 𝑖𝑝 ( 𝑥 ) → 𝑥 → 𝑗𝑞 ( 𝑥 ) whose composite admits an essentially unique null-homotopy, since 𝑗𝑞 ( 𝑥 ) ∈ ker( 𝑝 ) and hence hom C ( 𝑖𝑝 ( 𝑥 ) , 𝑗𝑞 ( 𝑥 )) =hom L ( 𝑝 ( 𝑥 ) , 𝑝𝑗𝑞 ( 𝑥 )) = 0 . This null-homotopy exhibits (186) as exact. Indeed, this sequence maps to anexact sequence by both 𝑝 and 𝑞 and these two functors are jointly conservative since ker( 𝑝 ) ∩ ker( 𝑞 ) ≃im( 𝑗 ) ∩ ker( 𝑞 ) = 0 . Since Φ( 𝑖𝑝 ( 𝑥 )) = 0 the shifted exact sequence Ω 𝑗𝑞 ( 𝑥 ) → 𝑖𝑝 ( 𝑥 ) → 𝑥 determines anexact square Φ( 𝑥 ) ΩΦ(Ω( 𝑗𝑞 ( 𝑥 ))ΩB Φ ( 𝑖𝑝 ( 𝑥 ) , Ω 𝑗𝑞 ( 𝑥 )) ΩB Φ (Ω 𝑗𝑞 ( 𝑥 ) , Ω 𝑗𝑞 ( 𝑥 )) which, having set Π( 𝑧 ) = ΩΦ(Ω 𝑗 ( 𝑧 )) , we can write as(187) Φ( 𝑥 ) Π( 𝑞 ( 𝑥 ))B Φ ( 𝑖𝑝 ( 𝑥 ) , 𝑗𝑞 ( 𝑥 )) B Π ( 𝑞 ( 𝑥 )) . Comparing the exact square (187) with the exact square (179) we then conclude that the equivalence D ≃Pair( L op , B Π ) refines to an equivalence of Poincaré ∞ -categories ( D , Φ) ≃ Pair( L op , Π) . By Corollary 7.3.8we then get that the natural transformation Φ ⇒ 𝑞 ∗ Π furnished by the top row of the square (187) exhibits Π as the left Kan extension of Φ along 𝑞 op ∶ D op → L . We hence an equivalence ( D , Φ) ≃ Pair( L , 𝑞 ! Φ) = Pair( L op , Π) compatible with the projections to (and hence also the embedding of) the ∞ -categories L and L op on bothsides. (cid:3) Given a Poincaré ∞ -category ( D , Φ) admitting a Lagrangian L D ⟂ 𝑝 with associated projection 𝑞 = 𝑝 D ∶ D → L op , Proposition 7.3.11 yields an equivalence of Poincaré ∞ -categories(188) ( D , Φ) ≃ Pair( L op , 𝑞 ! Φ) . In particular, the association ( C , Ϙ ) ↦ Pair( C , Ϙ ) takes values in metabolic Poincaré ∞ -categories, and everymetabolic Poincaré ∞ -category is obtained in this manner.In what follows, it will be useful to observe that the hermitian structure Π ∶= 𝑞 ! Φ on L op can also berecovered using the inclusion 𝑗 ∶ L op → D right adjoint to 𝑞 . To avoid a potential confusion we emphasizethat Π does not coincide with the restriction of Φ along 𝑗 . Instead, let Π pair ( 𝑥, 𝑦, 𝛽 ) = Π( 𝑥 ) × B( 𝑥,𝑥 ) hom( 𝑥, 𝑦 ) be the quadratic functor on Pair( L op , Π) as in Construction 7.3.1. Under the equivalence (188) the functor 𝑗 ∶ L op ↪ D sends 𝑥 to ( 𝑥, ,
0) ∈ Pair( L op , Ψ) and we have(189) Φ( 𝑗 ( 𝑥 )) = Π pair ( 𝑥, ,
0) = Π( 𝑥 ) × B Π ( 𝑥,𝑥 ) hom( 𝑥,
0) = f ib[Π( 𝑥 ) → B Π ( 𝑥, 𝑥 )] ≃ ΣΠ(Σ 𝑥 ) . where the last equivalence is issued from Lemma 1.1.19 and Example 1.1.21. It will consequently beconvenient to introduce the following terminology:7.3.13. Definition.
Let C be a stable ∞ -category and Ϙ a quadratic functor on C . We will denote by Ϙ [ 𝜎 ] ( 𝑥 ) ∶= Ω Ϙ (Ω 𝑥 ) and Ϙ [− 𝜎 ] ( 𝑥 ) ∶= Σ Ϙ (Σ 𝑥 ) the quadratic functors obtained by pre- and post-composingwith Ω and Σ , respectively. We note that the operations Ϙ ↦ Ϙ [ 𝜎 ] and Ϙ ↦ Ϙ [− 𝜎 ] are inverse to each other,and in particular adjoint (in both directions). By Lemma 1.1.19 we have natural equivalences Ϙ [− 𝜎 ] ( 𝑥 ) ≃ f ib[ Ϙ ( 𝑥 ) → B Ϙ ( 𝑥, 𝑥 )] and Ϙ ( 𝑥 ) ≃ f ib[ Ϙ [ 𝜎 ] ( 𝑥 ) → B Ϙ [ 𝜎 ] ( 𝑥, 𝑥 )] ,
42 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE yielding in particular a natural transformation Ϙ ⇒ Ϙ [ 𝜎 ] and an adjoint transformation Ϙ [− 𝜎 ] ⇒ Ϙ .7.3.14. Remark.
For a hermitian ∞ -category ( C , Ϙ ) , the underlying symmetric bilinear form of Ϙ [ 𝜎 ] is givenby ( 𝑥, 𝑦 ) ↦ ΩB(Ω 𝑥, Ω 𝑦 ) ≃ Σ 𝜎 B( 𝑥, 𝑦 ) , where Σ 𝜎 is the operation of tensoring by the sign representationsphere, see discussion in §3.4. In other words, B Ϙ [ 𝜎 ] has as underlying bilinear form ΣB Ϙ , but the symmetricstructure is twisted by a sign, see Remark 3.4.3. Similarly, B Ϙ [− 𝜎 ] = Σ − 𝜎 B Ϙ = Ω 𝜎 B Ϙ has underlying bilinearform ΩB , but the symmetric structure is twisted by a sign.The identification (189) can then be succinctly stated as 𝑗 ∗ Φ ≃ Π [− 𝜎 ] = 𝑞 ! Φ [− 𝜎 ] , or equivalently, 𝑗 ! Φ ≃ 𝑗 ∗ Φ [ 𝜎 ] . We may summarize this discussion by extending (188) to(190) ( D , Φ) ≃ Pair( L op , 𝑞 ! Φ) ≃ Pair( L op , 𝑗 ∗ Φ [ 𝜎 ] ) . We now use the pairing construction in order to form left and right adjoints to the forgetful functor
Cat p∞ → Cat h∞ .7.3.15. Proposition.
For every hermitian ∞ -category ( C , Ϙ ) and Poincaré ∞ -category ( E , Ψ) , the map (191) Map
Cat p∞ (( E , Ψ) , Pair( C , Ϙ )) → Map
Cat h∞ (( E , Ψ) , C ) induced by post-composition with the hermitian functor ( 𝑞, 𝜂 ) ∶ Pair( C , Ϙ ) → ( C , Ϙ ) of (182) , is an equiv-alence of spaces. In particular, the association ( C , Ϙ ) ↦ Pair( C , Ϙ ) assembles to form a functor Cat h∞ → Cat p∞ which is right adjoint to Cat p∞ → Cat h∞ . Remark.
Though one can show directly that the association ( C , Ϙ ) ↦ Pair( C , Ϙ ) organizes intoa functor Cat h∞ → Cat p∞ , Proposition 7.3.15 is formulated in a way that does not require knowing thisin advance, and on the other hand implies this functoriality via general principles of adjunctions; indeed,knowing that the comma category Cat p∞ × Cat h∞ (Cat h∞ ) ∕( C , Ϙ ) has a final object is enough to imply the existenceof the desired adjoints, which then must coincide with the given formula on objects.7.3.17. Remark.
In the situation of Proposition 7.3.15, if ( C , Ϙ ) is also Poincaré then we have a naturalequivalence Pair( C , Ϙ ) ≃ Ar( C , Ϙ ) (see Example 7.3.3iii)) under which the hermitian functor of (182) be-comes the domain projection Ar( C , Ϙ ) → ( C , Ϙ ) . The unit of the adjunction furnished by Proposition 7.3.15then corresponds to the essentially unique Poincaré functor ( C , Ϙ ) → Ar( C , Ϙ ) for which the composite withthe domain projection is the identity on ( C , Ϙ ) . In particular, the unit must coincide with the fully-faithfulinclusion ( C , Ϙ ) → Ar( C , Ϙ ) 𝑥 ↦ [id ∶ 𝑥 → 𝑥 ] endowed with the natural equivalence Ϙ ( 𝑥 ) ≃ Ϙ ar ([id ∶ 𝑥 → 𝑥 ]) .7.3.18. Remark.
By [Lur17, Corollary 7.3.2.7] the functor
Pair(−) inherits a lax symmetric monoidal struc-ture by virtue of being right adjoint to the symmetric monoidal functor 𝜄 ∶ Cat p∞ → Cat h∞ (see Theo-rem 5.2.7). In particular, the adjunction 𝜄 ⊣ Pair(−) of Proposition 7.3.15 is a symmetric monoidal adjunc-tion. It then follows from the equivalence of Example 7.3.3(ii) that the functor
Ar(−) ∶ Cat p∞ → Cat p∞ islax symmetric monoidal as well.7.3.19. Remark.
In the situation of Proposition 7.3.15, if ( E , Ψ) = ( E , Ϙ qB ) is the quadratic Poincaré ∞ -category associated to a symmetric bilinear form B ∈ Fun s ( E ) , then by Propositions 7.2.16 and 7.2.17 thearrow (191) identifies with the arrow Map
Cat ps∞ (( E , B) , (Pair( C , B Ϙ ) , B pair )) → Map
Cat sb∞ (( E , B) , ( C , B Ϙ )) , and so we may conclude that the association ( C , B) ↦ (Pair( C , B) , B pair ) assembles to form a right ad-joint to the inclusion Cat ps∞ → Cat sb∞ . Identifying
Cat ps∞ with (Cat ex∞ ) hC via Corollary 7.2.15 and usingLemma 7.3.2 we may also reformulate this as saying that the association ( C , B) ↦ (Pair( C , B) , D pair ) givesa right adjoint to the functor (Cat ex∞ ) hC → Cat sb∞ sending ( C , D) to ( C , B D ) . This last conclusion could alsobe obtained from the opposite direction by showing that the association ( A , B , B) ↦ Pair( A , B , B) givesa C -equivariant right adjoint to the functor Cat ex∞ → Cat b sending C to ( C , C , 𝑚 C ) , and hence induces aright adjoint on the level of C -fixed objects on both sides. In fact, Pair(− , − , −) being right adjoint to C ↦ ( C , C , 𝑚 C ) is a statement that holds also in the non-stable setting and can be proven using the settingof bifibrations as described in §7.1. Alternatively, an argument in the stable setting can be mounted alongthe lines of the proof of Proposition 7.3.15 below, using Remark 7.3.12 in place of Proposition 7.3.11. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 143 Proof of Proposition 7.3.15.
Fix a hermitian ∞ -category ( C , Ϙ ) and a Poincaré ∞ -category ( E , Ψ) , and let ( D , Φ) ∶= Fun ex (( E , Ψ) , Pair( C , Ϙ )) = (Fun ex ( E , Pair( C , B Ϙ )) , nat Ψ Ϙ pair ) be the corresponding internal hom Poincaré ∞ -category constructed in §6.2. We will use Proposition 7.3.11in order to identify ( D , Ϙ ) with the pairing Poincaré ∞ -category associated to the internal hom hermitian ∞ -category Fun ex (( E , Ψ) , ( C , Ϙ )) ∶= (Fun ex ( E , C ) , nat Ψ Ϙ ) , thus reducing Proposition 7.3.15 to the algebraicThom isomorphism of Proposition 7.3.5. Indeed, define L ∶= Fun ex ( E , C op ) and let 𝑖 ∗ ∶ L → D stand for post-composition with 𝑖 ∶ C op ↪ Pair( C , B Ϙ ) . Since 𝑖 is fully-faithful andadmits a right adjoint 𝑝 ∶ Pair( C , B Ϙ ) → C op we have that 𝑖 ∗ is fully-faithful and admits a right adjoint 𝑝 ∗ ∶ D → L obtained by post-composing with 𝑝 . In addition, the restriction of Φ = nat Ψ Ϙ pair to L van-ishes because if 𝑓 ∶ E → Pair( C , B Ϙ ) is an exact functor which factors through C op then 𝑓 ∗ Ϙ pair = 0 (since Ϙ pair vanishes on C op ) and so nat Ψ Ϙ pair ( 𝑓 ) = nat(Ψ , 𝑓 ∗ Ϙ pair ) ≃ 0 . Finally, the orthogonal complement L ⟂ ⊆ D consists of those exact functors 𝑓 ∶ E → Pair( C , B Ϙ ) such that 𝑝 ∗ D D ( 𝑓 ) = 0 , that is, such that 𝑝 D pair 𝑓 D E ( 𝑥 ) = 0 for every 𝑥 ∈ E . This is just equivalent to saying that 𝑓 takes values in the orthogonalcomplement ( 𝑖 C op ) ⟂ , which coincides with 𝑖 C op itself since 𝑖 C op is a Lagrangian. We may then concludethat L is a Lagrangian in D .Now, if we identify L op = Fun ex ( E , C op ) op = Fun ex ( E op , C ) with Fun ex ( E , C ) via pre-composition withthe duality of E , then the inclusion L op → D sending 𝑓 to D D ( 𝑖 ∗ 𝑓 ) identifies with 𝑓 ↦ 𝑗 ∗ 𝑓 , where 𝑗 ∗ denotes post-composition with 𝑗 ∶ C ↪ Pair( C , B Ϙ ) . The left adjoint of 𝑗 ∗ is then given by post-compositionwith 𝑞 ∶ Pair( C , B Ϙ ) → C , which we denote by 𝑞 ∗ . Let Π = ( 𝑞 ∗ ) ! Φ ∈ Fun q ( L op ) be the quadratic functorobtained by left Kan extending Φ along 𝑞 ∗ ∶ D → L op . As in (190) we may also identify Π with thequadratic functor Π( 𝑔 ) = Φ [ 𝜎 ] ( 𝑗𝑔 ) . We may then compute Π( 𝑔 ) = Φ [ 𝜎 ] ( 𝑗𝑔 ) ≃ nat(Ψ , 𝑔 ∗ 𝑗 ∗ Ϙ [ 𝜎 ]pair ) ≃ nat(Ψ , 𝑔 ∗ Ϙ ) , and identify the natural map Φ( 𝑓 ) → Π( 𝑞𝑓 ) for 𝑓 ∈ D with the map 𝜂 ∗ ∶ nat(Ψ , 𝑓 ∗ Ϙ pair ) → nat(Ψ , 𝑓 ∗ 𝑞 ∗ Ϙ ) obtained post-composition with 𝑓 ∗ 𝜂 ∶ 𝑓 ∗ Ϙ pair ⇒ 𝑓 ∗ 𝑞 ∗ Ϙ . Invoking Proposition 7.3.11 we now get an iden-tification Fun ex (( E , Ψ) , ( C , Ϙ )) Pair(Fun ex (( E , Ψ) , ( C , Ϙ ))) (Fun( E , C op ) , ex (( E , Ψ) , ( C , Ϙ )) Fun ex (( E , Ψ) , Pair( C , Ϙ )) (Fun ex ( E , C op ) , ≃( 𝑞 ∗ ,𝜂 ∗ ) ( 𝑝 ∗ , of ( D , Φ) ∶= Fun ex (( E , Ψ) , Pair( C , Ϙ )) as the pairings poincaré category of the hermitian ∞ -category Fun ex (( E , Ψ) , ( C , Ϙ )) , under which the associated cartesian fibration Pair(Fun ex (( E , Ψ) , ( C , Ϙ ))) → Fun ex (( E , Ψ) , ( C , Ϙ )) identifies with post-composition with ( 𝑞, 𝜂 ) ∶ Pair( C , Ϙ ) → ( C , Ϙ ) . Proposition 7.3.15 consequently followsfrom Proposition 7.3.5. (cid:3) To obtain a left adjoint to the forgetful functor
Cat p∞ → Cat h∞ we first promote 𝑗 to a hermitian functor(192) ( 𝑗, 𝜗 ) ∶ ( C , Ϙ [− 𝜎 ] ) → Pair( C , Ϙ ) 𝑗 ( 𝑥 ) = ( 𝑥, , , 𝜗 ∶ Ϙ [− 𝜎 ] ( 𝑥 ) ≃ Ϙ pair ( 𝑗 ( 𝑥 )) . see Definition 7.3.13.7.3.20. Proposition.
For every Poincaré ∞ -category ( E , Ψ) , the map (193) Map
Cat p∞ (Pair( C , Ϙ ) , ( E , Ψ)) → Map
Cat h∞ (( C , Ϙ [− 𝜎 ] ) , ( E , Ψ))
44 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE induced by pre-composition with (192) , is an equivalence of spaces. In particular, substituting Ϙ [ 𝜎 ] insteadof Ϙ we deduce that the association ( C , Ϙ ) ↦ Pair( C , Ϙ [ 𝜎 ] ) assembles to form a functor Cat h∞ → Cat p∞ whichis left adjoint to the forgetful functor Cat p∞ → Cat h∞ . Remark.
In the situation of Proposition 7.3.20, if ( C , Ϙ ) is also Poincaré then by Example 7.3.3iii)we have a natural equivalence Pair( C , Ϙ [ 𝜎 ] ) ≃ Ar( C , Ϙ [ 𝜎 ] ) ≃ Met( C , Ϙ ◦ Ω) ≃ Met( C , Ϙ ) , where the second equivalence covers the exact functor [ 𝑤 → 𝑥 ] ↦ [Ω 𝑤 → Ω 𝑥 ] . Under this equivalencethe hermitian functor (192) becomes the inclusion triv ∶ ( C , Ϙ ) → Met( C , Ϙ ) 𝑥 ↦ [0 → 𝑥 ] . The counit of the adjunction furnished by Proposition 7.3.20 then corresponds to the essentially uniquePoincaré functor
Met( C , Ϙ ) → ( C , Ϙ ) for which the pre-composing with this unit gives the identity on ( C , Ϙ ) .In particular, the counit must coincide with the projection met ∶ Met( C , Ϙ ) → ( C , Ϙ ) [ 𝑤 → 𝑥 ] ↦ 𝑥, of Lemma 2.3.7.7.3.22. Remark.
For a Poincaré ∞ -category ( C , Ϙ ) , applying the functor Pair(− , (−) [ 𝜎 ] ) to the canonicalhermitian functors ( C , Ϙ ) → ( C , and ( C , → ( C , Ϙ ) yields, using Remark 7.3.21, the Poincaré functors dlag ∶ Met( C , Ϙ ) → Hyp( C ) and dcan ∶ Hyp( C , Ϙ ) → Met( C ) of Construction 2.3.8, respectively. This gives, in particular, a certain abstract justification for the appear-ance of these Poincaré functors.7.3.23. Remark.
It follows from Proposition 7.3.20 and Remark 7.3.21 that the association ( C , Ϙ ) ↦ Met( C , Ϙ ) carries the structure of a comonad on Cat p∞ , with the Poincaré functors met ∶ Met( C , Ϙ ) → ( C , Ϙ ) assem-bling to form the counit of this monad. A formal consequence of this which we record here for later use isthat the resulting comultiplication Poincaré functor Met( C , Ϙ ) → Met(Met( C , Ϙ )) gives a section for eitherof the two projections Met(Met( C , Ϙ )) → Met( C , Ϙ ) , the first being the counit evaluated at Met( C , Ϙ ) and the second obtained by applying Met to the counitevaluated at ( C , Ϙ ) .7.3.24. Remark.
In the situation of Proposition 7.3.20, if ( E , Ψ) = ( E , Ϙ sB ) is the symmetric Poincaré ∞ -category associated to a symmetric bilinear form B ∈ Fun s ( E ) , then by Propositions 7.2.16 and 7.2.17 andRemark 7.3.14, the arrow (193) identifies with the arrow Map
Cat ps∞ ((Pair( C , Σ 𝜎 B Ϙ ) , B pair ) , ( E , B)) → Map
Cat sb∞ (( C , B Ϙ ) , ( E , B)) . As in Remark (7.3.19) we may then conclude that the association ( C , B) ↦ (Pair( C , Σ 𝜎 B) , D pair ) assemblesto form a left adjoint to the functor (Cat ex∞ ) hC → Cat sb∞ sending ( C , D) to ( C , C , 𝑚 C ) . This conclusioncould also be obtained differently by showing first that the association ( A , B , B) ↦ Pair( A , B , ΣB) gives C -equivariant left adjoint to the functor Cat ex∞ → Cat b sending C to ( C , C , 𝑚 C ) (though the C -equivariantstructure here involves a somewhat subtle sign). The last claim can be proven using an argument similar tothat of the proof of Proposition 7.3.20 below, by replacing the recognition principle of Proposition 7.3.11by its bilinear version (see Remark 7.3.12). We leave the details to the motivated reader. Proof of Proposition 7.3.20.
Fix a hermitian ∞ -category ( C , Ϙ ) and a Poincaré ∞ -category ( E , Ψ) , and let ( D , Φ) ∶= Fun ex (Pair( C , Ϙ ) , ( E , Ψ)) = (Fun ex (Pair( C , B Ϙ ) , E ) , nat Ϙ pair Ψ ) be the corresponding internal hom Poincaré ∞ -category. As in the proof of Proposition 7.3.15 we will useProposition 7.3.11 in order to identify D with a pairing Poincaré ∞ -category associated to the internal hom ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 145 hermitian ∞ -category Fun ex (( C , Ϙ ) , ( E , Ψ)) ∶= (Fun ex ( C , E ) , nat Ϙ Ψ ) , thus reducing Proposition 7.3.15 to thealgebraic Thom isomorphism of Proposition 7.3.5. For this, define L ∶= Fun ex ( C op , E ) and let 𝑝 ∗ ∶ L → D stand for pre-composition with 𝑝 ∶ Pair( C , B Ϙ ) → C op . Since 𝑝 has a fully-faithfulleft adjoint 𝑖 ∶ C op → Pair( C , B Ϙ ) we get that 𝑝 ∗ is fully-faithful and admits a right adjoint 𝑖 ∗ ∶ D → L given by pre-composition with 𝑖 . In addition, the restriction of Φ = nat Ϙ pair Ψ to L vanishes because if 𝑓 = 𝑔 ◦ 𝑝 ∶ Pair( C , B Ϙ ) → E for some 𝑔 ∶ C op → E then nat( Ϙ pair , 𝑓 ∗ Ψ) = nat( Ϙ pair , 𝑝 ∗ 𝑔 ∗ Ψ) = nat( 𝑝 ! Ϙ pair , 𝑔 ∗ Ψ) = nat( 𝑖 ∗ Ϙ pair , 𝑔 ∗ Ψ) = 0 where the identification 𝑝 ! Ϙ pair ≃ 𝑖 ∗ Ϙ pair is since 𝑖 op is right adjoint to 𝑝 op . Finally, the orthogonal comple-ment L ⟂ ⊆ D consists of those exact functors 𝑓 ∶ Pair( C , B Ϙ ) → E such that 𝑖 ∗ D D ( 𝑓 ) = 0 , that is, such that D E 𝑓 D pair ( 𝑖 ( 𝑥 )) = 0 for every 𝑥 ∈ C op . This is just equivalent to saying that 𝑓 vanishes on im( 𝑗 ) = ker( 𝑝 ) ,which is equivalent to saying that 𝑓 factors through 𝑝 . We may then conclude that L is a Lagrangian in D .Let us now identify L op = Fun ex ( C op , E ) op = Fun ex ( C , E op ) with Fun ex ( C , E ) via post-composition withthe duality of E . Then the inclusion L op → D sending 𝑓 to D D ( 𝑝 ∗ 𝑓 ) identifies with 𝑓 ↦ 𝑞 ∗ 𝑓 , where 𝑞 ∗ denotes pre-composition with the cartesian projection 𝑞 ∶ Pair( C , B Ϙ ) → C , and left adjoint of 𝑞 ∗ isgiven by pre-composition 𝑗 ∗ with 𝑗 ∶ C → Pair( C , B Ϙ ) . Let Π ∶= ( 𝑗 ∗ ) ! Ψ ∈ Fun q ( L op ) be the quadraticfunctor obtained by left Kan extension Ψ along 𝑗 ∗ ∶ D → L op , so that by (190) we can also write as Π( 𝑔 ) = Φ [ 𝜎 ] ( 𝑞 ∗ ( 𝑔 )) for 𝑔 ∈ L op = Fun ex ( C , E ) . Using Corollary 7.3.8 we then compute Π( 𝑔 ) = Φ [ 𝜎 ] ( 𝑔𝑞 ) ≃ nat( Ϙ pair , 𝑞 ∗ 𝑔 ∗ Ψ [ 𝜎 ] ) ≃≃ nat( 𝑞 ! Ϙ pair , 𝑔 ∗ Ψ [ 𝜎 ] ) ≃ nat( Ϙ , 𝑔 ∗ Ψ [ 𝜎 ] ) ≃ nat( Ϙ [− 𝜎 ] , 𝑔 ∗ Ψ) . The canonical map Φ( 𝑓 ) → Π( 𝑓 𝑗 ) for 𝑓 ∈ D then identifies with the map 𝜗 ∗ ∶ nat( Ϙ pair , 𝑓 ∗ Ψ) → nat( Ϙ [− 𝜎 ] , 𝑗 ∗ 𝑓 ∗ Ψ) obtained by restricting along 𝑗 using the equivalence 𝜃 ∶ Ϙ [− 𝜎 ] ≃ 𝑗 ∗ Ϙ pair . Invoking Proposition 7.3.11 wenow get an identification Fun ex (( C , Ϙ [− 𝜎 ] ) , ( E , Ψ)) Pair(Fun ex (( C , Ϙ [− 𝜎 ] ) , ( E , Ψ))) (Fun( C op , E ) , ex (( C , Ϙ [− 𝜎 ] ) , ( E , Ψ)) Fun ex (Pair( C , Ϙ ) , ( E , Ψ)) (Fun ex ( C op , E ) , ≃( 𝑗 ∗ ,𝜗 ∗ ) ( 𝑖 ∗ , of ( D , Φ) ∶= Fun ex (Pair( C , Ϙ ) , ( E , Ψ)) as the pairings poincaré category of the hermitian ∞ -category Fun ex (( C , Ϙ [− 𝜎 ] ) , ( E , Ψ)) , under which the associated cartesian fibration
Pair(Fun ex (( C , Ϙ [− 𝜎 ] ) , ( E , Ψ))) → Fun ex (( C , Ϙ [− 𝜎 ] ) , ( E , Ψ)) identifies with pre-composition with ( 𝑗, 𝜗 ) ∶ ( C , Ϙ ) → Pair( C , Ϙ ) . Proposition 7.3.15 consequently followsfrom Proposition 7.3.5. (cid:3) We take the point of view that the functor ( C , Ϙ ) ↦ Pair( C , Ϙ [ 𝜎 ] ) is a categorical analogue of the al-gebraic Thom construction studied in §2.3. Recall that the latter takes a hermitian object and returns ametabolic Poincaré object with respect to a shifted Poincaré structure. In the algebraic setting we saw thatthis association determines an equivalence Fm( C , Ϙ ) ≃ Pn 𝜕 ( C , Ϙ [1] ) between hermitian object in ( C , Ϙ ) and Poincaré objects in ( C , Ϙ [1] ) equipped with a prescribed Lagrangian.To make the categorical analogue complete we would like to argue that Pair(−) determines an equivalencebetween
Cat h∞ and a suitable ∞ -category whose objects are Poincaré ∞ -categories ( D , Φ) equipped with aLagrangian L ⊆ D , and whose maps are Poincaré functors which preserve the given Lagrangians. Whilewe will not make this completely precise, the gist of this claim amounts to the following two facts:
46 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE i) The essential image of the functor ( C , Ϙ ) ↦ Pair( C , Ϙ [ 𝜎 ] ) consists of the metabolic Poincaré ∞ -categories. This follows from Proposition 7.3.11.ii) Given two hermitain ∞ -categories ( C , Ϙ ) , ( C ′ , Ϙ ′ ) , Poincaré functors Pair( C , Ϙ [ 𝜎 ] ) → Pair( C ′ , Ϙ ′[ 𝜎 ] ) sending the Lagrangian C op ⊆ Pair( C , Ϙ [ 𝜎 ] ) to the Lagrangian C ′op ⊆ Pair( C , Ϙ [ 𝜎 ] ) are in bijection withhermitian functors ( C , Ϙ ) → ( C ′ , Ϙ ′ ) . Indeed, by Proposition 7.3.15 Poincaré functors Pair( C , Ϙ [ 𝜎 ] ) → Pair( C ′ , Ϙ ′[ 𝜎 ] ) correspond to hermitian functors ( C , Ϙ ) → Pair( C ′ , Ϙ ′[ 𝜎 ] ) , and such a hermitian functortakes values in the full subcategory C ′ ⊆ Pair( C ′ , Ϙ ′[ 𝜎 ] ) if and only if the corresponding Poincaréfunctor Pair( C , Ϙ [ 𝜎 ] ) → Pair( C ′ , Ϙ ′[ 𝜎 ] ) sends C to C ′ , which is equivalent to sending the Lagrangian C op ⊆ Pair( C , Ϙ [ 𝜎 ] ) to the Lagrangian C ′op ⊆ Pair( C ′ , Ϙ ′[ 𝜎 ] ) since C and C op are two full subcategoriesof Pair( C , B Ϙ ) which are switched by the duality, and the same holds for C ′ and C ′op .7.4. Genuine semi-additivity and spectral Mackey functors.
In this section we will explain how variouscategorical structures appearing in the theory of Poincaré ∞ -categories can be neatly encoded in frameworkof C -categories and Mackey functors, as developed by Barwick and collaborators in the setting of param-eterized higher category theory , see [Bar17], [BDG + hyperbolic Mackey functor constructed in Corollary 7.4.18 below, will form the basisto the formation of the real K -theory spectrum in Paper [II].To begin, let 𝐎 C be the orbit category of C , that is, the category of transitive C -sets and C -equivariantmaps. We note that 𝐎 C has two objects, C ∕C =∗ and C ∕ 𝑒 = C , such that ∗ is terminal, Hom 𝐎 C2 (C , C ) =C and there are no maps from ∗ to C . A C -category is by definition a cocartesian fibration 𝜋 ∶ E → 𝐎 opC , which by the straightening-unstraightening equivalence is the same data as a functor 𝐎 opC → Cat ∞ . A C -functor between C -categories is then a functor over 𝐎 C which preserves cocartesian edges. We note thatby the above explicit description of 𝐎 C we see that it is isomorphic to the categorical cone on the category BC . As a result, the data of a functor 𝐎 opC → C is equivalent to that of an ∞ -category E ∗ (the image ofthe “cone point” ∗ ), an ∞ -category E C with C -action (the image of C with the C -action induced by itsautomorphisms) and a C -equivariant map E ∗ → E C where the domain is considered with the trivial C -action. Since Cat ∞ admits limits the data of a C -equivariant map E ∗ → E C can equivalently be encodedvia a map E ∗ → E hC C .7.4.1. Example. If C is an ∞ -category with a C -action then we can right Kan extend the functor BC → Cat ∞ encoding this action to a functor 𝐎 opC → Cat ∞ , which we can then straighten to obtain a C -category E → 𝐎 opC with fibres E C ≃ C and E ∗ ≃ C hC , and structure map E ∗ → E hC C the identity. This constructionembeds Fun(BC , Cat ∞ ) as a full subcategory of C -categories.7.4.2. Example.
In the situation of Example 7.4.1, if C is of the form D × D with the flip action then C hC ≃ D and the C -equivariant functor E ∗ → E C is the diagonal D → D × D .The examples we will be interested in are the following:7.4.3. Examples. i) For a stable ∞ -category C the functor of taking symmetric bilinear parts B (−) ∶ Fun q ( C ) → Fun s ( C ) =Fun b ( C ) hC determines a C -category Fun q ( C ) → 𝐎 opC whose fibre over ∗ is Fun q ( C ) and whose fibre over C is Fun b ( C ) .ii) The functor Cat h∞ → Cat sb∞ = (Cat b ) hC sending a hermitian ∞ -category ( C , Ϙ ) to its underlyingsymmetric category ( C , B) determines a C -category Cat h → 𝐎 opC whose fibre over ∗ is Cat h∞ and whose fibre over C is Cat b . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 147 iii) The functor Cat p∞ → Cat ps∞ = (Cat ex∞ ) hC sending a Poincaré ∞ -category ( C , Ϙ ) to its underlying ∞ -category with perfect duality determines a C -category Cat p → 𝐎 opC whose fibre over ∗ is Cat p∞ and whose fibre over C is Cat ex∞ .For a C -category E → 𝐎 opC one may consider the C -variants of the usual notions of limits and colimits,defined for a given C -functor 𝑝 ∶ I → E . If I → 𝐎 opC is equivalent to a projection 𝐾 × 𝐎 opC → 𝐎 opC forsome 𝐾 then a C -colimit of 𝑝 ∶ I → E is given by a cocartesian section 𝑠 ∶ 𝐎 opC → E together witha natural transformation 𝜂 ∶ 𝑝 ⇒ 𝑠 | 𝐾 which exhibits 𝑠 as a colimit fibrewise, and dually for limits. Forexample, a C -initial object is given by a section 𝑠 ∶ 𝐎 opC → C which is fibrewise initial. We shall refer tothese as fibrewise C -colimits. By [Nar16, Proposition 2.11] a C -category E has all fibrewise C -colimitsindexed by 𝐾 × 𝐎 opC → 𝐎 opC if and only if the fibres E C and E ∗ both have 𝐾 -indexed colimits and thefunctor E ∗ → E C preserves 𝐾 -indexed colimits. In particular, all the examples in 7.4.3 have all fibrewise C -limits and C -colimits, since in all three cases the individual fibres have all limits and colimits and thecocartesian transition functor preserves all limits and colimits.For an indexing C -category I → 𝐎 opC which is not of the form 𝐾 × 𝐎 opC the notion of a C -colimit is abit more involved. The underlying data is still given by that of 𝑠 and 𝜂 , but the condition these are requiredto satisfy is more complicated, and is neither weaker nor stronger than being a fibrewise colimit. To avoid atechnical digression let us avoid giving the general definition, referring the reader to [Sha18, Definition 5.2].The simplest type of non-fibrewise C -(co)limits are finite C -(co)products, and these will be the only typeof non-fibrewise C -(co)limits that we will consider here. These are C -(co)limits indexed by finite C -sets,that is, C -categories I → 𝐎 opC which are finite direct sums of corepresentable left fibrations. Explicitly,these are encoded by the data of two finite sets 𝐴, 𝐵 , a C -action on 𝐵 , and an injective C -equivariant map 𝐴 → 𝐵 . We may then decompose them as a disjoint union of the finite C -set [∗ → ∗] (standing for theleft fibration over 𝐎 opC corepresented by ∗ ) and the finite C -set [∅ → C ] (standing for the left fibrationcorepresented by C ) ).We wish to verify that our examples of interest 7.4.3 all have finite C -products and coproducts. Forthis we will use a convenient criterion from [Nar16]. Before we can state it, we point out the followingobservation: if E → 𝐎 opC is a C -category with associated C -equivariant functor 𝑓 ∶ E ∗ → E C , and 𝑔 ∶ E C → E ∗ is a left or right adjoint to 𝑓 , then 𝑔 inherits a canonical C -equivariant structure. In fact, theentire adjunction carries a C -action, so that the unit and counit are C -equivariant natural transformations.This essentially follows from the uniqueness of adjoints given their existence. Otherwise put, the functorthat forgets an adjunction to its left adjoint is fully-faithful and hence any C -action can be lifted along it.One can also see this as follows. If 𝑓 admits a right adjoint then the cocartesian fibration E → 𝐎 opC is alsolocally cartesian (since C → ∗ is the only arrow that is not an isomorphism in 𝐎 C ), and hence a cartesianfibration. This cartesian fibration then encodes the data of a C -equivariant functor E C → E ∗ , which isright adjoint to 𝑓 . If a left adjoint to 𝑓 is considered then the same argument can be made using the dualcartesian fibration ̂ E → 𝐎 C , that is, the cartesian fibration classified by the same functor as E → 𝐎 opC .The following lemma is just an adaptation of [Nar16, Proposition 2.11] to the case at hand:7.4.4. Lemma.
Let E → 𝐎 C be a C -category with such that the fibres E C , E ∗ admit finite coproducts andthe functor E ∗ → E C preserves finite coproducts. Write 𝜎 ∶ E C → E C for the action of the generator of C . Then the following are equivalent:i) E admits all finite C -coproducts.ii) E admits C -colimits for C -diagrams indexed by the corepresentable C -set [∅ → C ] .iii) The functor 𝑓 ∶ E ∗ → E C admits a left adjoint 𝑔 ∶ E C → E ∗ such that for 𝑥 ∈ E C the map 𝑥 ∐ 𝜎 ( 𝑥 ) → 𝑓 𝑔 ( 𝑥 ) adjoint to the fold map 𝑔 ( 𝑥 ∐ 𝜎 ( 𝑥 )) ≃ 𝑔 ( 𝑥 ) ∐ 𝑔 ( 𝜎 ( 𝑥 )) ≃ 𝑔 ( 𝑥 ) ∐ 𝑔 ( 𝑥 ) → 𝑔 ( 𝑥 ) is an equivalence.Proof. The equivalence of i) and ii) follows from the fact that under the assumptions of the lemma E → 𝐎 C has fibrewise products by [Nar16, Proposition 2.11] and so the existence of C -coproducts follows from the
48 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE case of the corepresentable ones. The copresentable C -set [∗ → ∗] trivially has a C -colimit, and so theequivalence of i) and ii) follows. The equivalence of i) and iii) follows from [Nar16, Proposition 2.11]since iii) is simply a reformulation of the Beck-Chevalley criterion given there in the case of the uniquenon-invertible edge C → ∗ of 𝐎 C . (cid:3) Remark.
Lemma 7.4.4 has a dual version which is proven exactly the same way. It says that E has aall finite C -products if and only if it has C -limits for diagrams indexed by [∅ → C ] , and that the latter isequivalent to 𝑓 having a right adjoint 𝑔 ∶ E C → E ∗ such that for 𝑥 ∈ E C the map 𝑓 𝑔 ( 𝑥 ) → 𝑥 × 𝜎 ( 𝑥 ) adjoint to the diagonal 𝑔 ( 𝑥 ) → g( 𝑥 ) × 𝑔 ( 𝑥 ) ≃ 𝑔 ( 𝑥 ) × 𝑔 ( 𝜎 ( 𝑥 )) ≃ 𝑔 ( 𝑥 × 𝜎 ( 𝑥 )) is an equivalence.7.4.6. Remark.
Diagrams in E indexed by [∅ → C ] are determined by the data of an object 𝑥 ∈ E C .When the equivalent conditions of Lemma 7.4.4 hold then the C -coproduct of such a diagram is given bythe cocartesian section 𝑠 ∶ 𝐎 C → E whose value at ∗ is 𝑔 ( 𝑥 ) and whose value at C is 𝑥 ∐ 𝜎 ( 𝑥 ) . Theexistence of such a cocartesian section is insured by iii) above. Similarly, the C -product of such a diagram,when exists, is given by a cocartesian section 𝑠 ∶ 𝐎 C → E whose value at ∗ is 𝑔 ( 𝑥 ) and whose value at C is 𝑥 × 𝜎 ( 𝑥 ) .7.4.7. Example.
In the situation of Example 7.4.2, the resulting C -category E → 𝐎 opC has finite products(resp. coproducts) if and only if D has finite products (resp. coproducts).7.4.8. Proposition.
The C -categories of Examples 7.4.3 all have finite C -products and coproducts.Proof. We will use the criterion of Lemma 7.4.4. For this we first verify that the fibres over C and ∗ havefinite (co)products. In Example i) the fibres are stable, and in particular admit finite products and coproducts.For Example ii) the existence of limits of colimits is established in Proposition 6.1.2 and Remark 7.2.5, whilefor Example iii) it is established in Proposition 6.1.1 and Proposition 6.1.4. We now verify that all threeexamples satisfy Criterion iii) of Lemma 7.4.4. In the case of Example i) it follows from Lemma 1.1.7 andRemark 1.1.18 that the functor B ↦ B Δ gives both a left and a right adjoint to the bilinear part functor Ϙ ↦ B Ϙ . Unwinding the definitions, we now need to verify that for B ∈ Fun b ( C ) the maps B( 𝑥, 𝑦 ) ⊕ B( 𝑦, 𝑥 ) → f ib[B( 𝑥 ⊕ 𝑦 ) → B( 𝑥, 𝑥 ) ⊕ B( 𝑦, 𝑦 )] and cof[B( 𝑥, 𝑥 ) ⊕ B( 𝑦, 𝑦 ) → B( 𝑥 ⊕ 𝑦 )] → B( 𝑥, 𝑦 ) ⊕ B( 𝑦, 𝑥 ) are equivalences. Indeed, this follows directly from the bilinearity of B . For Example ii) we have byProposition 7.2.19 that the association ( A , B , B) ↦ ( A × B op , B) gives both a left and a right adjoint tothe functor Cat h∞ → Cat b . Criterion iii) can then be deduced from its validity for Example i) and forExample 7.4.7 with D = Cat ex∞ . Finally, the case of Example iii) follows from that of ii) since the formermaps to the latter via C -functor which is a fibrewise a replete subcategory inclusion the two-sided adjointof Cat h∞ → Cat b restricts to give and two sided adjoints for Cat p∞ → Cat ex∞ by Proposition 7.2.19. (cid:3)
It will be important for us in subsequent instalments of this project to know that the C -categoriesof Examples 7.4.3 don’t just admit finite C -products and coproducts but that they are furthermore C -semiadditive , see [Nar16, Definition 5.3]. To explain what this means let E → 𝐎 C be a C -category whichadmits finite C -products and coproducts, so that by Lemma 7.4.4 𝑓 admits both a left adjoint 𝑔 ∶ E C → E ∗ and a right adjoint ℎ ∶ E C → E ∗ , and these satisfy 𝑓 𝑔 ( 𝑥 ) ≃ 𝑥 ∐ 𝜎 ( 𝑥 ) and 𝑓 ℎ ( 𝑥 ) ≃ 𝑥 × 𝜎 ( 𝑥 ) . Supposethat the fibres E C and E ∗ are both semiadditive, so that we may identify products and coproducts and writethem as direct sums 𝑓 𝑔 ( 𝑥 ) ≃ 𝑥 ⊕ 𝜎 ( 𝑥 ) ≃ 𝑓 ℎ ( 𝑥 ) . Then we have natural candidate for a comparison map(194) 𝑔 ( 𝑥 ) → ℎ ( 𝑥 ) which is adjoint to the map 𝑥 → 𝑓 ℎ ( 𝑥 ) ≃ 𝑥 ⊕ 𝜎 ( 𝑥 ) , corresponding to the inclusion of the component 𝑥 . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 149 The following definition is an adaptation of [Nar16, Definition 5.3] to the particular case where the baseis 𝐎 C , making use of the fact that 𝐎 C has a unique non-invertible arrow, given by C → ∗ .7.4.9. Definition.
Let E → 𝐎 C be a C -category which admits finite C -products and finite C -coproducts.Then E is called C -semiadditive if the following holds:i) The fibres E C and E ∗ are semiadditive.ii) The comparison map (194) between the left and right adjoints of 𝑓 is an equivalence.7.4.10. Example.
In the situation of Example 7.4.2, the resulting C -category E → 𝐎 opC is semiadditive ifand only if D is semiadditive.7.4.11. Proposition.
The C -categories of Examples 7.4.3 are all C -semiadditive.Proof. We first verify that in all the examples in 7.4.3 the fibres are semi-additive. For Example i) the fibresare stable and in particular semiadditive. For Examples ii) and iii) this was established in Proposition 6.1.7and Remark 7.2.7.We now establish the second condition of Definition 7.4.9. Arguing as in the proof of Proposition 7.4.8using Example 7.4.10 in place of Example 7.4.7 we see that it will suffice to establish the second conditionfor Example i). Now by Remark 1.1.18 the diagonal restriction functor Δ ∗ ∶ Fun b ( C ) → Fun q ( C ) is bothleft and right adjoint to the cross effect functor B (−) ∶ Fun q ( C ) → Fun b ( C ) and we already saw in the proofof Proposition 7.4.8 that the composite Fun b ( C ) → Fun q ( C ) → Fun b ( C ) is naturally equivalent to the functor B ↦ B ⊕ B swap . Unwinding the definitions, to establish the second condition it will suffice to show thatunit of the adjunction Δ ∗ ⊣ B (−) is given by the component inclusion B( 𝑥, 𝑦 ) → B( 𝑥, 𝑦 ) ⊕ B( 𝑦, 𝑥 ) , and the counit of the adjunction B (−) ⊣ Δ ∗ is given by the component projection B( 𝑥, 𝑦 ) ⊕ B( 𝑦, 𝑥 ) → B( 𝑥, 𝑦 ) . Indeed, this is established in Remark 1.1.18. (cid:3)
Remark. If E → 𝐎 C is a C -semiadditive C -category then the C -product and C -coproduct of a [∅ → C ] -indexed C -diagram in E corresponding to an object 𝑥 ∈ E C are both given by the cocartesiansection 𝑠 ∶ 𝐎 C → E whose value at ∗ is 𝑔 ( 𝑥 ) and whose value at C is 𝑥 ⊕ 𝜎 ( 𝑥 ) , cf. Remark 7.4.6.7.4.13. Remark. If E → 𝐎 C is a C -semiadditive C -category then the functor 𝑓 ∶ E ∗ → E C admits a twosided adjoint 𝑔 . The C -equivariant structure induces a C -equivariant structure on 𝑔 in apriori two differentways: one by the uniqueness of 𝑔 as a left adjoint of 𝑓 and once by its uniqueness as a right adjoint. Thecomparison map (194) is however a natural transformation of C -equivariant functors (since the componentinclusion 𝑥 → 𝑥 ⊕ 𝜎 ( 𝑥 ) is such), and so it identifies the left and right adjoints of 𝑓 also as C -equivariantfunctors. Similarly, the induced functor 𝑔 ∶ E hC C → Fun(BC , E ∗ ) is a two sided adjoint to the inducedfunctor 𝑓 ∶ Fun(BC , E ∗ ) → E hC C .7.4.14. Remark.
Specializing to the case of the C -category Cat p → 𝐎 C we now get that the func-tor Hyp inherits a C -equivariant structure making it a two-sided C -equivariant adjoint to U ∶ Cat p∞ → Cat ex∞ , and similarly the induced functor
Hyp hC ∶ (Cat ex∞ ) hC → Fun(BC , Cat p∞ ) is a two-sided adjoint to U hC ∶ Fun(BC , Cat p∞ ) → (Cat ex∞ ) hC . The composed functor Hyp∶ Cat p∞ → (Cat ex∞ ) hC Hyp hC2 ←←←←←←←←←←←←←←←←←←←←←←←←←←←←→
Fun(BC , Cat p∞ ) then determines a C -action on Hyp(U( C , Ϙ )) = Hyp( C ) for a Poincaré ∞ -category. For a fixed ( C , Ϙ ) , thisis the C -action of Construction 2.2.7, but now promoted to be natural in C . Similarly, the C -equivarianceof the maps Hyp( C ) hyp ←←←←←←←←←←←←←←→ ( C , Ϙ ) fgt ←←←←←←←←←←←←→ Hyp( C ) constructed in Lemma 2.2.9 is now exhibited as the components of two natural transformations of C -equivariant functors in ( C , Ϙ ) .
50 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
Remark.
In the situation of Remark 7.4.14, the functors U and U hC participate in a commutativesquare of forgetful functors Fun(BC , Cat p∞ ) Cat p∞ (Cat ex∞ ) hC Cat ex∞ . U hC2 U Passing to left adjoints, we obtain a commutative square(195)
Cat ex∞ (Cat ex∞ ) hC Cat p∞ Fun(BC , Cat p∞ ) . Hyp Hyp hC2
At the same time, since
Cat ex∞ is semi-additive the top horizontal functor in (195) is given by the symmetriza-tion C ↦ C × C op associated to the op -action on Cat ex∞ . But
Cat p having finite C -coproducts (Proposi-tion 7.4.8) means that this symmetrization is identified with U ◦ Hyp as a functor
Cat ex∞ → (Cat ex∞ ) C . Since Cat p∞ is also semi-additive (Proposition 6.1.7) it then follows that from the commutativity of the abovesquare that the functor Cat p∞ → Fun(BC , Cat p∞ ) ( C , Ϙ ) ↦ Hyp(Hyp( C )) is naturally equivalent to the functor sending ( C , Ϙ ) to Hyp( C ) × Hyp( C ) , equipped with the flip C -action.We now wish to use the C -semiadditivity of the Examples in 7.4.3 in order to extract extra structuresin terms of Mackey functors. For this, let Span(C ) be the span ∞ -category of finite C -sets, as definedin [Bar17, Df. 3.6]. A Mackey object in an additive ∞ -category A is by definition a product preservingfunctor from Span(C ) → A . If A is taken to be S 𝑝 , the results of [GM11] and [Nar16] show that thearising ∞ -category underlines the model category classically used for the definition of genuine spectra. Wewill treat spectral Mackey functors as the definition of the latter objects and therefore put S 𝑝 gC = Fun × (Span(C ) , S 𝑝 ) . Evaluation at the finite C -sets C then defines the functor 𝑢 ∶ S 𝑝 gC → S 𝑝 C , by retaining the action of thespan C ←←←←←←←←← C ←←←←←←←←←←←→ C . Evaluation at the one-point C -set defines the genuine fixed points − gC ∶ S 𝑝 gC → S 𝑝 . The datum of a genuine C -spectrum thus is equivalent to the datum of the pair of spectra ( 𝐸 𝑔 C , 𝐸 ) ,together with a C -action on 𝐸 and restriction and transfer maps res ∶ 𝐸 gC → 𝐸 tr ∶ 𝐸 → 𝐸 gC coming from the spans(196) ∗ ← C ←←←←←←←←→ C and C ←←←←←←←←← C → ∗ with a host of compatibility data, and similarly for other target categories.7.4.16. Proposition.
Let E → 𝐎 C be a C -semiadditive ∞ -category with transition functor 𝑓 ∶ E ∗ → E C and two-sided adjoint 𝑔 ∶ E C → E ∗ . Then the identity functor E ∗ → E ∗ canonically lifts to functor Fun × 𝐎 C2 (Span(C ) , E ∗ ) E ∗ E ∗ where the vertical arrow is given by evaluation at ∗ . In addition the composed functor E ∗ → Fun × 𝐎 C2 (Span(C ) , E ∗ ) ev C2 ←←←←←←←←←←←←←←←←←←→ E ∗ , where ev C denotes evaluation at C ∈ Span(C ) , is naturally equivalent to the functor 𝑥 ↦ 𝑔𝑓 ( 𝑥 ) . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 151 Proof.
Let A eff (C ) → 𝐎 opC be the C -Burnside ∞ -category of [Nar16, Df. 4.12]. The objects of A eff (C ) are given by arrows 𝑈 → 𝑉 where 𝑈 is finite C -set and 𝑉 ∈ 𝐎 C is a C -orbit, and morphisms in A eff (C ) from [ 𝑈 → 𝑉 ] to [ 𝑈 ′ → 𝑉 ′ ] are given by diagrams of the form 𝑈 𝑈 ′′ 𝑈 ′ 𝑉 𝑉 ′ 𝑉 ′ . The functor A eff (C ) → 𝐎 opC is then given by [ 𝑈 → 𝑉 ] ↦ 𝑉 , and is a cocartesian fibration whose fibre over 𝑉 ∈ 𝐎 C is the span ∞ -category of finite C -sets over 𝑉 . Combining [Nar16, Pr. 5.11] and [Nar16, Th. 6.5]we have the evaluation at the object [∗ → ∗] ∈ A eff (C ) yields an equivalence Fun ×C (A eff (C ) , E ) ≃ ⟶ E ∗ . Now the action of every C -product preserving C -functor A eff (C ) → E on fibres over ∗ is again a product-preserving functor from Span(C ) to E ∗ . Base change along {∗} ⊆ 𝐎 opC then determines a functor R ∶ E ∗ ≃ Fun ×C (A eff (C ) , E ) → Fun × (Span(C ) , E ∗ ) . equipped with a natural equivalence R ( 𝑥 )(∗) ≃ 𝑥 , by construction. Furthermore, for 𝑥 ∈ E ∗ the Mackey functor R ( 𝑥 ) ∶ Span(C ) → E ∗ is obtained inparticular by restricting a C -functor R ( 𝑥 ) ∶ A eff (C ) → E , i.e., a functor over 𝐎 opC which preservescocartesian edges. Since R ( 𝑥 ) sends [∗ → ∗] to 𝑥 ∈ E ∗ ⊆ E by construction it must send the object [C → C ] ∈ A eff (C ) to 𝑓 ( 𝑥 ) ∈ E C ⊆ E . Since R ( 𝑥 ) furthermore preserves C -biproducts it musttherefore send [C → ∗] ∈ A eff (C ) to 𝑔𝑓 ( 𝑥 ) , see Remark 7.4.12. (cid:3) Applying Proposition 7.4.16 in the case of the C -category Fun q ( C ) → 𝐎 C of Examples 7.4.3i) weobtain7.4.17. Corollary.
The inclusion
Fun q ( C ) ⊆ Fun( C op , S 𝑝 ) admits a canonical lift to a functor Fun q ( C ) → Fun( C op , S 𝑝 gC ) . In particular, every quadratic functor Ϙ ∶ C → S 𝑝 lifts canonically to a functor ̃ Q ∶ C op → S 𝑝 gC valued ingenuine C -spectra, such that ̃ Ϙ ( 𝑥 ) has underlying spectrum B Ϙ ( 𝑥, 𝑥 ) and genuine fixed points Ϙ ( 𝑥 )) . Corollary (The hyperbolic Mackey functor) . The construction of hyperbolic categories canonicallyrefines to a functor gHyp∶ Cat p∞ ⟶ Fun × (Span(C ) , Cat p∞ ) together with natural equivalences of Poincaré ∞ -categories [gHyp( C , Ϙ )](∗) ≃ ( C , Ϙ ) , and a natural C -equivariant equivalence of Poincaré ∞ -categories [gHyp( C , Ϙ )](C ) ≃ Hyp C . In addition, the resulting C -equivariant functors Hyp( C ) ( C , Ϙ ) Hyp( C ) [ gHyp( C , Ϙ ) ] (C ) [ gHyp( C , Ϙ ) ] (∗) [ gHyp( C , Ϙ ) ] (C ) ≃ ≃ ≃ associated to the spans of (196) are given by the functors hyp and f gt of (42) which are the unit and counitof the two-sided adjunctions between Cat p∞ and Cat ex∞ of Corollary 7.2.20.
52 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
Multiplicativity of Grothendieck-Witt and L -groups. In this section we will prove that the invari-ants L (−) and GW (−) defined in §2.3 and §2.4 are lax symmetric monoidal functors. In addition, thehigher L -groups organize into a symmetric monoidal functor to the category of graded abelian groups (withits Koszul symmetric monoidal structure). As a result, they carry a graded-commutative algebra structurewhen applied to symmetric monoidal Poincaré ∞ -categories, such as those described in §5.4.To begin, we first note that by Corollary 5.2.8 the functor 𝜋 Pn ∶ Cat p∞ → Set admits a canonical laxsymmetric monoidal structure. This lax symmetric monoidal structure can be made quite explicit. Indeed,consider the assignment 𝜋 Pn( C , Ϙ ) × 𝜋 Pn( C ′ , Ϙ ′ ) → 𝜋 Pn( C ⊗ C ′ , Ϙ ⊗ Ϙ ′ ) . For a pair of Poincaré objects ( 𝑥, 𝑞 ) in ( 𝑥 ′ , 𝑞 ′ ) we get the Poincaré object ( 𝑥 ⊗ 𝑥 ′ , 𝑞 ⊗ 𝑞 ′ ) in ( C ⊗ C ′ , Ϙ ⊗ Ϙ ′ ) ,where 𝑥⊗𝑥 ′ ∈ C ⊗ C ′ is the image of ( 𝑥, 𝑥 ′ ) ∈ C × C ′ under the universal bilinear functor 𝛽 ∶ C × C ′ → C ⊗ C ′ ,and 𝑞 ⊗ 𝑞 ′ denotes the map 𝕊 → [ Ϙ ⊗ Ϙ ′ ]( 𝑥 ⊗ 𝑥 ′ ) obtained as the composite 𝕊 = 𝕊 ⊗ 𝕊 𝑞⊗𝑞 ′ ←←←←←←←←←←←←←←←←←←←→ Ϙ ( 𝑥 ) ⊗ Ϙ ′ ( 𝑥 ′ ) = [ Ϙ ⊠ Ϙ ′ ]( 𝑥, 𝑥 ′ ) → P 𝛽 ! [ Ϙ ⊠ Ϙ ′ ]( 𝑥 ⊗ 𝑥 ′ ) = [ Ϙ ⊗ Ϙ ′ ]( 𝑥 ⊗ 𝑥 ′ ) . This object is Poincaré because its underlying bilinear form is given in light of Proposition 5.1.3 by thecombination of the underlying bilinear forms of 𝑞 and 𝑞 ′ .We now wish to upgrade the above lax symmetric monoidal structure to the level of E ∞ -spaces. Forthis, first note that since Cat p∞ and Cat h∞ are semi-additive (Proposition 6.1.7) the corepresentable functors Pn and Fm canonically refine to functors with values in E ∞ -spaces. Recall (see, e.g., [Nik16, Proposition5.6]) that the ∞ -category Mon E ∞ of E ∞ -monoids carries a canonical symmetric monoidal structure suchthat the free-forgetful adjunction F ∶ S ⟂ Mon E ∞ ∶ U becomes symmetric monoidal (that is, its left adjoint is symmetric monoidal from which the right adjointinherits a lax symmetric monoidal structure). We now claim that the E ∞ -refinement ̃ Pn ∶ Cat p∞ → Mon E ∞ and ̃ Fm ∶ Cat h∞ → Mon E ∞ also carry lax symmetric monoidal structures. This is in fact a completely formalconsequence of the fact that the monoidal structure on Cat p∞ and Cat h∞ preserves direct sums:7.5.1. Lemma.
Let E be a small semi-additive ∞ -category equipped with a symmetric monoidal structure ⊗ which preserves direct sums in each variable. Then the lax symmetric monoidal structure of Map E (1 E , −) canonically lifts to its E ∞ -refinement ̃ Map E (1 E , −) ∶ E → Mon E ∞ . Corollary.
The lax symmetric monoidal structure of Pn and Fm canonically lifts to their E ∞ -refinements ̃ Pn ∶ Cat p∞ → Mon E ∞ and ̃ Fm ∶ Cat h∞ → Mon E ∞ .Proof. By possibly enlarging the universe we may assume that
Cat p∞ and Cat h∞ are small. The claim thenfollows from Lemma 7.5.1. (cid:3) Proof of Lemma 7.5.1.
The full subcategory
Fun × ( E , Mon E ∞ ) ⊆ Fun( E , Mon E ∞ ) spanned by the product-preserving functors is an accessible localization of Fun( E , Mon E ∞ ) with a left adjoint which we will denoteby 𝐿 ∶ Fun( E , Mon E ∞ ) → Fun × ( E , Mon E ∞ ) . By Lemma 5.3.4 this localization is compatible with Dayconvolution and extends to a symmetric monoidal localization 𝐿 ⊗ ∶ Fun( E , Mon E ∞ ) ⊗ → Fun × ( E , Mon E ∞ ) ⊗ , where the codomain is endowed with the structure inherited from being a full suboperad of Fun( E , Mon E ∞ ) ⊗ .Now the identification Map E (1 E , −) ≃ U ̃ Map E (1 E , −) transposes to give a natural transformation of theform F ◦ Map E (1 E , −) ⇒ ̃ Map E (1 E , −) . Since ̃ Map E (1 E , −) is product preserving this natural transforma-tion induces a natural transformation 𝐿 (F ◦ Map E (1 E , −)) ⇒ ̃ Map E (1 E , −) . We claim that this last transformation is an equivalence. Note that this implies the desired claim via thesymmetric monoidal structures of 𝐿 and F . Now, to prove the claim, it will suffice to show that for everyproduct-preserving functor G ∶ E → Mon E ∞ the induced map Nat( ̃ Map E (1 E , −) , G ) → Nat(F ◦ Map E (1 E , −) , G ) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 153 is an equivalence of spaces. Indeed, by adjunction we may also identify this map with the map Nat( ̃ Map E (1 E , −) , G ) → Nat(U ◦ ̃ Map E (1 E , −) , U ◦ G ) induced by U ◦ (−) . This last map is an equivalence since the forgetful functor U ◦ (−) ∶ Fun × ( E , Mon E ∞ ) → Fun × ( E , S ) is an equivalence on product-preserving functors. (cid:3) We now come to the main result of this subsection:7.5.3.
Proposition.
The functors L , GW ∶ Cat p∞ → CMon admit unique lax symmetric monoidal struc-tures such that the transformations 𝜋 Pn ⇒ GW ⇒ L are symmetric monoidal, where CMon stands for the (ordinary) symmetric monoidal category of commu-tative monoids.
Remark.
The full subcategory A 𝑏 ⊆ CMon spanned by abelian groups is a full suboperad and a sym-metric monoidal localisation of
CMon . Since GW and L take values in A 𝑏 the lax monoidal structures on GW , L and the map GW ⇒ L equally applies if we consider GW and L as functors to A 𝑏 . The reasonfor working with the larger category of commutative monoids is to be able to make arguments pertainingto the natural transformation from 𝜋 Pn .The proof of Proposition 7.5.3 will require knowing certain multiplicative properties of the adjunctions Cat h∞ ⟂ Cat p∞ and Cat ex∞ ⟂ Cat p∞ , which we now verify.7.5.5. Lemma.
The adjunction
Cat h∞ ⟂ Cat p∞ of Proposition 7.3.15, in which the right adjoint Cat p∞ → Cat h∞ is symmetric monoidal, satisfies the projection formula: for ( C , Ϙ ) ∈ Cat p∞ and ( C ′ , Ϙ ′ ) ∈ Cat h∞ thePoincaré functor (197) Pair( C ⊗ C ′ , ( Ϙ ⊗ Ϙ ′ ) [ 𝜎 ] ) → ( C , Ϙ ) ⊗ Pair( C ′ , Ϙ ′[ 𝜎 ] ) associated under this adjunction to the hermitian functor ( C ⊗ C ′ , Ϙ ⊗ Ϙ ′ ) = ( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ ) → ( C , Ϙ ) ⊗ Pair( C ′ , Ϙ ′[ 𝜎 ] ) induced by the unit hermitian functor ( C ′ , Ϙ ′ ) → Pair( C ′ , Ϙ ′[ 𝜎 ] ) , is an equivalence of Poincaré ∞ -categories.Proof. This is a formal consequence of (and in fact equivalent to) the fact that the inclusion
Cat p∞ → Cat h∞ is closed symmetric monoidal, see Remark 6.2.18. Explicitly, in light of Proposition 7.3.15 it will sufficeto show that for every Poincaré ∞ -category ( E , Ψ) the restriction map Map
Cat p∞ (( C , Ϙ ) ⊗ Pair( C ′ , Ϙ ′[ 𝜎 ] ) , ( E , Ψ)) → Map
Cat h∞ (( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ ) , ( E , Ψ)) is an equivalence of spaces. Indeed, we may identify this map with the map
Map
Cat p∞ (Pair( C ′ , Ϙ ′[ 𝜎 ] ) , Fun ex (( C , Ϙ ) , ( E , Ψ))) → Map
Cat h∞ (( C ′ , Ϙ ′ ) , Fun ex (( C , Ϙ ) , ( E , Ψ))) which is an equivalence by another application of Proposition 7.3.15. (cid:3)
Taking Ϙ = 0 in Lemma 7.5.5 and using Remark 7.2.21 we immediately find:7.5.6. Corollary.
The adjunction
Hyp ⊣ U satisfies the projection formula: for ( C , Ϙ ) ∈ Cat p∞ and C ′ ∈Cat h∞ ) the Poincaré functor Hyp( C ⊗ C ′ ) → ( C , Ϙ ) ⊗ Hyp( C ′ ) induced by the component inclusion C ⊗ C ′ → C ⊗ U Hyp( C ′ ) = C ⊗ [ C ′ ⊕ C ′op ] = [ C ⊗ C ′ ] ⊕ [ C ⊗ C ′op ] is an equivalence of Poincaré ∞ -categories. Remark.
One can also deduce Corollary 7.5.6 from the fact that the forgetful functor
U ∶ Cat p∞ → Cat ex∞ is closed symmetric monoidal, being the composition of the inclusion
Cat p∞ → Cat h∞ and the closedsymmetric monoidal projection Cat h∞ → Cat ex∞ , see Remark 6.2.11 and the final part of Remark 6.2.18.In particular, the adjunction
Cat ex∞ ⟂ Cat h∞ also satisfies the projection formula, as is visible from theequivalence ( C , Ϙ ) ⊗ ( C ′ ,
0) ≃ ( C ⊗ C ′ , .
54 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
Remark.
In the situation of Lemma 7.5.5, when ( C , Ϙ ) is also Poincaré then by the triangle identitiesthe projection formula equivalence (197) fits into a commutative triangle Pair( C ⊗ C ′ , ( Ϙ ⊗ Ϙ ′ ) [ 𝜎 ] ) ( C , Ϙ ) ⊗ Pair( C ′ , Ϙ ′[ 𝜎 ] )( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ ) , ≃ where the diagonal arrows are obtained from the counit of the adjunction Cat h∞ ⟂ Cat p∞ at ( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ ) and ( C ′ , Ϙ ′ ) , respectively. Using Example (7.3.3)iii) and Remark 7.3.21 we may also write this commutativetriangle as(198) Met(( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ )) ( C , Ϙ ) ⊗ Met( C ′ , Ϙ ′ )( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ ) . met ≃ ( C , Ϙ ) ⊗ met Applying the same argument for the projection formula of Corollary 7.5.6 we similarly have the commuta-tive triangle(199)
Hyp(( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ )) ( C , Ϙ ) ⊗ Hyp( C ′ , Ϙ ′ )( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ ) . hyp ≃ ( C , Ϙ ) ⊗ hyp Remark.
The commutative triangle (198) taken with ( C ′ , Ϙ ′ ) = ( S 𝑝 f , Ϙ u ) yields an equivalence Met( C , Ϙ ) ≃ ( C , Ϙ ) ⊗ Met( S 𝑝 f , Ϙ u ) of Poincaré ∞ -categories over ( C , Ϙ ) . It then follows that when ( C , Ϙ ) isa symmetric monoidal Poincaré ∞ -category the Poincaré ∞ -category Met( C , Ϙ ) acquires the structure ofa module object over ( C , Ϙ ) (specifically, the free ( C , Ϙ ) -module generated from Met( S 𝑝 f , Ϙ u ) ) such that thefunctor met ∶ Met( C , Ϙ ) → ( C , Ϙ ) is a map of (free) ( C , Ϙ ) -modules. Similarly, the commutative triangle (199) taken with ( C ′ , Ϙ ′ ) = ( S 𝑝 f , Ϙ u ) yields an equivalence Hyp( C , Ϙ ) ≃ ( C , Ϙ ) ⊗ Hyp( S 𝑝 f , Ϙ u ) over ( C , Ϙ ) , and so when ( C , Ϙ ) is symmetricmonoidal Poincaré we get that Hyp( C ) acquires the structure of a module object over ( C , Ϙ ) (freely generatedby Hyp( S 𝑝 f ) ) and hyp ∶ Hyp( C ) → ( C , Ϙ ) is a map of free ( C , Ϙ ) -modules. Proof of Proposition 7.5.3.
The uniqueness is clear since the maps 𝜋 Pn( C , Ϙ ) → GW ( C , Ϙ ) → L ( C , Ϙ ) are surjective (and surjectivity is stable under tensor products in CMon ). It will hence suffice to show thatfor every pair of Poincaré ∞ -categories ( C , Ϙ ) , ( C ′ , Ϙ ′ ) the dotted arrows(200) 𝜋 Pn( C , Ϙ ) ⊗ 𝜋 Pn( C ′ , Ϙ ′ ) GW ( C , Ϙ ) ⊗ GW ( C ′ , Ϙ ′ ) L ( C , Ϙ ) ⊗ L ( C ′ , Ϙ ′ ) 𝜋 Pn(( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ )) GW (( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ )) L (( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ )) exists to make the diagram commute, where the tensor products in the top row is that of commutativemonoids.Let us first treat the case of the functor L . For this it will suffice to show that for a pair of Poincaré objects ( 𝑥, 𝑞 ) and ( 𝑥 ′ , 𝑞 ′ ) such that ( 𝑥 ′ , 𝑞 ′ ) is metabolic the associated Poincaré object ( 𝑥, 𝑞 ) ⊗ ( 𝑥 ′ , 𝑞 ′ ) is metabolicin ( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ ) (the rest follows by symmetry). But this follows directly from the commutativity of the ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES I: FOUNDATIONS 155 diagram(201) 𝜋 Pn Met(( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ )) 𝜋 Pn(( C , Ϙ ) ⊗ Met( C ′ , Ϙ ′ )) 𝜋 Pn( C , Ϙ ) ⊗ 𝜋 Pn Met( C ′ , Ϙ ′ ) 𝜋 Pn(( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ )) 𝜋 Pn( C , Ϙ ) ⊗ 𝜋 Pn( C ′ , Ϙ ′ ) [met]≃ id ⊗ [met] given by the commutative triangle (198).We now turn to GW . For the well-definedness of the middle dotted arrow in (200) we have to show thatthe relation [hyp( 𝑤 )] ∼ [ 𝑥, 𝑞 ] for a Langrangian 𝑤 → 𝑥 is preserved under tensoring with some Poincaréobject ( 𝑥 ′ , 𝑞 ′ ) . Given the commutativity of the diagram (201) above, it will suffice to show that the diagram 𝜋 Pn Met(( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ )) 𝜋 Pn(( C , Ϙ ) ⊗ Met( C ′ , Ϙ ′ )) 𝜋 Pn( C , Ϙ ) ⊗ 𝜋 Pn Met( C ′ , Ϙ ′ ) 𝜋 Pn Hyp( C ⊗ C ′ ) [hyp] 𝜋 Pn(( C , Ϙ ) ⊗ Hyp( C ′ )) 𝜋 Pn( C , Ϙ ) ⊗ 𝜋 Pn Hyp( C ′ , Ϙ ′ ) 𝜋 Pn(( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ )) 𝜋 Pn( C , Ϙ ) ⊗ 𝜋 Pn( C ′ , Ϙ ′ ) dlag ∗ ≃ (id ⊗ dlag) ∗ id ⊗ [dlag]≃ id ⊗ [hyp] is commutative as well, where the vertical arrows in the top rows are induced by the Poincaré functor dlag ∶ Met(−) → Hyp(−) of Construction 2.3.8. Here, the squares on the right hand side are commutativeas they are given by the lax monoidal structure on 𝜋 Pn , and the bottom left triangle is induced by thecommutative triangle (199). It will suffice to show that the top left square commutes. Indeed, this square isobtained by applying 𝜋 Pn to the square of Poincaré ∞ -categories Met(( C , Ϙ ) ⊗ ( C ′ , Ϙ ′ )) ( C , Ϙ ) ⊗ Met( C ′ , Ϙ ′ )Hyp( C ⊗ C ′ ) ( C , Ϙ ) ⊗ Hyp( C ′ ) , dlag ≃ ( C , Ϙ ) ⊗ dlag≃ which commutes since it is obtained by evaluating the natural transformation (197) of the projection formulaat the arrow ( C , Ϙ ) → ( C , in Cat h∞ , see Remark 7.3.22. (cid:3) Corollary.
The functor L ∗ ∶ Cat p∞ → gr A 𝑏 admits a lax symmetric monoidal structure, where gr A 𝑏 denotes the category of ℤ -graded abelian groups with the symmetric monoidal structure using the Koszulsign rules.Proof. By definition we have that L 𝑛 ( C , Ϙ ) = L ( C , Ϙ [− 𝑛 ] ) = L ( C , Ϙ ⊗ 𝕊 − 𝑛 ) . Thus the claim follows bycombining Proposition 7.5.3, Remark 2.3.13, and the fact that 𝕊 −∙ is a graded commutative algebra in thehomotopy category of spectra. Concretely, the structure maps are simply given by L 𝑛 ( C , Ϙ ) ⊗ L 𝑚 ( C ′ , Ϙ ′ ) = L ( C , Ϙ [− 𝑛 ] ) ⊗ L ( C ′ , Ϙ ′[− 𝑚 ] ) → L ( C ⊗ C ′ , Ϙ [− 𝑛 ] ⊗ Ϙ ′[− 𝑚 ] ) = L ( C ⊗ C , ( Ϙ ⊗ Ϙ ′ ) [− 𝑛 − 𝑚 ] ) = L 𝑛 + 𝑚 ( C ⊗ C , Ϙ ⊗ Ϙ ′ ) and the fact that it is symmetric follows as explained above. (cid:3) Corollary. If ( C , Ϙ ) is a monoidal ∞ -category then GW ( C , Ϙ ) acquires a ring structure and L ∙ ( C , Ϙ ) a graded ring structure such that the natural map GW ( C , Ϙ ) → L ( C , Ϙ ) is a ring homomorphism. If the monoidal structure is symmetric then GW ( C , Ϙ ) is commutative and L ∙ ( C , Ϙ ) is graded commutative. Example.
Let 𝐴 ∈ Alg hC E ∞ be a commutative ring spectrum equipped with a C -action. Thenthe Grothendieck-Witt groups GW (Mod 𝜔𝐴 , Ϙ s 𝐴 ) , GW (Mod 𝜔𝐴 , Ϙ ≥ 𝐴 ) and GW (Mod 𝜔𝐴 , Ϙ t 𝐴 ) of the symmetric
56 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE monoidal Poincaré ∞ -categories of Examples 5.4.9 carry natural commutative ring structures, and simi-larly the corresponding graded L -groups L ∙ (Mod 𝜔𝐴 , Ϙ s 𝐴 ) , L ∙ (Mod 𝜔𝐴 , Ϙ ≥ 𝐴 ) and L ∙ (Mod 𝜔𝐴 , Ϙ t 𝐴 ) carry canonicalgraded-commutative ring structures.Combining Proposition 7.5.3 with Example 2.4.5, Example 5.4.2 and Remark 7.5.9 we also get:7.5.13. Corollary. If ( C , Ϙ ) is a symmetric monoidal ∞ -category then [f gt] ∶ GW( C , Ϙ ) → K ( C ) is a map of rings and [hyp] ∶ K ( C ) → GW ( C , Ϙ ) is a map of GW ( C ) -modules. Remark.
In the situation of Corollary 7.5.13, the GW ( C , Ϙ ) -module structure on K ( C ) could beconsidered ambiguous: on the one hand we have the module structure determined by the ring structure of K ( C ) via the ring map f gt ∶ GW ( C , Ϙ ) → K ( C ) , and on the other we have the module structure inducedby the ( C , Ϙ ) -module structure on Hyp( C ) of Remark 7.5.9 via the identification K ( C ) ≅ GW (Hyp( C )) .These two modules structures however coincide. Indeed, unwinding the definitions we see that the formerstructure is induced via the lax monoidal structure of GW and the symmetric monoidal structure of ( C , Ϙ ) by the Poincaré functor ( C , Ϙ ) ⊗ Hyp( C ) → Hyp( C ⊗ C ) , corresponding via the adjunction U ⊣ Hyp to the exact functor C ⊗ [ C ⊕ C op ] → C ⊗ C induced by theprojection C ⊕ C op → C , while the latter module structure is induced in the same manner by the inverse ofthe Poincaré equivalence Hyp( C ⊗ C ) ≃ ←←←←←←←→ ( C , Ϙ ) ⊗ Hyp( C ) of Corollary 7.5.6, corresponding via the adjunction Hyp ⊣ U to the exact functor C ⊗ C → C ⊗ [ C ⊕ C op ] induced by the inclusion C → C ⊕ C op . It will hence suffice to verify that these Poincaré functors determinesinverse equivalences between Hyp( C ⊗ C ) and ( C , Ϙ ) ⊗ Hyp( C ) . However, since Poincaré functors from (orto) Hyp are determines by their underlying exact functors, it suffices to check that these underlying exactfunctors determine inverse equivalences between C ⊗ [ C ⊕ C op ] and [ C ⊗ C ] ⊕ [ C op ⊗ C op ] . The latteris however a formal consequence of the fact that the monoidal structure on Cat ex∞ preserves direct sums ineach variable.Invoking Examples 5.4.10 and Example 5.4.11 we also have the following two corollaries:7.5.15.
Corollary.
Let ( C , Ϙ ) be a symmetric monoidal hermitian ∞ -category with underlying bilinear part B = B Ϙ . Then the quadratic Grothendieck-Witt group GW ( C , Ϙ qB ) is canonically a module over the ring GW ( C , Ϙ ) and the map GW ( C , Ϙ q ) → GW ( C , Ϙ ) is a map of GW ( C , Ϙ ) -modules. Similarly, the quadratic L -groups L ∙ ( C , Ϙ qB ) form a graded module overthe graded ring L ∙ ( C , Ϙ ) and the map L ∙ ( C , Ϙ q ) → L ∙ ( C , Ϙ ) is a map of graded L ∙ ( C , Ϙ ) -modules. Corollary.
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