Hermitian K-theory for stable ∞ -categories II: Cobordism categories and additivity
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 II:COBORDISM CATEGORIES AND ADDITIVITY BAPTISTE CALMÈS, EMANUELE DOTTO, YONATAN HARPAZ, FABIAN HEBESTREIT, MARKUS LAND,KRISTIAN MOI, DENIS NARDIN, THOMAS NIKOLAUS, AND WOLFGANG STEIMLE
To Andrew Ranicki. A BSTRACT . We define Grothendieck-Witt spectra in the setting of Poincaré ∞ -categories, show that they fitinto an extension with a K - and an L -theoretic part and deduce localisation sequences for Verdier quotients. Asspecial cases we obtain generalisations of Karoubi’s fundamental and periodicity theorems for rings in which need not be invertible.A novel feature of our approach is the systematic use of ideas from cobordism theory by interpreting thehermitian Q -construction as an algebraic cobordism category. We also use this to give a new description of the LA -spectra of Weiss and Williams. C ONTENTS
Introduction Recollection Q -construction and algebraic cobordism categories Q -construction 442.2 The cobordism category of a Poincaré ∞ -category 472.3 Algebraic surgery 522.4 The additivity theorem 572.5 Fibrations between cobordism categories 582.6 Additivity in K -Theory 65 ∞ -categories 733.3 The group-completion of an additive functor 793.4 The spectrification of an additive functor 863.5 Bordism invariant functors 913.6 The bordification of an additive functor 933.7 The genuine hyperbolisation of an additive functor 100 L -theory and the fundamental fibre square 1094.5 The real algebraic K -theory spectrum and Karoubi periodicity 1184.6 LA -theory after Weiss and Williams 120 Date : September 16, 2020.
AppendixA Verdier sequences, Karoubi sequences and stable recollements
AppendixB Comparisons to previous work ∞ -category with duality 144B.2 Schlichting’s Grothendieck-Witt-spectrum of a ring with invertible 146 References
NTRODUCTION
Overview.
Unimodular symmetric and quadratic forms are ubiquitous objects in mathematics appearing incontexts ranging from norm constructions in number theory to surgery obstructions in geometric topology.Their classification, however, even over simple rings such as the integers, remains out of reach. A sim-plification, following ideas of Grothendieck for the study of projective modules, suggests to consider for acommutative ring 𝑅 (for ease of exposition) the abelian group GW q0 ( 𝑅 ) given as the group completion of themonoid of isomorphism classes of finitely generated projective 𝑅 -modules 𝑃 , equipped with a unimodularquadratic (say) form 𝑞 , with addition the orthogonal sum [ 𝑃 , 𝑞 ] + [ 𝑃 ′ , 𝑞 ′ ] = [ 𝑃 ⊕ 𝑃 ′ , 𝑞 ⟂ 𝑞 ′ ] . This group, commonly known as the Grothendieck-Witt group of 𝑅 , was given a homotopy-theoreticalrefinement at the hands of Karoubi and Villamayor in [KV71], by adapting Quillen’s approach to higheralgebraic K -theory.For this, one organizes the collection of pairs ( 𝑃 , 𝑞 ) of unimodular quadratic forms into a groupoid Unimod q ( 𝑅 ) , which may be viewed as an E ∞ -space using the symmetric monoidal structure on Unimod q ( 𝑅 ) arising from the orthogonal sum considered above. One can then take the group completion to obtain an E ∞ -group GW qcl ( 𝑅 ) = Unimod q ( 𝑅 ) grp , the classical Grothendieck-Witt space, those group of components is the Grothendieck-Witt group describedabove. By definition the higher Grothendieck-Witt groups of 𝑅 are the homotopy groups of GW cl ( 𝑅 ) .There are variants for symmetric bilinear and even forms, and instead of starting with a commutativering, one can study unimodular hermitian forms valued in an invertible 𝑅⊗ ℤ 𝑅 -module 𝑀 equipped with aninvolution (subject to an invertibility condition) also for non-commutative 𝑅 ; this generality includes boththe case of a ring 𝑅 with involution by considering 𝑀 = 𝑅 , and also skew-symmetric and skew-quadraticforms by changing the involution on 𝑀 by a sign. Polarisation in general produces maps GW qcl ( 𝑅, 𝑀 ) ⟶ GW evcl ( 𝑅, 𝑀 ) ⟶ GW scl ( 𝑅, 𝑀 ) which are equivalences if is a unit in 𝑅 .In the present paper we establish a general decomposition of the Grothendieck-Witt space into a K -theoretic and an L -theoretic part, the latter of which is closely related to Witt groups of unimodular forms:For 𝑟 ∈ {q , ev , s} the Witt group W 𝑟 ( 𝑅, 𝑀 ) of the pair ( 𝑅, 𝑀 ) is given by dividing isomorphism classes ofunimodular 𝑀 -valued forms by those admitting a Lagrangian. In low degrees the relation takes the formof an exact sequence K ( 𝑅 ) C hyp ←←←←←←←←←←←←←←→ GW 𝑟 ( 𝑅, 𝑀 ) ⟶ W 𝑟 ( 𝑅, 𝑀 ) ⟶ here the map labelled hyp assigns to a projective module 𝑃 it hyperbolisation 𝑃 ⊕
Hom 𝑅 ( 𝑃 , 𝑀 ) equippedwith the evaluation form and the C -coinvariants on the left are formed with respect to the action 𝑃 ↦ Hom 𝑅 ( 𝑃 , 𝑀 ) . The first goal of the paper is to extend this to a long exact sequence with L -groups play-ing the role of higher Witt groups. Such results are well-known principally from the work of Karoubi andSchlichting if is a unit in 𝑅 , and have lead to a good understanding of Grothendieck-Witt theory relative ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 3 to K -theory for two reasons: Firstly, Witt groups are rather accessible. As an example let us mention thatVoevodsky’s solution to the Milnor conjecture provides a complete filtration of the Witt group W( 𝑘 ) forany field 𝑘 not of characteristic with filtration quotients H ∗ (Gal( 𝑘 ∕ 𝑘 ) , ℤ ∕2) , and an older result of Katoachieves a similar description in characteristic , see [Kat82, Voe03, OVV07]. Secondly, by work of Ran-icki [Ran92] the higher L -groups satisfy L 𝑖 +2 ( 𝑅, 𝑀 ) = L 𝑖 ( 𝑅, − 𝑀 ) if is invertible in 𝑅 and are thus inparticular -periodic, which greatly reduces the computational complexity. The second goal of the presentpaper series is to describe the extent to which such periodicity statements still hold if is not invertible in 𝑅 . Let us also mention that the K -theoretic part of the description is rather indifferent to the invertibility of in 𝑅 , so from an understanding of the L -theoretic term, one can often deduce absolute statements aboutGrothendieck-Witt theory by appealing to the recent progress in the understanding of algebraic K -theory.We will take up this thread in the third installment of the series. History and main result.
To state our results, let us give a more detailed account of the ingredients. Thestudy of Grothendieck-Witt spaces begins by comparing them to Quillen’s algebraic K -theory space K ( 𝑅 ) defined as the group completion of the groupoid of finitely generated projective modules over 𝑅 . To thisend one has f gt ∶ GW scl ( 𝑅, 𝑀 ) → K ( 𝑅 ) and hyp ∶ K ( 𝑅 ) → GW qcl ( 𝑅, 𝑀 ) , the former extracting the underlying module of a unimodular form, the latter induced by the hyperbolisationconstruction.In his fundamental papers [Kar80a, Kar80b] Karoubi analysed the case in which is a unit in 𝑅 (so nodistinction between the three flavours of Grothendieck-Witt groups is necessary). He considered the spaces U cl ( 𝑅, 𝑀 ) = f ib( K ( 𝑅 ) hyp ←←←←←←←←←←←←←←→ GW cl ( 𝑅, 𝑀 )) and V cl ( 𝑅, 𝑀 ) = f ib( GW cl ( 𝑅, 𝑀 ) fgt ←←←←←←←←←←←←→ K ( 𝑅 )) , produced equivalences Ω U cl ( 𝑅, − 𝑀 ) ≃ V cl ( 𝑅, 𝑀 ) , and moreover showed that the cokernels W 𝑖 ( 𝑅, 𝑀 ) of K 𝑖 ( 𝑅 ) hyp ←←←←←←←←←←←←←←→ GW 𝑖 ( 𝑅, 𝑀 ) satisfy W 𝑖 ( 𝑅, 𝑀 )[ ] ≅ 𝑊 𝑖 +2 ( 𝑅, − 𝑀 )[ ] and are in particular -periodic up to -torsion. In fact, Karoubi shows that this latter statement also holdswithout the assumption that be invertible in 𝑅 ; in other words, the additional difficulties of Grothendieck-Witt theory as compared to K -theory are concentrated at the prime . These results are nowadays knownas Karoubi’s fundamental and periodicity theorems and form one of the conceptual pillars of hermitian 𝐾 -theory; they permit one to inductively deduce results on higher Grothendieck-Witt groups from informationabout algebraic K -theory on the one hand and about W 𝑖 ( 𝑅, ± 𝑀 ) for 𝑖 = 0 , on the other.To control the behaviour of the -torsion in the cokernel of the hyperbolisation map Kobal in [Kob99]introduced refinements of the hyperbolic and forgetful maps: By the invertibility assumption on 𝑀 , taking 𝑀 -valued duals induces an action of the group C on the algebraic K -theory spectrum and we denote thearising C -spectrum by K( 𝑅, 𝑀 ) and similarly for the K -theory space. The maps above then refine to asequence K ( 𝑅, 𝑀 ) hC hyp ←←←←←←←←←←←←←←→ GW qcl ( 𝑅, 𝑀 ) ⟶ GW scl ( 𝑅, 𝑀 ) fgt ←←←←←←←←←←←←→ K ( 𝑅, 𝑀 ) hC , whose composite is the norm on K ( 𝑅, 𝑀 ) . Kobal used these refinements to show that, if is invertible in 𝑅 , the cofibre of hyp ∶ K ( 𝑅, 𝑀 ) hC → GW cl ( 𝑅, 𝑀 ) is -periodic on the nose.The next major steps forward were then taken by Schlichting in [Sch17], who introduced (non-connective)Grothendieck-Witt spectra for differential graded categories with duality in which is invertible. He usedthese to give a new proof of Karoubi’s fundamental theorem by first establishing the existence of a fibresequence GW cl ( 𝑅, 𝑀 [−1]) fgt ←←←←←←←←←←←←→ K( 𝑅, 𝑀 ) hyp ←←←←←←←←←←←←←←→ GW cl ( 𝑅, 𝑀 ) , which he termed the Bott sequence; here GW cl ( 𝑅, 𝑀 [ 𝑖 ]) is the Grothendieck-Witt spectrum of the category Ch b (Proj( 𝑅 )) with its duality determined by 𝑀 [ 𝑖 ] . For 𝑖 = 0 (in which case we suppress it from notation)Schlichting shows that indeed Ω ∞ GW cl ( 𝑅, 𝑀 ) ≃ GW cl ( 𝑅, 𝑀 ) . The salient feature that relates this se-quence to Karoubi’s theorem is the existence of an equivalence GW cl ( 𝑅, 𝑀 [−2]) ≃ GW cl ( 𝑅, − 𝑀 ) . Still CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE assuming invertible in 𝑅 he, furthermore, showed that the ( -periodic) homotopy groups of the cofibreof the refined hyperbolic map hyp ∶ K( 𝑅, 𝑀 ) hC → GW cl ( 𝑅, 𝑀 ) are indeed given by the Witt groups W( 𝑅, 𝑀 ) and W( 𝑅, − 𝑀 ) in even degrees and by Witt groups of formations in odd degrees.This lead to the folk theorem that if is a unit in 𝑅 the cofibre of hyp ∶ K( 𝑅, 𝑀 ) hC → GW cl ( 𝑅, 𝑀 ) isgiven by Ranicki’s L -theory spectum L( 𝑅, 𝑀 ) from [Ran92], whose homotopy groups are well-known tomatch Schlichting’s results, though as far as we are aware no account at the level of spectra has appeared inthe literature.Let us not fail to mention that Schlichting also introduced a variant of symmetric Grothendieck-Wittspectra without the assumption that is invertible in 𝑅 in [Sch10a], that satisfy localisation results by thecelebrated [Sch10b]. These are, however, of slightly different flavour in that they should relate to non-connective K -theory, though to the best of our knowledge this is not developed in the literature. To differ-entiate we will refer to them as Karoubi-Grothendieck-Witt spectra, and relegate a thorough discussion tothe fourth part of this series of papers.As an example, let us mention that the strategy described above lead to an almost complete computationof the Grothendieck-Witt groups of ℤ [ ] in [BK05] and to great structural insight, for example by controllingthe -adic behaviour of the forgetful map GW cl → K hC in [BKSØ15] under the assumption that is a unit.Without this assumption, however, many of the methods employed break down. In particular, the relationto L -groups remained mysterious: If is not invertible in 𝑅 there are many flavours of L -groups and as faras we are aware not even a precise conjecture has been put forward. In contrast to this situation, Karoubiconjectured in [Kar09] that his fundamental theorems should have an extension to general rings, where itis not only the sign that changes when passing from U( 𝑅, 𝑀 ) to V( 𝑅, − 𝑀 ) but also the form parameter; asimilar suggestion was made by Giffen, see [Wil05]. In what is hopefully evident notation they predicted Ω U qcl ( 𝑅, − 𝑀 ) ≃ V evcl ( 𝑅, 𝑀 ) and Ω U evcl ( 𝑅, − 𝑀 ) ≃ V scl ( 𝑅, 𝑀 ) . In this paper series along with its companion [HS20] we entirely resolve these questions. In the presentpaper we obtain the extensions of Karoubi’s periodicity and fundamental theorem, affirming in particularthe conjecture of Karoubi and Griffen, and also determine the cofibre of the hyperbolisation map in terms ofan L -theory spectrum. In distinction with the variants usually employed for example in geometric topology,the L -spectra appearing are generally not -periodic. Part three of this series is devoted to a detailed studyof these spectra and in particular, an investigation of their periodicity properties. While the results of thatpaper are largely specific to the case of discrete rings, the results of the present paper also apply much moregenerally to schemes, E ∞ -rings, parametrised spectra among others.Our approach is based on placing Grothendieck-Witt- and L -theory into a common general framework,namely the setting of Poincaré ∞ -categories, introduced by Lurie in his approach to L -theory [Lur11], anddeveloped in detail in the first part of this series. A Poincaré ∞ -category is a small stable ∞ -category C together with a certain kind of functor Ϙ ∶ C op → Sp which encodes the type of form (such as, quadratic,even or symmetric) under consideration. The requirements on Ϙ are such, that it, in particular, yields anassociated duality equivalence D Ϙ ∶ C op → C .As mentioned, Lurie defined L -theory for general Poincaré ∞ -categories, and it is by now standard toview K -theory as a functor on stable ∞ -categories. The duality D Ϙ induces a C -action on the K -spectrumof a Poincaré ∞ -category and we will denote the resulting C -spectrum by K( C , Ϙ ) . Adapting the hermitian Q -construction, we here also produce a Grothendieck-Witt spectrum GW( C , Ϙ ) in this generality. To explainhow this generalises the Grothendieck-Witt theory of discrete rings, take C = D p ( 𝑅 ) , the stable subcate-gory of the derived ∞ -category D ( 𝑅 ) spanned by the perfect complexes over 𝑅 . As part of Paper [I] weconstructed Poincaré structures Ϙ q 𝑀 ⟹ Ϙ gq 𝑀 ⟹ Ϙ ge 𝑀 ⟹ Ϙ gs 𝑀 ⟹ Ϙ s 𝑀 connected by maps as indicated: Roughly, the outer two assign to a chain complex its spectrum of homotopycoherent quadratic or symmetric 𝑀 -valued forms, whereas the middle three are the more subtle animations,or in more classical terminology non-abelian derivations, of the functors Quad 𝑀 , Ev 𝑀 , Sym 𝑀 ∶ Proj( 𝑅 ) op → A 𝑏 parametrising ordinary 𝑀 -valued quadratic, even and symmetric forms, respectively. The comparison mapsbetween these are equivalences, if is a unit in 𝑅 , but in general they are five distinct Poincaré structures ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 5 on D p ( 𝑅 ) . Now, essentially by construction the spectra L( D p ( 𝑅 ) , Ϙ q 𝑀 ) = L q ( 𝑅, 𝑀 ) and L( D p ( 𝑅 ) , Ϙ s 𝑀 ) = L s ( 𝑅, 𝑀 ) are Ranicki’s -periodic L -spectra, but from the main result of [HS20] we find that it is the middle threePoincaré structures which give rise to the classical Grothendieck-Witt spaces, i.e. we have Ω ∞ GW( D p ( 𝑅 ) , Ϙ gq 𝑀 ) ≃ GW qcl ( 𝑅, 𝑀 ) , Ω ∞ GW( D p ( 𝑅 ) , Ϙ ge 𝑀 ) ≃ GW evcl ( 𝑅, 𝑀 ) and Ω ∞ GW( D p ( 𝑅 ) , Ϙ gs 𝑀 ) ≃ GW scl ( 𝑅, 𝑀 ) . This mismatch (which is also the reason for carrying the subscript cl through the introduction) explainsmuch of the subtlety that arose in previous attempts to connect Grothendieck-Witt- and L -theory.In case is invertible in 𝑅 the identification extends to GW cl ( 𝑅, 𝑀 ) ≃ GW( D p ( 𝑅 ) , Ϙ gs 𝑀 ) and we willtherefore use the names GW qcl , GW evcl and GW scl also for the Grothendieck-Witt spectra of the Poincaré ∞ -categories considered above.As the main result of the present paper we provide extensions of Karoubi’s periodicity theorem andSchlichting’s extension of his fundamental theorem in complete generality: Main Theorem.
For every Poincaré ∞ -category ( C , Ϙ ) , there is a fibre sequence K( C , Ϙ ) hC hyp ←←←←←←←←←←←←←←→ GW( C , Ϙ ) bord ←←←←←←←←←←←←←←←←←→ L( C , Ϙ ) , which canonically splits after inverting and a fibre sequence GW( C , Ϙ [−1] ) fgt ←←←←←←←←←←←←→ K( C ) hyp ←←←←←←←←←←←←←←→ GW( C , Ϙ ) . Here, we have used Ϙ [ 𝑖 ] to denote the shifted Poincaré structure 𝕊 𝑖 ⊗ Ϙ . As in Schlichting’s set-up thisoperation satisfies ( D p ( 𝑅 ) , ( Ϙ q 𝑀 ) [2] ) ≃ ( D p ( 𝑅 ) , Ϙ q− 𝑀 ) and ( D p ( 𝑅 ) , ( Ϙ s 𝑀 ) [2] ) ≃ ( D p ( 𝑅 ) , Ϙ s− 𝑀 ) , so if is a unit in 𝑅 we, in particular, recover the results of Karoubi and Schlichting mentioned above, andextend the identification of the cofibre of the hyperbolisation map to the spectrum level. More importantly,however, if is not invertible we find ( D p ( 𝑅 ) , ( Ϙ gs 𝑀 ) [2] ) ≃ ( D p ( 𝑅 ) , Ϙ ge− 𝑀 ) and ( D p ( 𝑅 ) , ( Ϙ ge 𝑀 ) [2] ) ≃ ( D p ( 𝑅 ) , Ϙ gq− 𝑀 ) , whence the second part settles the conjecture of Giffen and Karoubi. Explicitly, we obtain: Corollary.
For a discrete ring 𝑅 and an invertible 𝑅 -module 𝑀 with involution there are canonical equiv-alences U qcl ( 𝑅, − 𝑀 ) ≃ 𝕊 ⊗ V evcl ( 𝑅, 𝑀 ) and U evcl ( 𝑅, − 𝑀 ) ≃ 𝕊 ⊗ V scl ( 𝑅, 𝑀 ) . As a consequence of the first part of our Main Theorem we obtain a direct relation between the Grothendieck-Witt spectra for different form parameters. As an implementation of Ranicki’s L -theoretic periodicity resultsLurie produced canonical equivalences L( C , Ϙ [1] ) ≃ 𝕊 ⊗ L( C , Ϙ ) . Applying this twice we obtain a stabilisation map stab ∶ 𝕊 ⊗ L( D p ( 𝑅 ) , Ϙ gs 𝑀 ) ≃ L( D p ( 𝑅 ) , Ϙ gq 𝑀 ) ⟶ L( D p ( 𝑅 ) , Ϙ gs 𝑀 ) and as another articulation of periodicity we have: Corollary.
The natural map GW qcl ( 𝑅, 𝑀 ) → GW scl ( 𝑅, 𝑀 ) fits into a commutative diagram K( 𝑅 ) hC GW qcl ( 𝑅, 𝑀 ) 𝕊 ⊗ L( D p ( 𝑅 ) , Ϙ gs 𝑀 )K( 𝑅 ) hC GW scl ( 𝑅, 𝑀 ) L( D p ( 𝑅 ) , Ϙ gs 𝑀 ) id hyp stabhyp bord of fibre sequences, i.e. the cofibres of the two hyperbolisation maps differ by a fourfold shift. CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE If is invertible in 𝑅 this shift on the right hand side is invisible since in that case L( D p ( 𝑅 ) , Ϙ gs 𝑀 ) =L( D p ( 𝑅 ) , Ϙ s 𝑀 ) is -periodic. Outlook.
As mentioned, the main content of the third paper in this series is a detailed investigation of thespectra L( D p ( 𝑅 ) , Ϙ gs 𝑀 ) . We show there that 𝜋 ∗ L( D p ( 𝑅 ) , Ϙ gs 𝑀 ) is Ranicki’s original version of symmetric L -theory from [Ran81], which he eventually abandoned in favour of L s ( 𝑅, 𝑀 ) precisely because in gen-eral it lacks the -periodicity exhibited by the latter. In particular, the cofibre of the hyperbolisation map K( 𝑅 ) hC → GW gs ( 𝑅, 𝑀 ) is not generally -periodic if is not invertible in 𝑅 .Furthermore, improving a previous bound of Ranicki’s we show there that for 𝑅 commutative and noe-therian of global dimension 𝑑 the comparison maps L( D p ( 𝑅 ) , Ϙ gq 𝑀 ) ⟶ L( D p ( 𝑅 ) , Ϙ ge 𝑀 ) ⟶ L( D p ( 𝑅 ) , Ϙ gs 𝑀 ) ⟶ L( D p ( 𝑅 ) , Ϙ s 𝑀 ) are equivalences in degrees past 𝑑 − 2 , 𝑑 and 𝑑 + 2 , respectively. Thus in sufficiently high degrees theperiodic behaviour of the cofibre of the hyperbolisation map is restored and surprisingly there is also nodifference between the various flavours of Grothendieck-Witt groups. This allows one to use the inductivemethods previously only available if is invertible also in more general situations. We demonstrate this bygiving a solution for number rings of Thomasson’s homotopy limit problem [Tho83], asking when the map GW s ( 𝑅, 𝑀 ) → K( 𝑅, 𝑀 ) hC is a -adic equivalence, and an essentially complete computation of GW 𝑟 ( ℤ ) where 𝑟 ∈ {±s , ±q} (over theintegers, quadratic and even forms happen to agree), affirming a conjecture of Berrick and Karoubi from[BK05].Before explaining the strategy of proof in the next section let us finally mention that feeding Poincaré ∞ -categories of parametrised spectra into our machinery produces, by our Main Theorem, another set ofinteresting objects, the LA -spectra introduced by Weiss and Williams in their study of automorphism groupsof manifolds [WW14]. In this case, the results of the next section allow for an entirely new interpretationof these spectra, which sheds light on their geometric meaning. In particular, this furthers the programsuggested by Williams in [Wil05] to connect the study manifold topology more intimately with hermitian K -theory. We will spell this out in the third section of this introduction along with further results concerningdiscrete rings, that require a bit of preparation. Hermitian K -theory of Poincaré ∞ -categories. Let us now sketch in greater detail the road to our mainresults. Besides the setup of Poincaré ∞ -categories the main novelty of our approach is its direct connectionto the theory of cobordism categories of manifolds. To facilitate the discussion recall that Cob 𝑑 has asobjects 𝑑 − 1 closed oriented manifolds, and cobordisms thereof as morphisms. The celebrated equivalence | Cob 𝑑 | ≃ Ω ∞−1 MTSO( 𝑑 ) , established by Galatius, Madsen, Tillmann and Weiss in [GTMW09] then lies at the heart of much mod-ern work on the homotopy types of diffeomorphism groups [GRW14]; here MTSO( 𝑑 ) denotes the Thomspectrum of − 𝛾 𝑑 → BSO( 𝑑 ) .Now, a Poincaré ∞ -category ( C , Ϙ ) determines a space of Poincaré objects Pn( C , Ϙ ) to be thought of asthe higher categorical generalisation of the groupoid Unimod(
𝑅, 𝑀 ) of unimodular forms considered in thecase of discrete rings above. Along with the Grothendieck-Witt spectrum we produce for every Poincaré ∞ -category ( C , Ϙ ) an analogous cobordism category Cob( C , Ϙ ) ∈ Cat ∞ with objects given by Pn( C , Ϙ [1] ) and morphisms given by spaces of Poincaré cobordisms, Ranicki style; here our dimension conventionsadhere to those of the geometric setting.As the technical heart of the present paper we show the following version of the additivity theorem: Theorem A. If ( C , Ϙ ) ⟶ ( D , Φ) ⟶ ( E , Ψ) is a split Poincaré-Verdier sequence, then the second map induces a bicartesian fibration of ∞ -categories Cob( D , Φ) ⟶ Cob( E , Ψ) , ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 7 whose fibre over E , Ψ) is Cob( C , Ϙ ) . In particular, one obtains a fibre sequence | Cob( C , Ϙ ) | ⟶ | Cob( D , Φ) | ⟶ | Cob( E , Ψ) | of spaces. Here, a Poincaré-Verdier sequence is a null-composite sequence, which is both a fibre sequence and acofibre sequence in
Cat p∞ , the ∞ -category of Poincaré ∞ -categories; we call it split if both underlying func-tors admit both adjoints. This requirement precisely makes the underlying sequence of stable ∞ -categories C → C ′ → C ′′ into a stable recollement. The simplest (and in fact universal) example of such a recollementis the sequence C 𝑥 ↦ [ 𝑥 → ←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ Ar( C ) [ 𝑥 → 𝑦 ] ↦ 𝑦 ←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ C . By the non-hermitian version of Theorem A due to Barwick [Bar17] (which can in fact also be extracted asspecial case of Theorem A), it gives rise to a fibre sequence | Span( C ) | ⟶ | Span(Ar( C )) | ⟶ | Span( C ) | which is split by the functor C → Ar( C ) taking 𝑥 to id 𝑥 . Taking loopspaces thus results in an equivalence K(Ar( C )) ≃ K( C ) × K( C ) , since K( C ) ≃ Ω | Span( C ) | , which makes Theorem A is a hermitian analogue of Waldhausen’s additivitytheorem.The simplest example of a split Poincaré-Verdier sequence arises similarly: If C has a Poincaré structure Ϙ , then the arrow category of C refines to a Poincaré ∞ -category Met( C , Ϙ ) , whose Poincaré objects encodePoincaré objects in ( C , Ϙ ) equipped with a Lagrangian, or in other words a nullbordism. There results themetabolic Poincaré-Verdier sequence ( C , Ϙ [−1] ) ⟶ Met( C , Ϙ ) 𝜕 ←←←←←←→ ( C , Ϙ ) , refining the recollement above. We interpret the cobordism category Cob 𝜕 ( C , Ϙ ) of its middle term as thatof Poincaré objects with boundary in ( C , Ϙ ) . From the additivity theorem we then find a fibre sequence | Cob( C , Ϙ ) | ⟶ | Cob 𝜕 ( C , Ϙ ) | 𝜕 ←←←←←←→ | Cob( C , Ϙ [1] ) | , that is entirely analogous to Genauer’s fibre sequence | Cob 𝑑 | ⟶ | Cob 𝜕𝑑 | 𝜕 ←←←←←←→ | Cob 𝑑 −1 | from geometric topology [Gen12]. Note, however, that neither of these latter sequences are split (the adjointfunctors in a split Poincaré-Verdier sequence need not be compatible with the Poincaré structures), so thename additivity is maybe slightly misleading, but we will stick with it.Our proof of Theorem A is in fact modelled on the recent proof of Genauer’s fibre sequence at thehands of the 9’th author [Ste18] and is new even in the context of algebraic K -theory. Similar resultsare known in varying degrees of generality, see for example [Sch17, HSV19]. The actual additivity theo-rem we prove is, however, quite a bit more general than Theorem A: We show that in fact every additivefunctor F ∶ Cat p∞ → S , a mild strenghthening of the requirement that split Poincaré-Verdier sequencesare taken to fibre sequences, gives rise to an F -based cobordism category Cob F ( C , Ϙ ) and that the functor | Cob F | ∶ Cat p∞ → S is then also additive. Applied to F = Pn this gives the result above, but the statementcan now be iterated. Since the functor GW ∶ Cat p∞ → S 𝑝 (and thus also GW = Ω ∞ GW ) is defined by aniterated hermitian Q -construction, this generality gives sufficient control to establish: Theorem B. i) There is a natural equivalence | Cob( C , Ϙ ) | ≃ Ω ∞−1 GW( C , Ϙ ) , and in particular Ω | Cob( C , Ϙ ) | ≃ GW ( C , Ϙ ) ii) The functors GW ∶ Cat p∞ → S 𝑝 and GW ∶ Cat p∞ → Grp E ∞ ( S ) are the initial additive functors equippedwith a transformation Pn → GW ≃ Ω ∞ GW , respectively.iii) The functor L ∶ Cat p∞ → S 𝑝 is the initial additive, bordism invariant functor equipped with a trans-formation Pn → Ω ∞ L . CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
Here, we call an additive functor
Cat p∞ → S 𝑝 bordism invariant if it vanishes when evaluated on meta-bolic categories, though there are many other characterisations. Theorem B simultaneously gives the her-mitian analogue of the theorem of Blumberg, Gepner and Tabuada from [BGT13], that K ∶ Cat ex∞ → S 𝑝 is the initial additive functor with a transformation Cr → Ω ∞ K , and of the theorem of Galatius, Madsen,Tillmann and Weiss concerning the homotopy type of the cobordism category. Just as for the additivity the-orem, our cobordism theoretic methods provide a more direct proof of the universal property of algebraic K -theory avoiding all mention of non-commutative motives.From Theorem B, it is straight-forward to obtain our Main Theorem: The first assertion of the Main Theoremmay be restated as the formula cof(hyp∶ K( C , Ϙ ) hC ⟶ GW( C , Ϙ )) ≃ L( C , Ϙ ) , and it is somewhat tautologically true that the left hand side is the initial bordism invariant functor under GW , whence the universal properties of GW and L from Theorem B give the claim. For the second statementwe take another queue from geometric topology and use Ranicki’s algebraic Thom construction to producean equivalence | Cob 𝜕 ( C , Ϙ ) | ≃ | Span( C ) | = Ω ∞−1 K( C ) which extends to an identification GW(Met( C , Ϙ )) ≃ K( C ) for all Poincaré ∞ -categories ( C , Ϙ ) . Via Theorem B the metabolic Poincaré-Verdier sequence then givesrise to the fibre sequence GW( C , Ϙ ) fgt ←←←←←←←←←←←←→ K( C ) hyp ←←←←←←←←←←←←←←→ GW( C , Ϙ [1] ) , which we term the Bott-Genauer-sequence, bearing witness to its relation with the fibre sequence MTSO( 𝑑 ) ⟶ 𝕊 [BSO( 𝑑 )] ⟶ MTSO( 𝑑 − 1) first established by Galatius, Madsen, Tillmann and Weiss, and beautifully explained by Genauer’s theoremthat | Cob 𝜕𝑑 | ≃ Ω ∞−1 𝕊 [BSO( 𝑑 )] . As explained in the previous section, if is invertible (and the input is sufficiently strict) this sequence isdue to Schlichting, but as far as we are aware its connection with the fibre sequence of Thom spectra abovehad not been noticed before.With the Main Theorem established, we observe that since both L - and K -theory are well-known to takearbitrary bifibre sequence in Cat p∞ to fibre sequences (and not just split ones) we obtain: Corollary C.
The functor
GW ∶ Cat p∞ → S 𝑝 is Verdier localising, i.e. it takes arbitrary Poincaré-Verdiersequences ( C , Ϙ ) ⟶ ( D , Φ) ⟶ ( E , Ψ) to bifibre sequences GW( C , Ϙ ) ⟶ GW( D , Φ) ⟶ GW( E , Ψ) of spectra. This result is a full hermitian analogue of the localisation theorems available for algebraic K -theory,and (as far as we are aware) subsumes and extends all known localisation sequences for Grothendieck-Wittgroups, in particular the celebrated results of [Sch10b]. We explicitly spell out some consequences for lo-calisations of discrete rings in Corollary F below.The fibre sequence of the Main Theorem can also be neatly repackaged using equivariant homotopytheory: The assignment ( C , Ϙ ) ↦ K( C , Ϙ ) tC is another example of a bordism invariant functor, whenceTheorem B produces a natural map Ξ ∶ L( C , Ϙ ) → K( C , Ϙ ) tC . A version of this map first appeared in thework of Weiss and Williams on automorphisms of manifolds [WW14], and we show that our construction ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 9 agrees with theirs. With it one can reexpress the fibre sequence from the Main Theorem as a cartesiansquare GW( C , Ϙ ) L( C , Ϙ )K( C , Ϙ ) hC K( C , Ϙ ) tC , bordfgt Ξ which we term the fundamental fibre square.Now, in [HM] Hesselholt and Madsen promoted the Grothendieck-Witt spectrum GW scl ( 𝑅, 𝑀 ) into thegenuine fixed points of what they termed the real algebraic K -theory KR scl ( 𝑅, 𝑀 ) , a genuine C -spectrum.We similarly produce a functor KR ∶ Cat p∞ ⟶ Sp gC using the language of spectral Mackey functors, withthe property that the isotropy separation square of KR( C , Ϙ ) is precisely the fundamental fibre square above,so that in particular KR( C , Ϙ ) gC ≃ GW( C , Ϙ ) and KR( C , Ϙ ) 𝜑 C ≃ L( C , Ϙ ); here (−) gC and (−) 𝜑 C ∶ S 𝑝 gC → S 𝑝 denote the genuine and geometric fixed points functors, respectively.Combined with the comparison results of [HS20] this affirms the conjecture of Hesselholt and Madsen,that the geometric fixed points of the real algebraic K -theory spectrum of a discrete ring are a version ofRanicki’s L -theory.As the ultimate expression of periodicity, we then enhance our extension of Karoubi’s periodicity to thefollowing statement in the language of genuine homotopy theory: Theorem D.
The boundary map of the metabolic Poincaré-Verdier sequence provides a canonical equiva-lence
KR( C , Ϙ [1] ) ≃ 𝕊 𝜎 ⊗ KR( C , Ϙ ) . Passing to geometric fixed points recovers the result of Lurie that L( C , Ϙ [1] ) ≃ 𝕊 ⊗ L( C , Ϙ ) whereas theabstract version of Karoubi periodicty, i.e U( C , Ϙ [2] ) ≃ 𝕊 ⊗ V( C , Ϙ ) , and thus in particular the periodicityresults for the classical Grothendieck-Witt spectra of discrete rings, is obtained by considering the normmap from the underlying spectrum to the genuine fixed points. Further applications to rings and parametrised spectra.
We start by specialising the abstract resultsof the previous section to Grothendieck-Witt spectra of (discrete) rings. In the body of the paper, we willderive these for E -ring spectra satisfying appropriate assumptions, but aside from a few comments werefrain from engaging with this generality here.Given a ring 𝑅 , and an invertible 𝑅 -module with involution 𝑀 , we constructed in Paper [I] a sequenceof Poincaré structures Ϙ q 𝑀 = Ϙ ≥ ∞ 𝑀 ⟹ ⋯ ⟹ Ϙ ≥ 𝑚𝑀 ⟹ Ϙ ≥ 𝑚 −1 𝑀 ⟹ ⋯ ⟹ Ϙ ≥ −∞ 𝑀 = Ϙ s 𝑀 on the stable ∞ -category D p ( 𝑅 ) , ultimately coming from the Postnikov filtration of 𝑀 tC . The genuinePoincaré structures from the first section appear as Ϙ gs = Ϙ ≥ 𝑀 , Ϙ ge 𝑀 = Ϙ ≥ 𝑀 and Ϙ gq 𝑀 = Ϙ ≥ 𝑀 . Recall that thesegive rise to the classical symmetric, even and quadratic Grothendieck-Witt spectra of ( 𝑅, 𝑀 ) , whereasessentially by construction Ϙ q 𝑀 and Ϙ s 𝑀 give rise to the classical L -spectra of ( 𝑅, 𝑀 ) . This sequence ofPoincaré structures collapses in to a single one if is invertible in 𝑅 .Now, extending the discussion after the Main Theorem we constructed in Paper [I] equivalences of theform ( D p ( 𝑅 ) , ( Ϙ ≥ 𝑚𝑀 ) [2] ) ≃ ( D p ( 𝑅 ) , Ϙ ≥ 𝑚 +1− 𝑀 ) . and entirely similar results hold for the stable subcategory D f ( 𝑅 ) of D p ( 𝑅 ) generated by 𝑅 [0] . On the L -theory side, the switch between these categories corresponds to the change of decoration from projective tofree and the distinction will be important momentarily. Applying Theorem D we find the following result,which at least for invertible in 𝑅 proves an unpublished conjecture of Hesselholt-Madsen: Corollary E (Genuine Karoubi periodicity) . For a (discrete) ring 𝑅 and an invertible 𝑅 -module 𝑀 withinvolution and 𝑚 ∈ ℤ ∪ {±∞} there are canonical equivalences KR( D p ( 𝑅 ) , Ϙ ≥ 𝑚𝑀 ) ≃ 𝕊 𝜎 ⊗ KR( D p ( 𝑅 ) , Ϙ ≥ 𝑚 −1− 𝑀 ) . In particular, the genuine C -spectra KR( D p ( 𝑅 ) , Ϙ s 𝑀 ) and KR( D p ( 𝑅 ) , Ϙ q 𝑀 ) are (4 − 4 𝜎 ) -periodic and even (2 − 2 𝜎 ) -periodic if 𝑅 has characteristic . The same results hold for D f ( 𝑅 ) in place of D p ( 𝑅 ) . We also find
KR( D p ( 𝑅 ) , Ϙ gq 𝑀 ) ≃ 𝕊 𝜎 ⊗ KR( D p ( 𝑅 ) , Ϙ gs 𝑀 ) for any discrete ring 𝑅 and invertible 𝑅 -module with involution 𝑀 . In a different direction, the (4 − 4 𝜎 ) -or (2 − 2 𝜎 ) -fold periodicity of KR( D p ( 𝑅 ) , Ϙ s 𝑀 ) and KR( D p ( 𝑅 ) , Ϙ q 𝑀 ) in fact holds for any complex oriented or real oriented E -ring 𝑅 , respectively; we will deduce it in thisgenerality in the body of the paper.Let us now turn to the behaviour of Grothendieck-Witt spectra under localisations of rings. As oneapplication of Corollary C we find: Corollary F.
Let 𝑅 be a (discrete) ring, 𝑀 an invertible 𝑅 -module with involution and 𝑓 , 𝑔 ∈ 𝑅 elementsspanning the unit ideal. Then the square GW( D p ( 𝑅 ) , Ϙ ≥ 𝑚𝑀 ) GW( D p 𝑅 ( 𝑅 [ 𝑓 ]) , Ϙ ≥ 𝑚𝑀 [1∕ 𝑓 ] )GW( D p 𝑅 ( 𝑅 [ 𝑔 ]) , Ϙ ≥ 𝑚𝑀 [1∕ 𝑔 ] ) GW( D p 𝑅 ( 𝑅 [ 𝑓𝑔 ]) , Ϙ ≥ 𝑚𝑀 [1∕ 𝑓𝑔 ] ) is cartesian, where for an 𝑅 -algebra 𝑆 the term D p 𝑅 ( 𝑆 ) denotes the full subcategory of D p ( 𝑆 ) spanned bythose complexes 𝐶 such that [ 𝐶 ] ∈ K ( 𝑆 ) lies in the image of K ( 𝑅 ) → K ( 𝑆 ) .A similar result holds for the categories D f in place of D p (even without further decorations). The case 𝑚 = 0 recovers the affine case of Schlichting’s celebrated Mayer-Vietoris principle for Grothendieck-Witt groups of schemes [Sch10b] and the case 𝑚 = 1 , extends these results from symmetric to even andquadratic Grothendieck-Witt groups. Outlook.
We will not consider the Grothendieck-Witt theory of schemes in the present paper, as it worksmore smoothly when considering Karoubi-Grothendieck-Witt spectra, i.e. the variant of Grothendieck-Witttheory that is invariant under idempotent completion, just as non-connective K -spectra are better suitedfor the study of schemes than connective ones; as explained previously it is this variant which Schlichtingconsiders in [Sch10b] as well. We will develop this extension in Paper [IV] and give a proof of Nisnevichdescent in another upcoming paper [CHN].We shall use our main result in the third instalment of this paper series, to deduce dévissage results forthe fibres of localisation maps as in the above square if 𝑚 = 0 , i.e. for symmetric Grothendieck-Witt groups,under the additional assumption 𝑅 is a Dedekind domain. In fact, dévissage statements hold naturally for 𝑚 = −∞ and we transport them to other classical Grothendieck-Witt spectra by a detailed analysis of the L -theory spectra involved.Lastly, we turn to another class of examples of Poincaré ∞ -categories, namely those formed by compactparametrised spectra over a space 𝐵 . The relevance of these examples is already visible in the equivalences A( 𝐵 ) ≃ K(( S 𝑝 ∕ 𝐵 ) 𝜔 ) describing Waldhausen’s K -theory of spaces in the present framework. Given a stable spherical fibration 𝜉 over 𝐵 , there are three important Poincaré structures on ( S 𝑝 ∕ 𝐵 ) 𝜔 , the quadratic, symmetric and visibleone, all of whose underlying duality is the Costenoble-Waner functor 𝐸 ↦ Hom 𝐵 ( 𝐸 ⊠ 𝐸, Δ ! 𝜉 ); here ⊠ is the exterior tensor product, Δ ∶ 𝐵 → 𝐵 × 𝐵 is the diagonal map, and the subscript denotes theleft adjoint functor to its associated pullback. Then from the isotropy separation square of KR(( S 𝑝 ∕ 𝐵 ) 𝜔 , Ϙ 𝑟𝜉 ) with 𝑟 ∈ {q , s , v} we find: ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 11 Corollary G.
There are canonical equivalences
GW((Sp∕ 𝐵 ) 𝜔 , Ϙ 𝑟𝜉 ) ≃ LA 𝑟 ( 𝐵, 𝜉 ) and in particular Ω ∞−1 LA 𝑟 ( 𝐵, 𝜉 ) ≃ | Cob((Sp∕ 𝐵 ) 𝜔 , Ϙ 𝑟𝜉 ) | for 𝑟 ∈ {q , s , v} . Here LA 𝑟 ( 𝐵, 𝜉 ) denotes the spectra constructed by Weiss and Williams (under the names LA ∙ , LA ∙ ,and VLA ) in their pursuit of a direct combination of surgery theory and pseudo-isotopy theory into a di-rect description of the spaces G ( 𝑀 )∕Top( 𝑀 ) for closed manifolds 𝑀 , see [WW14]. This result unitestheir work with the recent approaches to the study of diffeomorphism groups at the hands of Galatius andRandal-Williams [GRW14]. In particular, the second part provides a cycle model for the previously rathermysterious spectra LA 𝑟 ( 𝐵, 𝜉 ) that can be used to give a new construction of Waldhausen’s map ̃ Top( 𝑀 )∕Top( 𝑀 ) ⟶ Wh( 𝑀 ) hC along with a new proof of the index theorems of Weiss and Williams. These results will appear in futurework.In the present paper we only give a small application of the above equivalence in another direction. Weuse computations of Weiss and Williams for 𝐵 =∗ together with the universal properties of GW and L todetermine the automorphism groups of these functors. The result is that 𝜋 Aut(GW) ≅ (C ) and 𝜋 Aut(L) ≅ C the former spanned by −id GW and id − (hyp ◦ f gt) and the latter by −id L . Remark.
During the completion of this work on the one hand Schlichting announced results similar to thecorollaries of our main theorem, and some of the applications we pursue in the third installement of thisseries in [Sch19b], though as far as we are aware no proofs have appeared as of yet. On the other handthe draft [HSV19] contains a construction of the real algebraic K -theory spectrum in somewhat greatergenerality than in the present paper (in particular, not necessarily stable ∞ -categories), with a version ofB part ii) as their main result, albeit using a slightly weaker notion of additivity than the one we use here(resulting in a logically incomparable result).However, as far as we are aware, neither of these systematically relates Grothendieck-Witt theory to L -theory, the main thread of our work. Organisation of the paper.
In the next section we briefly summarise the necessary results of Paper [I],providing in particular a guide to the requisite parts. In §1 we study (co)fibre sequences in
Cat p∞ in detailand introduce additive and localising functors. The analogous results in the setting of stable ∞ -categories,on which our results are based, are well-known but seem difficult to locate coherently in the literature. Wetherefore give a systematic account in Appendix A, without any claim of originality.The real work of the present paper then starts in §2. It contains the definition of the hermitian Q -construction and the algebraic cobordism category and proves Theorem A as 2.4.1 and 2.4.3. In §3 wethen generally analyse the behaviour or additive functors Cat p∞ → S 𝑝 and Cat p∞ → S . This leads to verygeneral versions of Theorem B in 3.3.6, 3.4.5, our Main Theorem in 3.6.7 and Theorem C in 3.7.7. We thenobtain all other results of this introduction as simple consequences in §4, where we specialise the discussionto the Grothendieck-Witt functor.Finally, there is a second appendix which establishes two comparison results to other Grothendieck-Wittspectra, not covered by [HS20]. They are not used elsewhere in the paper. 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 andDN 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. R ECOLLECTION
In the present section we briefly recall the parts of Paper [I] that are most relevant for the considerations ofthe present paper. We first summarise the abstract features of the theory, and then spell out some examples.
Poincaré ∞ -categories and Poincaré objects. Recall from §[I].1.2 that a hermitian structure on a smallstable ∞ -category C is a reduced, quadratic functor Ϙ ∶ C op → S 𝑝 , see Diagram (1) for a characterisation ofsuch functors. A pair ( C , Ϙ ) consisting of this data we call a hermitian ∞ -category. These organise into an ∞ -category Cat h∞ whose morphisms consist of what we term hermitian functors, that is pairs ( 𝑓 , 𝜂 ) where 𝑓 ∶ C → D is an exact functor and 𝜂 ∶ Ϙ ⇒ Φ ◦ 𝑓 op is a natural transformation.To such a hermitian ∞ -category is associated its category of hermitian forms He( C , Ϙ ) , whose objectsconsist of pairs ( 𝑋, 𝑞 ) where 𝑋 ∈ C and 𝑞 is a Ϙ -hermitian form on 𝑋 , i.e. a point in Ω ∞ Ϙ ( 𝑋 ) , see §[I].2.1.Morphisms are maps in C preserving the hermitian forms. The core of the category He( C , Ϙ ) is denoted Fm( C , Ϙ ) and these assemble into functors He ∶ Cat h∞ → Cat ∞ and Fm ∶ Cat h∞ → S . In order to impose a non-degeneracy condition on the forms in
Fm( C , Ϙ ) , one needs a non-degeneracycondition on the hermitian ∞ -category ( C , Ϙ ) itself. To this end recall the classification of quadratic functorsfrom Goodwillie calculus: Any reduced quadratic functor uniquely extends to a cartesian diagram(1) Ϙ ( 𝑋 ) L Ϙ ( 𝑋 )B Ϙ ( 𝑋, 𝑋 ) hC B Ϙ ( 𝑋, 𝑋 ) tC where L Ϙ ∶ C op → S 𝑝 is linear (i.e. exact) and B Ϙ ∶ C op × C op → S 𝑝 is bilinear (i.e. exact in each variable)and symmetric (i.e. comes equipped with a refinement to an element of Fun( C op × C op , S 𝑝 ) hC , with C acting by flipping the input variables), see §[I].1.3. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 13 A hermitian structure Ϙ is called Poincaré if there exists an equivalence D ∶ C op → C such that B Ϙ ( 𝑋, 𝑌 ) ≃ Hom C ( 𝑋, D 𝑌 ) naturally in 𝑋, 𝑌 ∈ C op . By Yoneda’s lemma, such a functor D is uniquely determined if it exists, so werefer to it as D Ϙ . By the symmetry of B Ϙ the functor D Ϙ then automatically satisfies D Ϙ ◦ D op Ϙ ≃ id C . Anyhermitian functor ( 𝐹 , 𝜂 ) ∶ ( C , Ϙ ) → ( D , Φ) between Poincaré ∞ -categories (i.e. hermitian ∞ -categorieswhose hermitian structure is Poincaré) induces a tautological map 𝐹 ◦ D Ϙ ⟹ D Φ ◦ 𝐹 op , see §[I].1.2. We say that ( 𝐹 , 𝜂 ) is a Poincaré functor if this transformation is an equivalence, and Poincaré ∞ -categories together with Poincaré functors form a (non-full) subcategory Cat p∞ of Cat h∞ .Now, if ( C , Ϙ ) is Poincaré, then to any hermitian form ( 𝑋, 𝑞 ) ∈ Fm( C , Ϙ ) there is tautologically associateda 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 ) and we say that ( 𝑋, 𝑞 ) is Poincaré if 𝑞 ♯ is an equivalence. The full subspace of Fm( C , Ϙ ) spanned by thePoincaré forms is denoted by Pn( C , Ϙ ) and provides a functor Pn ∶ Cat p∞ → S , which we suggest to view in analogy with the functor Cr ∶ Cat ex∞ → S taking a stable ∞ -category to itsgroupoid core. Details about this functor are spelled out in §[I].2.1.The simplest example of a Poincaré ∞ -category to keep in mind is C = D p ( 𝑅 ) , where 𝑅 is a discretecommutative ring and D p ( 𝑅 ) is the ∞ -category of perfect complexes over 𝑅 (i.e. finite chain complexesof finitely generated projective 𝑅 -modules), together with the symmetric and quadratic Poincaré structuresgiven by Ϙ q 𝑅 ( 𝑋 ) ≃ hom 𝑅 ( 𝑋 ⊗ 𝕃 𝑅 𝑋, 𝑀 ) hC and Ϙ s 𝑅 ( 𝑋 ) ≃ hom 𝑅 ( 𝑋 ⊗ 𝕃 𝑅 𝑋, 𝑀 ) hC , where hom 𝑅 denotes the mapping spectrum of the category D p ( 𝑅 ) (in other words the spectrum underlyingderived mapping complex ℝ Hom 𝑅 ). In either case the bilinear part and duality are given by B( 𝑋, 𝑌 ) ≃ hom 𝑅 ( 𝑋 ⊗ 𝕃 𝑅 𝑌 , 𝑅 ) and D( 𝑋 ) ≃ ℝ Hom 𝑅 ( 𝑋, 𝑅 ) , which makes both Ϙ s 𝑅 and Ϙ q 𝑅 into Poincaré structures on D p ( 𝑅 ) .We will discuss further examples in detail below. Constructions of Poincaré ∞ -categories. We next collect a few important structural properties of the ∞ -categories Cat h∞ and Cat p∞ . First of all, by the results of §[I].6.1 they are both complete and cocomplete,and the inclusion Cat p∞ → Cat h∞ is conservative, i.e. it detects equivalences among Poincaré ∞ -categories.Furthermore, the forgetful functors Cat p∞ ⟶ Cat h∞ ⟶ Cat ex∞ both possess both adjoints, so preserve both limits and colimits; these are constructed in §[I].7.2 and §[I].7.3.For the right hand functor the adjoints simply equip a stable ∞ -category C with the trivial hermitian struc-ture . For the left hand functor, the left and right adjoints are related by a shift: Denoting the right adjointfunctor by ( C , Ϙ ) ↦ Pair( C , Ϙ ) , the left adjoint is given by ( C , Ϙ ) ↦ Pair( C , Ϙ [−1] ) , where generally Ϙ [ 𝑖 ] denotes the hermitian structure Σ 𝑖 S 𝑝 ◦ Ϙ . We refrain at this place from giving the explicit construction ofcategory Pair( C , Ϙ ) since it is somewhat involved, and we shall not need it here.The following two special cases of this construction will be of great importance. By the above discussionthe left and right adjoint of the composite Cat p∞ → Cat ex∞ agree. They are given by the hyperbolic construc-tion C → Hyp( C ) with underlying category C × C op and Poincaré structure hom C ∶ ( C × C op ) op → S 𝑝 , see§[I].2.2. The associated duality is given by ( 𝑋, 𝑌 ) ↦ ( 𝑌 , 𝑋 ) , and there is a natural equivalence Cr C ≃ Pn Hyp( C ) implemented by 𝑋 ↦ ( 𝑋, 𝑋 ) . We denote by 𝑓 hyp ∶ Hyp( C ) → ( D , Ϙ ′ ) and 𝑓 hyp ∶ ( C , Ϙ ) → Hyp( D ) the Poincaré functors obtained through these adjunctions from a exact functor 𝑓 ∶ C → D .The other important case is the composite of the inclusion Cat p∞ → Cat h∞ with its left adjoint. Thisassigns to a Poincaré ∞ -category ( C , Ϙ ) the metabolic category Met( C , Ϙ ) , whose underlying category is thearrow category Ar( C ) of C and whose Poincaré structure is given by Ϙ met ( 𝑋 → 𝑌 ) ≃ f ib( Ϙ ( 𝑌 ) → Ϙ ( 𝑋 )) , see §[I].2.3. The associated duality is D Ϙ met ( 𝑋 → 𝑌 ) ≃ f ib(D Ϙ ( 𝑌 ) → D Ϙ ( 𝑋 )) ⟶ D Ϙ ( 𝑌 ) . The Poincaré objects in
Met( C , Ϙ ) are best thought of as Poincaré objects with boundary in the Poincaré ∞ -category ( C , Ϙ [−1] ) , which embeds into Met( C , Ϙ ) via 𝑋 ↦ ( 𝑋 → , i.e. as the objects with trivialboundary.From the various adjunction units and counits there then arises a commutative diagram Hyp( C )Hyp( C ) ( C , Ϙ )Met( C , Ϙ ) hypcanlag met in Cat p∞ for every Poincaré ∞ -category; the underlying functors pointing to the right are given by met( 𝑋 → 𝑌 ) = 𝑌 and hyp( 𝑋, 𝑌 ) =
𝑋 ⊕ D Ϙ 𝑌 , whereas the other two are given by extending the source and identity functors 𝑠 ∶ Ar( C ) ⟶ C and id ∶ C ⟶ Ar( C ) using the adjunction properties of Hyp . Regarding the induced maps after applying Pn , one finds that anelement in ( 𝑋, 𝑞 ) ∈ 𝜋 Pn( C , Ϙ ) is in the image of met if it admits a Lagrangian, that is a map 𝑓 ∶ 𝐿 → 𝑋 such that there is an equivalence 𝑓 ∗ 𝑞 ≃ 0 , whose associated nullhomotopy of the composite 𝐿 𝑓 ←←←←←←←→ 𝑋 ≃ D Ϙ 𝑋 D Ϙ 𝑓 ←←←←←←←←←←←←←←←←←←→ D Ϙ 𝐿 makes this sequence into a fibre sequence in C . Similarly, ( 𝑋, 𝑞 ) lies in the image of hyp if there is anequivalence 𝑋 ≃ 𝐿 ⊕ D Ϙ 𝐿 which translates the form 𝑞 into the tautological evaluation form on the target.Thus the categories Hyp( C ) and Met( C , Ϙ ) encode the theory of metabolic and hyperbolic forms in ( C , Ϙ ) and the remainder of the diagram witnesses that any hyperbolic form has a canonical Lagrangian, fromwhich it can be reconstructed.One further property of these constructions that we shall need is that the duality D Ϙ equips the underlying ∞ -category of ( C , Ϙ ) with the structure of a homotopy fixed point in Cat ex∞ under the C -action given bytaking C to C op , or in other words the forgetful functor Cat p∞ → Cat ex∞ is C -equivariant for the trivial actionon the source and the opponing action on the target, see §[I].7.2. As a formal consequence its adjoint Hyp is equivariant as well, and thus the composite
Cat p∞ fgt ←←←←←←←←←←←←→
Cat ex∞ Hyp ←←←←←←←←←←←←←←←←→
Cat p∞ lifts to a functor Hyp ∶ Cat p∞ → (Cat p∞ ) hC = Fun(BC , Cat p∞ ) , the category of (naive) C -objects in Cat p∞ .The action map on Hyp( C , Ϙ ) is given by the composite Hyp( C ) f lip ←←←←←←←←←←←←←←→ Hyp( C op ) Hyp(D Ϙ ) ←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ Hyp( C ) and the functor hyp ∶ Hyp( C ) → ( C , Ϙ ) is invariant under the action on the source.We recall from §[I].5.2 that the category Cat h∞ admits a symmetric monoidal structure making the func-tor f gt ∶ Cat h∞ → Cat ex∞ symmetric monoidal for Lurie’s tensor product of stable ∞ -categories on the ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 15 target. While we do not use the monoidal structure in the present paper, we heavily exploit the follow-ing: The monoidal structure on Cat h∞ is cartesian closed, i.e. Cat h∞ admits internal function objects, andalso both tensors and cotensors over Cat ∞ , see §[I].6.2, §[I].6.4 and §[I].6.3. More explicitly, to hermitian ∞ -categories ( C , Ϙ ) and ( D , Φ) and an ordinary category I there are associated hermitian ∞ -categories Fun ex (( C , Ϙ ) , ( D , Φ)) , ( C , Ϙ ) I and ( C , Ϙ ) I . connected by natural equivalences Fun ex (( C , Ϙ ) I , ( D , Φ)) ≃ Fun ex (( C , Ϙ ) , ( D , Φ)) I ≃ Fun ex (( C , Ϙ ) , ( D , Φ) I ) The underlying categories in the outer cases are given by
Fun ex ( C , D ) and Fun( I , C ) and their hermitian structures nat Φ Ϙ and Ϙ I are given by 𝑓 ⟼ nat( Ϙ , Φ ◦ 𝑓 op ) and 𝑓 ⟼ lim I op Ϙ ◦ 𝑓 op . This results in particular in equivalences
Fm Fun ex (( C , Ϙ ) , ( D , Φ)) ≃ Hom
Cat h∞ (( C , Ϙ ) , ( D , Φ)) , Pn Fun ex (( C , Ϙ ) , ( D , Φ)) ≃ Hom
Cat p∞ (( C , Ϙ ) , ( D , Φ)) and
He(( C , Ϙ ) I ) ≃ Fun( I , He( C , Ϙ )) , though Poincaré objects in ( C , Ϙ ) I are not generally easy to describe. Furthermore, the tensoring construc-tion is unfortunately far less explicit, and as we only need few concrete details let us refrain from spelling itout here; for I a finite poset it is described explicitely in Proposition [I].6.5.8. Finally, we note that neitherthe tensor nor cotensor construction generally preserve Poincaré ∞ -categories, though Lurie exstablishedsufficient criteria which we recorded in §[I].6.6. Examples of Poincaré ∞ -categories. Finally, we discuss the most important examples in detail: Poincaréstructures on module categories and parametrised spectra.We start with the former. Fix therefore an E -algebra 𝐴 over a base E ∞ -ring spectrum 𝑘 and considerits category of finitely presented or compact 𝐴 -module spectra Mod f 𝐴 ⊆ Mod 𝜔𝐴 . For the reader mostlyinterested in the applications to discrete rings we recall that any discrete ring 𝑅 gives rise to such data, viathe Eilenberg-Mac Lane functor H ∶ A 𝑏 ⟶ S 𝑝 , which is lax symmetric monoidal and therefore induces afunctor Ring ⟶ Alg E (Mod H ℤ ) In this way, any discrete ring may be regarded as an E -algebra over H ℤ . There are, furthermore, equiva-lences Mod 𝜔 H 𝑅 ≃ D p ( 𝑅 ) and Mod fH 𝑅 ≃ D f ( 𝑅 ) , where D p ( 𝑅 ) denotes the full subcategory of the derived ∞ -category D ( 𝑅 ) of 𝑅 spanned by the perfectcomplexes, i.e. finite chain complexes of finitely generated projective 𝑅 -modules and D f ( 𝑅 ) is the full sub-category spanned by the finite chain complexes of finite free 𝑅 -modules. In this regime the reader shouldkeep in mind, that terms such as ⊗ H ℤ or Hom H 𝑅 will evaluate to the functors ⊗ 𝕃ℤ and ℝ Hom 𝑅 .Hermitian structures on the categories Mod 𝜔𝐴 and Mod f 𝐴 are generated by 𝐴 -modules with genuine in-volution ( 𝑀, 𝑁, 𝛼 ) ; let us go through these ingredients one by one, compare §[I].3.2. The first entry 𝑀 iswhat we term an 𝐴 -module with (naive) involution: An 𝐴 ⊗ 𝑘 𝐴 -module, equipped with the structure of ahomotopy fixed point in the category Mod 𝐴⊗ 𝑘 𝐴 under the C -action flipping the two factors, see §[I].3.1.In the case of a discrete ring 𝑅 , the simplest examples of such a structure is given by a discrete 𝑅 ⊗ ℤ 𝑅 -module 𝑀 , and a selfmap 𝑀 → 𝑀 , that squares to the identity on 𝑀 and is semilinear for the flip mapof 𝑅 ⊗ ℤ 𝑅 . If 𝑅 is a ring equipped with an anti-involution 𝜎 , then 𝑀 = 𝑅 is a valid choice by using 𝜎 toturn the usual 𝑅 ⊗ ℤ 𝑅 op -module structure on 𝑅 into an 𝑅 ⊗ ℤ 𝑅 -module structure. The involution on 𝑅 can then be chosen as 𝜎 or − 𝜎 (or 𝜖𝜎 for any other central unit 𝜖 with 𝜎 ( 𝜖 ) = 𝜖 −1 ).The additional data of a module with genuine involution consists of an 𝐴 -module spectrum 𝑁 , and an 𝐴 -linear map 𝛼 ∶ 𝑁 → 𝑀 tC ; to make sense of the latter term, note that upon forgetting the 𝐴 ⊗ 𝑘 𝐴 -action,the involution equips 𝑀 with the structure of a (naive) C -spectrum (or even 𝑘 -module spectrum). The spectrum 𝑀 tC then becomes an ( 𝐴 ⊗ 𝑘 𝐴 ) tC -module via the lax monoidality of the Tate construction andfrom here obtains an 𝐴 -module structure on 𝑀 tC by pullback along the Tate diagonal 𝐴 → ( 𝐴 ⊗ 𝑘 𝐴 ) tC ,which is a map of E -ring spectra, see [NS18, Chapter III.1] for an exposition of the Tate diagonal in thepresent language. Let us immediately warn the reader that the Tate diagonal is not generally 𝑘 -linear forthe 𝑘 -module structure on ( 𝐴 ⊗ 𝑘 𝐴 ) tC arising from the unit map 𝑘 → ( 𝑘 ⊗ 𝑘 𝑘 ) tC = 𝑘 tC , as this map isusually different from the Tate-diagonal of 𝑘 (in particular, this is the case for 𝑘 = H ℤ by [NS18, TheoremIII.1.10]).Even if only interested in discrete 𝑅 , one therefore has to leave not only the realm of discrete 𝑅 -modulesto form the Tate construction, but even the realm of derived categories, as no replacement for the Tate di-agonal can exist in that regime.The hermitian structure associated to a module with genuine involution ( 𝑀, 𝑁, 𝛼 ) as described above isgiven by the pullback Ϙ 𝛼𝑀 ( 𝑋 ) hom 𝐴 ( 𝑋, 𝑁 )hom 𝐴⊗ 𝑘 𝐴 ( 𝑋 ⊗ 𝑘 𝑋, 𝑀 ) hC hom 𝐴⊗ 𝑘 𝐴 ( 𝑋 ⊗ 𝑘 𝑋, 𝑀 ) tC hom 𝐴 ( 𝑋, 𝑀 tC ) 𝛼 ∗ ≃ where the C -action on hom 𝐴⊗ 𝑘 𝐴 ( 𝑋 ⊗ 𝑘 𝑋, 𝑀 ) is given by flipping the factors in the source and the involu-tion on 𝑀 . It is a Poincaré structure on Mod 𝜔𝐴 (or on Mod f 𝐴 ) if 𝑀 restricts to an object of Mod 𝜔𝐴 (or Mod f 𝐴 )under either inclusion 𝐴 → 𝐴 ⊗ 𝑘 𝐴 , and furthermore 𝑀 is invertible, i.e. the natural map 𝐴 → hom 𝐴 ( 𝑀, 𝑀 ) is an equivalence. In this case the associated duality is given by 𝑋 ↦ hom 𝐴 ( 𝑋, 𝑀 ) regarded as an 𝐴 -module via the extraneous 𝐴 -module structure on 𝑀 , see again §[I].3.1.With the preliminaries established let us give some concrete examples. We shall restrict to the specialcase of discrete rings here for ease of exposition. So assume given a discrete ring 𝑅 and a discrete invertible 𝑅⊗ ℤ 𝑅 -module 𝑀 with involution, that is finitely generated projective (or stably free, as appropriate) whenregarded as an element of D ( 𝑅 ) via either inclusion of 𝑅 into 𝑅 ⊗ ℤ 𝑅 ; note that invertibility includes thecondition that Ext 𝑖𝑅 ( 𝑀, 𝑀 ) = 0 for all 𝑖 > .Generalising the simple case discussed in the first part, associated to this data are most easily defined thequadratic and symmetric Poincaré structures Ϙ q 𝑀 and Ϙ s 𝑀 given by Ϙ q ( 𝑋 ) = hom 𝑅⊗ 𝕃ℤ 𝑅 ( 𝑋 ⊗ 𝕃ℤ 𝑋, 𝑀 ) hC and Ϙ s ( 𝑋 ) = hom 𝑅⊗ 𝕃ℤ 𝑅 ( 𝑋 ⊗ 𝕃ℤ 𝑋, 𝑀 ) hC which correspond to the modules with genuine involution ( 𝑀, , and ( 𝑀, 𝑀 tC , id) , respectively. Interpolating between these, we have the genuine family of Poincaré structures Ϙ ≥ 𝑖𝑀 corre-sponding to the modules with genuine involution ( 𝑀, 𝜏 ≥ 𝑖 𝑀 tC , 𝜏 ≥ 𝑖 𝑀 tC → 𝑀 tC ) for 𝑖 ∈ ℤ . As alreadydone in the introduction we shall often include the quadratic and symmetric structures via 𝑖 = ±∞ to fa-cilitate uniform statements. These intermediaries are important mostly since they contain the followingexamples: The functors Quad 𝑀 , Ev 𝑀 , and Sym 𝑀 ∶ Proj( 𝑅 ) op ⟶ A 𝑏 assigning to a finitely generated projective module its abelian group of 𝑀 -valued quadratic, even or symmet-ric forms, respectively, admit animations (or non-abelian derived functors in more classical terminology)which we term Ϙ gq 𝑀 , Ϙ ge 𝑀 and Ϙ gs 𝑀 ∶ D p ( 𝑅 ) op ⟶ S 𝑝, respectively. One of the main results of Paper [I] is that there are equivalences Ϙ gq 𝑀 ≃ Ϙ ≥ 𝑀 , Ϙ ge 𝑀 ≃ Ϙ ≥ 𝑀 and Ϙ gs 𝑀 ≃ Ϙ ≥ 𝑀 , ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 17 see §[I].4.2. It is also not difficult to see that no further members of the genuine family arise as animationsof functors Proj( 𝑅 ) → A 𝑏 .Turning to a different kind of example consider the categories S 𝑝 ∕ 𝐵 = Fun( 𝐵, S 𝑝 ) for some 𝐵 ∈ S .Entirely parallel to the discussion above, one can derive hermitian structures on the compact objects of Sp∕ 𝐵 from triples ( 𝑀, 𝑁, 𝛼 ) with 𝑀 ∈ (Sp∕ 𝐵 × 𝐵 ) hC and 𝛼 ∶ 𝑁 → (Δ ∗ 𝑀 ) tC a map in S 𝑝 ∕ 𝐵 , where Δ ∶ 𝐵 → 𝐵 × 𝐵 is the diagonal, §[I].4.4. The most important examples of such functors are the visiblePoincaré structures Ϙ v 𝜉 given by the triples (Δ ! 𝜉, 𝜉, 𝑢 ∶ 𝜉 → (Δ ∗ Δ ! 𝜉 ) tC ) , where 𝜉 ∶ 𝐵 → Pic( 𝕊 ) is some stable spherical fibration over 𝐵 , where Δ ! ∶ S 𝑝 ∕ 𝐵 → S 𝑝 ∕( 𝐵 × 𝐵 ) is theleft adjoint to Δ ∗ and where 𝑢 is the unit of this adjunction (which factors through 𝜉 → (Δ ∗ Δ ! 𝜉 ) hC since Δ is invariant under the C -action on 𝐵 × 𝐵 ). These hermitian structures are automatically Poincaré withassociated duality given by 𝑋 ⟼ hom 𝐵 ( 𝑋, Δ ! 𝜉 ) , the Costenoble-Waner duality functor twisted by 𝜉 .As a common special case, let us finally mention the universal Poincaré structure Ϙ u on S 𝑝 𝜔 = Mod 𝜔 𝕊 from §[I].4.1: It is associated to the triple ( 𝕊 , 𝕊 , 𝕊 → 𝕊 tC ) , with structure map the unit of 𝕊 tC , whichhappens to agree with the Tate diagonal in this special case. The Poincaré ∞ -category ( S 𝑝 𝜔 , Ϙ u ) representsthe functors Pn and Fm , i.e. for every Poincaré ∞ -category ( C , Ϙ ) and every hermitian ∞ -category ( D , Φ) there are equivalences Hom
Cat p∞ ((Sp 𝜔 , Ϙ u ) , ( C , Ϙ )) ≃ Pn( C , Ϙ ) and Hom
Cat h∞ ((Sp 𝜔 , Ϙ u ) , ( D , Φ)) ≃ Fm( D , Φ) natural in the input. 1. P OINCARÉ -V ERDIER SEQUENCES AND ADDITIVE FUNCTORS
In this section we study the analogue of (split) Verdier sequences in the context of Poincaré ∞ -categories,as well as their analogue for idempotent complete Poincaré ∞ -categories, which, following a suggestionof Clausen and Scholze, we call Karoubi sequences . In particular, our terminology differs from that ofBlumberg-Gepner-Tabuada [BGT13]; see Appendix A for a thorough discussion.After developing the example of module ∞ -categories in some detail, we proceed to introduce the no-tions of additive , Verdier-localising and
Karoubi-localising functors
Cat p∞ → E , encoding the preservationof an increasing number of such sequences, or rather, in the general not necessarily stable context, of a mildgeneralisation thereof in the form of certain cartesian and cocartesian squares in Cat p∞ . These three no-tions we introduce correspond loosely to satisfying Waldhausen’s additivity theorem, Quillen’s localisationtheorem and Bass’ strengthening thereof.The notion of an additive functor from Cat p∞ to S 𝑝 is central in our work, since it essentially abstractsthe additivity properties enjoyed by our main subject of interest, the functor GW ∶ Cat p∞ → S 𝑝 (onlyto be defined in Definition 4.2.1); it is the universal such additive functor with a transformation from thefunctor Pn , space of Poincaré forms. Analogously to K -theory, the functor GW turns out to be furthermoreVerdier-localising, justifying as well the study of that notion. Finally, just as non-connective K -theoryrelates to K -theory, the search for a Karoubi-localising approximation of GW will yield in Paper [IV] theKaroubi-Grothendieck-Witt spectrum functor 𝔾𝕎 .In the present section we only give the very basic properties of such functors, as the only immediatelyinteresting examples are the space valued functors Cr and Pn . After §2 introduces more interesting exam-ples, we return to a detailed study of additive functors in §3. The study of Karoubi-localising functors willbe taken up in Paper [IV].1.1. Poincaré-Verdier sequences.
As the basis for our study we require a rather detailed analysis of Verdiersequences in the setup of stable ∞ -categories. Essentially all of the results we need are well-known tothe experts. To keep the exposition brief we have largely collected such statements and their proofs intoAppendix A, the focus of the present section being on incorporating Poincaré structures. A sequence(2) C 𝑓 ←←←←←←←→ D 𝑝 ←←←←←←→ E in Cat ex∞ with vanishing composite is a
Verdier sequence (Definition A.1.1) if it is both a fiber and a cofibersequence in
Cat ex∞ , in which case we refer to 𝑓 as a Verdier inclusion and to 𝑝 as a Verdier projection . Wealso say that (2) is split (Definition A.2.4) if 𝑝 or equivalently 𝑓 admits both adjoints.1.1.1. Definition.
A sequence(3) ( C , Ϙ ) ( 𝑓,𝜂 ) ←←←←←←←←←←←←←←←←←←←→ ( D , Φ) ( 𝑝,𝜗 ) ←←←←←←←←←←←←←←←←←←→ ( E , Ψ) of Poincaré functors with vanishing composite is called a Poincaré-Verdier sequence if it is both a fibersequence and a cofiber sequence in
Cat p∞ , in which case we call ( 𝑓 , 𝜂 ) a Poincaré-Verdier inclusion and ( 𝑝, 𝜗 ) a Poincaré-Verdier projection . We shall say that (3) is split if the underlying Verdier sequence splits.1.1.2.
Remark.
As explained in Remark A.1.2, the (pointwise) condition of the composite vanishing impliesthat sequence (2) extends to a commutative square
C D {0} E 𝑓 𝑝 in an essentially unique manner and the condition that it forms a Verdier sequence amounts to this squarebeing both cartesian and cocartesian in Cat ex∞ . If Ϙ , Φ and Ψ are now Poincaré structures on C , D and E respectively, then, since Ψ(0) ≃ 0 ∈ S 𝑝 , any null functor carries an essentially unique hermitian structure,and this hermitian structure is automatically Poincaré since the duality on E preserves zero objects. Thus,a sequence of Poincaré functors with null composite uniquely extends to a commutative square as aboveof Poincaré ∞ -categories, and the condition of being Poincaré-Verdier is the condition that this square iscartesian and cocartesian in Cat p∞ .1.1.3. Observation.
Since the forgetful and hyperbolic functors are both-sided adjoints to one another, weimmediately find that the underlying sequence C → D → E of a (split) Poincaré-Verdier sequence is a (split)Verdier sequence, and that the hyperbolization of any (split) Verdier sequence is a (split) Poincaré-Verdiersequence. We now proceed to consider Poincaré-Verdier sequences more closely. To begin, recall that the inclusion
Cat p∞ → Cat h∞ preserves both limits and colimits (Proposition [I].6.1.4), and since it is also conservativewe get that it detects limits and colimits. We may hence test if a given sequence of Poincaré ∞ -categoriesis a (co)fibre sequence at the level of Cat h∞ . In addition, the projection Cat h∞ → Cat ex∞ preserves smalllimits and colimits (Lemma [I].6.1.2), and is a bicartesian fibration with backwards transition maps givenby restriction and forward transition maps given by left Kan extensions. This means that limits in
Cat h∞ arecomputed by first taking the limit D of underlying stable ∞ -categories, then pulling back all the quadraticfunctors to D op , and finally calculating the limit of the resulting diagram in the ∞ -category of quadraticfunctors on D op . Similarly, colimits are computed by first computing the colimit D of underlying stable ∞ -categories, then left Kan extending all the quadratic functors to D op , and finally calculating the colimit ofthe resulting diagram in the ∞ -category of quadratic functors on D op . We also note that limits and colimitsin Fun q ( D op , S 𝑝 ) , i.e. of quadratic functors, can be computed in Fun( D op , S 𝑝 ) , see Remark [I].1.1.15.1.1.4. Proposition.
Let (4) ( C , Ϙ ) ( 𝑓,𝜂 ) ←←←←←←←←←←←←←←←←←←←→ ( D , Φ) ( 𝑝,𝜗 ) ←←←←←←←←←←←←←←←←←←→ ( E , Ψ) be a sequence in Cat p∞ with vanishing composite. Then the following holds:i) The sequence (4) is a fiber sequence in Cat p∞ if and only if its image in Cat ex∞ is a fiber sequence and 𝜂 ∶ Ϙ → 𝑓 ∗ Φ is an equivalence.ii) The sequence (4) is a cofiber sequence in Cat p∞ if and only if its image in Cat ex∞ is a cofiber sequenceand 𝜗 ∶ Φ → 𝑝 ∗ Ψ exhibits Ψ ∶ E op → S 𝑝 as the left Kan extension of Φ along 𝑝 op . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 19 iii) It is a Poincaré-Verdier sequence if and only if its image in Cat ex∞ is a Verdier sequence, and thePoincaré structures on C and E are obtained from that of D by pullback and left Kan extension, re-spectively.Proof. Specializing the preceding discussion to the case of squares with one corner the zero Poincaré ∞ -category gives that (4) is a fiber sequence in Cat p∞ if and only if its image in Cat ex∞ is a fiber sequence and Ϙ → 𝑓 ∗ Φ → 𝑓 ∗ 𝑝 ∗ Ψ is a fiber sequence in Fun( C op , S 𝑝 ) , which, since 𝑓 ∗ 𝑝 ∗ Ψ ≃ 0 , just means that the map Ϙ → 𝑓 ∗ Φ is an equivalence. This proves i).Similarly, (4) is a cofiber sequence in Cat p∞ if and only if its image in Cat ex∞ is a cofiber sequence and 𝑝 ! 𝑓 ! Ϙ → 𝑝 ! Φ → Ψ is a cofiber sequence of quadratic functors, which, since 𝑝 ! 𝑓 ! Ϙ ′′ ≃ 0 just means that themap 𝑝 ! Φ → Ψ is an equivalence, so that we get ii). (cid:3) Combining this Proposition with Proposition A.1.9 which states that an exact functor C → D betweenstable ∞ -categories is a Verdier inclusion if and only if it is fully faithful and its essential image is closedunder retracts in D , we get:1.1.5. Corollary.
A Poincaré functor ( 𝑓 , 𝜂 ) ∶ ( C , Ϙ ) → ( D , Φ) is a Poincaré-Verdier inclusion if and only if 𝑓 is fully-faithful, its essential image is closed under retracts, and the map 𝜂 ∶ Ϙ → 𝑓 ∗ Φ is an equivalence. To state the analogous corollary concerning Poincaré-Verdier projections, let us first stress that we takethe localisation D [ 𝑊 −1 ] of an ∞ -category D at a set 𝑊 of morphisms to mean the initial ∞ -categoryunder D in which the morphisms from 𝑊 become invertible. Beware that we differ in our use of theterm localisation from Lurie, who requires the existence of adjoints to the functor D → D [ 𝑊 −1 ] . SeeLemma A.2.3 for the precise relationship between the two notions.Given an exact functor C → D , the Verdier quotient D ∕ C of D by C is the localisation of D with respectto the collection of maps whose fiber is in smallest stable subcategory containing the essential image of 𝑓 (see Definition A.1.3).By [NS18, Theorem I.3.3(i)] D ∕ C is again a stable ∞ -category and the tautological functor D → D ∕ C is exact. For a further discussion of Verdier quotients, we refer the reader to §A.1. The main output of thediscussion there is Proposition A.1.6, which shows that an exact functor is a Verdier projection if and onlyif it is a localisation. Combining this with Proposition 1.1.4, we get:1.1.6. Corollary.
A Poincaré functor ( 𝑝, 𝜗 ) ∶ ( D , Φ) → ( E , Ψ) is a Poincaré-Verdier projection if and onlyif 𝑝 ∶ D → E is a localisation and Φ → 𝑝 ∗ Ψ exhibits Ψ as the left Kan extension of Φ along 𝑝 . Example. If 𝑝 ∶ D → E is a Verdier projection and Φ is a Poincaré structure on D then the hermitianstructure 𝑝 ! Φ on E and the tautological hermitian refinement of 𝑝 are Poincaré if and only if ker( 𝑝 ) isinvariant under the duality, and in this case (ker( 𝑝 ) , Φ) ⟶ ( D , Φ) ⟶ ( E , 𝑝 ! Φ) . is a Poincaré-Verdier sequence.Indeed, if 𝑝 ! Φ and 𝑝 are Poincaré, then it is immediate that ker( 𝑝 ) is closed under the duality. Converselyif ker( 𝑝 ) is closed under the duality, then, since the forgetful functor Cat p∞ → Cat ex∞ preserves colimits, thecofiber of the inclusion (ker( 𝑝 ) , Φ) → ( D , Φ) in Cat p∞ must be equivalent to a Poincaré ∞ -category of theform ( E , Ψ) for some Poincaré structure on E equipped with a Poincaré functor ( 𝑝, 𝜗 ) ∶ ( D , Φ) → ( E , Ψ) .The latter is then a Poincaré-Verdier projection by construction and by Proposition 1.1.4 ii) the naturaltransformation 𝑝 ! Φ → Ψ determined by 𝜗 must be an equivalence, and so the desired properties of 𝑝 ! Φ follow.1.1.8. Remark.
The left Kan extension of a functor Ϙ ∶ C op → S 𝑝 along (the opposite of) an exact functor 𝑝 ∶ C → D is given by 𝑔 ∗ Ϙ (along with the transformation Ϙ → 𝑝 ∗ 𝑔 ∗ Ϙ induced by the co-unit), whenever 𝑝 admits a left adjoint 𝑔 .Even if this is not the case, however, the left Kan extension 𝑝 ! Ϙ can always be computed using thefollowing trick (cf. Lemma [I].1.4.1). Consider the commutative square C op Ind( C op ) D op Ind( D op ) 𝑝 op Ind( 𝑝 op ) Since 𝑝 is exact the functor Ind( 𝑝 op ) ∶ Ind( C op ) → Ind( D op ) preserves all colimits and as its target ispresentable it admits a right adjoint ̃𝑔 ∶ Ind( D op ) → Ind( C op ) and the left Kan extension 𝑝 ! Ϙ ∶ D → S 𝑝 isgiven by the composite D op ⟶ Ind( D op ) ̃𝑔 ←←←←←←→ Ind( C op ) Ind( Ϙ ) ←←←←←←←←←←←←←←←←←←←←←←←→ Ind( S 𝑝 ) colim ←←←←←←←←←←←←←←←←←←←←→ S 𝑝. Since D op → Ind( D op ) is fully faithful 𝑝 ! Ϙ is the restriction of left Kan extension of Ϙ to Ind( D op ) . Bycommutativity of the above square this is equivalent to the left Kan extension along Ind( 𝑝 op ) of the left Kanextension ̃ Ϙ ∶ Ind( C op ) → S 𝑝 of Ϙ to Ind( C op ) , which in turn is given explicitly as the composite ̃ Ϙ ∶ Ind( C op ) Ind( Ϙ ) ←←←←←←←←←←←←←←←←←←←←←←←→ Ind( S 𝑝 ) colim ←←←←←←←←←←←←←←←←←←←←→ S 𝑝. Finally, left Kan extensions along
Ind( 𝑝 op ) are given by restriction along ̃𝑔 by adjunction, which results inthe claimed formula.By [NS18, Theorem I.3.3] the composite D op → Ind( C op ) takes 𝑝 ( 𝑐 ) to colim 𝑥 ∈( 𝑘𝑒𝑟 ( 𝑝 ) 𝑐 ∕ ) op f ib( 𝑐 → 𝑥 ) ,where the fibre is formed in C (as opposed to C op ). Ultimately the above procedure therefore results in theformula ( 𝑝 ! Ϙ )( 𝑝 ( 𝑑 )) ≃ colim 𝑐 ∈(ker( 𝑝 ) 𝑑 ∕ ) op Ϙ (f ib( 𝑑 → 𝑐 )) for the left Kan extension of Ϙ .1.2. Split Poincaré-Verdier sequences and Poincaré recollements.
We turn to split Poincaré-Verdiersequences, which are by definition Poincaré-Verdier sequences in which the underlying Verdier sequence issplit. Let us therefore mention from Lemma A.2.5 that a sequence(5) C 𝑓 ←←←←←←←→ D 𝑝 ←←←←←←→ E in Cat ex∞ with vanishing composite is a split Verdier sequence if and only if it is a fiber sequence and 𝑝 admitsfully faithful left and right adjoints, if and only if it is a cofiber sequence and 𝑓 is fully faithful and admitsleft and right adjoints. Furthermore, this notion is equivalent to that of a stable recollement.In the context of Poincaré ∞ -categories, one of the adjoints in fact, implies the existence of the others:1.2.1. Observation.
The underlying functor 𝑝 of a Poincaré functor admits a left adjoint if and only if itadmits a right adjoint. For a left or right adjoint to 𝑝 gives a right or left adjoint to 𝑝 op , respectively, but 𝑝 and 𝑝 op are naturallyequivalent by means of the dualities in source and target.With this at hand, we derive the following criterion to recognize split Poincaré-Verdier sequences.1.2.2. Proposition.
Let (6) ( C , Ϙ ) ( 𝑓,𝜂 ) ←←←←←←←←←←←←←←←←←←←→ ( D , Φ) ( 𝑝,𝜗 ) ←←←←←←←←←←←←←←←←←←→ ( E , Ψ) be a sequence in Cat p∞ with vanishing composite. Then the following holds:i) Suppose that (6) is a fiber sequence in Cat p∞ . Then (6) is a split Poincaré-Verdier sequence if and onlyif 𝑝 admits a fully faithful left adjoint 𝑔 and the transformation 𝑔 ∗ Φ 𝑔 ∗ 𝜗 ⟹ 𝑔 ∗ 𝑝 ∗ Ψ 𝑢 ∗ ⟹ Ψ is an equivalence, where 𝑢 ∶ id C ⇒ 𝑝𝑔 denotes an adjunction unit.ii) Suppose that (6) is a cofiber sequence in Cat p∞ . Then (6) is a split Poincaré-Verdier sequence if andonly if 𝑓 is fully faithful, 𝜂 ∶ Ϙ → 𝑓 ∗ Φ is an equivalence, and 𝑓 admits a right adjoint.Proof. Assume that (6) is a fiber sequence in
Cat p∞ , hence its image in Cat ex∞ is a fiber sequence as well. Bythe previous observation, the existence of a left adjoint to 𝑝 implies that of a right adjoint, so the underlyingsequence of stable ∞ -categories is a split Verdier-sequence if and only if 𝑝 admits a fully faithful leftadjoint 𝑔 ∶ E → D . In this case it follows from Remark 1.1.8 that 𝑔 ∗ Φ is a left Kan extension of Φ and thetransformation from the statement is the extension of 𝜗 . Thus Ψ is a left Kan extension of Φ if and only ifit is an equivalence, which gives the claim by Proposition 1.1.4.The second item is immediate from Observation 1.2.1 and Proposition 1.1.4 i). (cid:3) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 21 Corollary. i) A Poincaré functor ( 𝑓 , 𝜂 ) ∶ ( C , Ϙ ) → ( D , Φ) is a split Poincaré-Verdier inclusion if and only if 𝑓 isfully faithful, admits a right adjoint, and the map 𝜂 ∶ Ϙ → 𝑓 ∗ Φ is an equivalence.ii) A Poincaré functor ( 𝑝, 𝜗 ) ∶ ( D , Φ) → ( E , Ψ) is a split Poincaré-Verdier projection if and only if 𝑝 admits a fully faithful left adjoint 𝑔 and the composite transformation 𝑔 ∗ Φ 𝑔 ∗ 𝜗 ⟹ 𝑔 ∗ 𝑝 ∗ Ψ 𝑢 ∗ ⟹ Ψ is anequivalence. Remark.
By means of the equivalence 𝑔 ∗ Φ ≃ Ψ the left adjoint 𝑔 to a Poincaré-Verdier projection 𝑝 ∶ ( D , Φ) → ( E , Ψ) automatically becomes a hermitian functor ( E , Ψ) → ( D , Φ) (which is usually notPoincaré). One readily checks that the unit gives an equivalence of hermitian functors id ( E , Ψ) ⇒ 𝑝𝑔 , making 𝑔 a section of 𝑝 in Cat h∞ .In fact, granting that the ∞ -categories He(Fun ex (( D , Φ) , ( E , Ψ))) provide a
Cat ∞ -enrichment to Cat h∞ (afact we will neither prove nor even make precise here), the adjunction between 𝑔 and 𝑝 is an enriched one,i.e. its unit id E ⇒ 𝑔𝑝 and counit 𝑝𝑔 ⇒ id D canonically promote to objects in He(Fun ex (( E , Ψ) , ( E , Ψ))) and
He(Fun ex (( D , Φ) , ( D , Φ))) , such that the triangle identities hold in these ∞ -categories.Conversely, the existence of such an enriched left adjoint to 𝑝 , whose unit is an equivalence, is readilychecked to amount precisely to the conditions of Corollary 1.2.3 ii).Similarly, the existence of an enriched right adjoint with counit an equivalence, boils down to preciselythe conditions in i) above, and therefore detects split Poincaré-Verdier inclusions; in particular, the counitalways provides the right adjoint to a Poincaré-Verdier inclusion with a hermitian structure (which is againusually not Poincaré).We warn the reader that the analogous statements involving the right adjoint to a Poincaré-Verdier projec-tion and the left adjoint to a Poincaré-Verdier inclusion fail entirely; for instance in the metabolic Poincaré-Verdier sequence of Example 1.2.5 below, the only hermitian refinement of the right adjoint to the projectionis null, and so certainly does not give rise to a splitting of 𝑝 .The following is the most important example of a split Poincaré-Verdier sequence. It is in fact universalby Theorem 1.2.9 below and will be fundamental to several results we prove:1.2.5. Example.
For any Poincaré ∞ -category ( C , Ϙ ) the sequence ( C , Ϙ [−1] ) ⟶ Met( C , Ϙ ) met ←←←←←←←←←←←←←←→ ( C , Ϙ ) is a split Poincaré-Verdier sequence, the metabolic fibre sequence ; the left hand Poincaré functor is givenby sending 𝑥 to 𝑥 → , together with the identification Ω Ϙ ( 𝑋 ) ≃ f ib( Ϙ (0) → Ϙ ( 𝑥 )) . Proof.
The underlying sequence of stable ∞ -categories, described in detail in Proposition A.2.11, is a splitVerdier sequence. The sequence is a fibre sequence in Cat p∞ by Proposition 1.1.4 i). To see that it is a splitPoincaré-Verdier sequence apply Proposition 1.2.2 using the fully faithful left adjoint to met given by theexact functor 𝑔 ∶ C → Met( C ) sending 𝑥 to → 𝑥 . (cid:3) In the remainder of this section, we provide an analogue of the classification of split Verdier-projections,i.e. that they arise as pullbacks of the target functor 𝑡 ∶ Ar( C ) → C . The role of this universal split Verdierprojection is played by the metabolic Poincaré-Verdier sequence above. To this end we first record:1.2.6. Corollary.
A pullback of a split Poincaré-Verdier projection is again a split Poincaré-Verdier pro-jection.Proof.
From Corollary A.2.7 we know that the underlying functor of the pullback is again a split Verdierprojection. Thus it remains to analyse the Poincaré structures, where the claim is a straight-forward conse-quence of Corollary 1.2.3. (cid:3)
Now, recall that for a split Verdier sequence C 𝑓 ←←←←←←←→ D 𝑝 ←←←←←←→ E with adjoints 𝑔 ⊣ 𝑓 ⊣ 𝑔 ′ and 𝑞 ⊣ 𝑝 ⊣ 𝑞 ′ , the (co)units fit into fibre sequences 𝑓 𝑔 ′ ⟹ id D ⟹ 𝑞 ′ 𝑝 and 𝑞𝑝 ⟹ id D ⟹ 𝑓 𝑔, see Lemma A.2.5. Furthermore, there is a canonical equivalence 𝑔𝑞 ′ ≃ Σ C 𝑔 ′ 𝑞 and denoting this functor 𝑐 ∶ E → C there results a cartesian square(7) D Ar( C ) E C , 𝑔 → 𝑐𝑝𝑝 t 𝑐 cf. Proposition A.2.11. We now set out to show that this diagram canonically upgrades to a pullback in Cat p∞ , when extracted from a Poincaré-Verdier sequence ( C , Ϙ ) 𝑓 ←←←←←←←→ ( D , Φ) 𝑝 ←←←←←←→ ( E , Ψ) . We need:1.2.7.
Lemma.
For a split Verdier sequence C 𝑓 ←←←←←←←→ D 𝑝 ←←←←←←→ E and a hermitian structure Φ on D such that B Φ ( 𝑞 ( 𝑒 ) , 𝑓 ( 𝑐 )) ≃ 0 for every 𝑐 ∈ C and 𝑒 ∈ E , the fibre sequence 𝑞𝑝 ( 𝑑 ) ⟶ 𝑑 ⟶ 𝑓 𝑔 ( 𝑑 ) induces a fibre sequence Φ( 𝑓 𝑔 ( 𝑑 )) ⟶ Φ( 𝑑 ) ⟶ Φ( 𝑞𝑝 ( 𝑑 )) of spectra. The assumption of the lemma is satisfied for all Poincaré-Verdier sequences ( C , Ϙ ) 𝑓 ←←←←←←←→ ( D , Φ) 𝑝 ←←←←←←→ ( E , Ψ) since then B Φ ( 𝑞 ( 𝑒 ) , 𝑓 ( 𝑐 )) ≃ Hom D ( 𝑞 ( 𝑒 ) , D Φ 𝑓 ( 𝑐 )) ≃ Hom D ( 𝑞 ( 𝑒 ) , 𝑓 (D Ϙ 𝑐 )) ≃ Hom E ( 𝑒, 𝑝𝑓 (D Ϙ 𝑐 )) ≃ 0 . Proof.
From Example [I].1.1.21 we find the fibre of Φ( 𝑑 ) → Φ( 𝑞𝑝 ( 𝑑 )) equivalent to the total fibre of thediagram Φ( 𝑑 ) Φ( 𝑞𝑝 ( 𝑑 ))B Φ ( 𝑞𝑝 ( 𝑑 ) , 𝑑 ) B Φ ( 𝑞𝑝 ( 𝑑 ) , 𝑞𝑝 ( 𝑑 )) . The fibre of the lower horizontal map is B Φ ( 𝑞𝑝 ( 𝑑 ) , 𝑓 𝑔 ( 𝑑 )) which vanishes by assumption. (cid:3) We will next equip the horizontal functors of (7) with hermitian structures.1.2.8.
Construction.
Given a split Verdier sequence C 𝑓 ←←←←←←←→ D 𝑝 ←←←←←←→ E and a hermitian structure Φ on D suchthat B Φ ( 𝑞 ( 𝑒 ) , 𝑓 ( 𝑐 )) ≃ 0 , denote by Ϙ its restriction to C and by Ψ its left Kan extension to E . We thus find Φ( 𝑓 𝑔 ( 𝑑 )) ≃ Ϙ ( 𝑔 ( 𝑑 )) and Φ( 𝑞𝑝 ( 𝑑 )) ≃ Ψ( 𝑝 ( 𝑑 )) so that the fibre sequence of lemma 1.2.7 gives a natural equivalence Φ ≃ f ib ( 𝑝 ∗ Ψ → 𝑔 ∗ Ϙ [1] ) . Applying the unit transformation 𝑢 ∶ id D → 𝑞 ′ 𝑝 we obtain a commutative diagram(8) Φ( 𝑞 ′ 𝑝 ( 𝑑 )) Ψ( 𝑝𝑞 ′ 𝑝 ( 𝑑 )) Ϙ [1] ( 𝑔𝑞 ′ 𝑝 ( 𝑑 ))Φ( 𝑑 ) Ψ( 𝑝 ( 𝑑 )) Ϙ [1] ( 𝑔 ( 𝑑 )) whose rows are fibre sequences. By the triangle identities the unit 𝑝 ( 𝑑 ) → 𝑝𝑞 ′ 𝑝 ( 𝑑 ) is an equivalence, sinceit is a one-sided inverse to the counit, which is an equivalence as 𝑞 ′ is fully faithful. Thus the middle verticalarrow in (8) is an equivalence.It follows that the natural transformation 𝑝 ∗ Ψ → 𝑔 ∗ Ϙ [1] factors naturally through the maps ( 𝑔𝑞 ′ 𝑝 ) ∗ Ϙ [1] → 𝑔 ∗ Ϙ [1] induced by the unit of 𝑝 ⊢ 𝑞 ′ . But since 𝑝 op is a localisation this factorisation Ψ ◦ 𝑝 op = 𝑝 ∗ Ψ ⟶ ( 𝑔𝑞 ′ 𝑝 ) ∗ Ϙ [1] = ( ( 𝑔𝑞 ′ ) ∗ Ϙ [1] ) ◦ 𝑝 op ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 23 can be regarded as a natural transformation 𝜂 ∶ Ψ → ( 𝑔𝑞 ′ ) ∗ Ϙ [1] providing the desired hermitian structure to the functor 𝑐 = 𝑔𝑞 ′ ∶ E → C .The diagram (8) also provides an equivalence cof [ ( 𝑞 ′ 𝑝 ) ∗ Φ ⇒ Φ ] ≃ f ib [ ( 𝑔𝑞 ′ 𝑝 ) ∗ Ϙ [1] ⇒ 𝑔 ∗ Ϙ [1] ] , so in particular, a natural transformation 𝜉 ∶ Φ → ( 𝑔 → 𝑐𝑝 ) ∗ ( Ϙ [1] ) met . Furthermore, the diagram(9)
Φ ( 𝑔 → 𝑐𝑝 ) ∗ ( Ϙ [1] ) met 𝑝 ∗ Ψ ( 𝑐𝑝 ) ∗ Ϙ [1] 𝜉 met 𝜂 commutes by construction.In total, we obtain a commutative diagram ( D , Φ) Met( C , Ϙ [1] )( E , Ψ) ( C , Ϙ [1] ) ( 𝑔 → 𝑐𝑝,𝜉 ) 𝑝 met( 𝑐,𝜂 ) in Cat h∞ . At the level of underlying stable ∞ -categories it is cartesian by Proposition A.2.11. Furthermore,the diagram (9) is also cartesian: By (8) both vertical cofibres are given by 𝑔 ∗ Ϙ [1] , connected by the identity.We conclude that the diagram above is cartesian in Cat h∞ .The following is then the main result of the present section:1.2.9. Theorem.
The commutative square ( D , Φ) Met( C , Ϙ [1] )( E , Ψ) ( C , Ϙ [1] ) ( 𝑔 → 𝑐𝑝,𝜉 ) 𝑝 met( 𝑐,𝜂 ) is a cartesian square in Cat p∞ for every split Poincaré-Verdier sequence ( C , Ϙ ) 𝑓 ←←←←←←←→ ( D , Φ) 𝑝 ←←←←←←→ ( E , Ψ) . Proof of Theorem 1.2.9.
As limits in
Cat p∞ are detected in Cat h∞ by Proposition [I].6.1.4 it only remains toshow that the horizontal arrows are Poincaré functors, i.e. that they preserve the dualities. It suffices to treatthe top arrow, since the lower one is obtained by forming cofibres (with respect to the canonical maps from ( C , Ϙ ) ), and Cat p∞ is closed under colimits in Cat h∞ by Proposition [I].6.1.4. Recall then that generally D Ϙ met ( 𝑓 ∶ 𝑥 → 𝑦 ) ≃ [ D Ϙ cof ( 𝑓 ) → D Ϙ 𝑦 ] , whence it remains to check that the maps 𝑔 (D Φ 𝑑 ) ⟶ D Ϙ [1] cof( 𝑔 ( 𝑑 ) → 𝑐𝑝 ( 𝑑 )) and 𝑐𝑝 (D Φ 𝑑 ) ⟶ D Ϙ [1] ( 𝑐𝑝 ( 𝑑 )) induced by 𝜉 are equivalences. But through the fibre sequence 𝑓 𝑔 ′ ⇒ id D ⇒ 𝑞 ′ 𝑝 the target of the left handmap becomes D Ϙ 𝑔𝑓 𝑔 ′ ( 𝑑 ) ≃ 𝑔 ′ 𝑓 𝑔 (D Φ 𝑑 ) , and unwinding definitions, the map induced by 𝜉 is given by the unit of 𝑓 ⊢ 𝑔 ′ , which is an equivalencesince 𝑓 is fully faithful. Similarly, the target of the second map is given by Σ C D Ϙ 𝑔𝑞 ′ 𝑝 ( 𝑑 ) ≃ Σ C 𝑔 ′ 𝑞𝑝 (D Φ 𝑑 ) and the map in question unwinds to an instance of the natural equivalence 𝑔𝑞 ′ ⇒ Σ C 𝑔 ′ 𝑞 constructed beforeProposition A.2.11. (cid:3) Remark.
Using the identification 𝑐 ≃ Σ C 𝑔 ′ 𝑞 a lengthy diagram chase shows that the composite Ψ 𝜂 ←←←←←←→ 𝑐 ∗ Ϙ [1] ≃ Σ S 𝑝 ◦ ( 𝑔 ′ 𝑞 ) ∗ Ϙ ◦ Ω C op ⟶ ( 𝑔 ′ 𝑞 ) ∗ Ϙ is given by the composite of the two canonical hermitian structures carried by the functors 𝑔 ′ and 𝑞 , seeRemark 1.2.4.The uniqueness of the classifying map in Theorem 1.2.9 is implied by the following hermitian analogueof Proposition A.2.13:1.2.11. Proposition.
Given a split Verdier sequence C → D → E and a hermitian structure Φ on D suchthat B Φ ( 𝑞 ( 𝑒 ) , 𝑓 ( 𝑐 )) ≃ 0 for all 𝑐 ∈ C and 𝑒 ∈ E . Then for every hermitian ∞ -category ( C ′ , Ϙ ′ ) thefull subcategory of Fun ex (( D , Φ) , Met( C ′ , Ϙ ′[1] )) spanned by the pairs ( 𝐹 , 𝜂 ) that give rise to adjointablesquares D Ar( C ′ ) E C ′ 𝐹 𝑡𝐹 on underlying ∞ -categories is equivalent to Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) as a hermitian ∞ -category via restrictionto horizontal fibres, where Ϙ denotes the restriction of Φ to C . Here, adjointability refers to the diagrams formed by passing to vertical left or right adjoints commuting,see [Lur09a, §7.3.1] for a detailed discussion of such squares.Given Proposition A.2.13 one might expect a hermitian version of adjointability to appear in the presentstatement; this is simply implied by the adjointability at the level of underlying categories, essentially sincea morphism in
Fun h (( C , Ϙ ) , ( D , Φ)) is invertible if and only if its image in
Fun ex ( C , D ) is.1.2.12. Corollary.
The horizontal maps in Theorem 1.2.9 are determined up to contractible choice by yield-ing a pullback on underlying stable ∞ -categories and inducing the identity functor on the vertical fibre ( C , Ϙ ) . Put differently met ∶ Met( C , Ϙ [1] ) → ( C , Ϙ [1] ) is the universal Poincaré-Verdier projection with fibre ( C , Ϙ ) . Proof.
Note first that the lower horizontal map in Theorem 1.2.9 is uniquely determined by the upper onethrough the universal property of Poincaré-Verdier quotients. Thus to apply Proposition 1.2.11 it only re-mains to note that cartesian squares with vertical Verdier projections are adjointable. This is easy to checkdirectly and also contained in Proposition A.3.15. (cid:3)
Proof of Proposition 1.2.11.
On underlying ∞ -categories the restriction functor is an equivalence by Propo-sition A.2.13. It therefore suffices to show that the restriction map nat ( Φ , Ϙ ′[1]met ◦ 𝐹 op ) ⟶ nat ( Ϙ , Ϙ ′ ◦ 𝐹 op ) is an equivalence of spectra for every 𝐹 ∶ D → Ar( C ′ ) . To see this, note that adjointability naturallyidentifies 𝐹 with the functor taking 𝑑 to the arrow 𝐺𝑔 ( 𝑑 ) → 𝐺𝑐𝑝 ( 𝑑 ) , where 𝐺 ∶ C → C ′ is the functorinduced by 𝐹 on vertical fibres. Since therefore Ϙ ′[1]met 𝐹 ( 𝑑 ) ≃ f ib ( Ϙ [1] ( 𝐺𝑐𝑝 ( 𝑑 )) → Ϙ [1] ( 𝐺𝑔 ( 𝑑 )) ) , the source of the map in question is equivalent to the fibre of nat ( Φ , Ϙ ′[1] ◦ ( 𝐺𝑐𝑝 ) op ) ⟶ nat ( Φ , Ϙ ′[1] ◦ ( 𝐺𝑔 ) op ) . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 25 Writing 𝑐𝑝 = cof ( 𝑔 ′ ⇒ 𝑔 ) we can use Example [I].1.1.21 to express Ϙ ′[1] 𝐺𝑐𝑝 ( 𝑑 ) op as the total fibre of Ϙ ′[1] 𝐺𝑔 ( 𝑑 ) Ϙ ′[1] 𝐺𝑔 ′ ( 𝑑 )B Ϙ ′[1] ( 𝐺𝑔 ( 𝑑 ) , 𝐺𝑔 ′ ( 𝑑 )) B Ϙ ′[1] ( 𝐺𝑔 ′ ( 𝑑 ) , 𝐺𝑔 ′ ( 𝑑 )) . 𝑡 This results in a cartesian square nat ( Φ , Ϙ ′[1]met ◦ 𝐹 op ) nat ( Φ , Ϙ ′ ◦ ( 𝐺𝑔 ′ ) op ) nat ( Φ , B Ϙ ′[1] ◦ ( 𝐺𝑔, 𝐺𝑔 ′ ) op ◦ Δ D op ) nat ( Φ , B Ϙ ′[1] ◦ ( 𝐺𝑔 ′ , 𝐺𝑔 ′ ) op ◦ Δ D op ) . Now by adjunction the top right corner is equivalent to nat ( Φ ◦ 𝑓 op , Ϙ ′ ◦ 𝐺 op ) ≃ nat ( Ϙ , Ϙ ′ ◦ 𝐺 op ) and unwinding definitions shows that this identifies the top horizontal map with the restriction in question.We there have to show that the lower horizontal map is an equivalence. By Lemma [I].1.1.7 this mapidentifies with nat ( B Φ , B Ϙ ′ ◦ ( 𝐺𝑔, 𝐺𝑔 ′ ) op ) ⟶ nat ( B Φ , B Ϙ ′ ◦ ( 𝐺𝑔 ′ , 𝐺𝑔 ′ ) op ) whose fibre is nat(B Φ , B Ϙ ′ ◦ ( 𝐺𝑐𝑝, 𝐺𝑔 ′ ) op ) , which we will show vanishes. Separating the variables using Fun( D op × D op , S 𝑝 ) ≃ Fun( D op , Fun( D op , S 𝑝 )) yields equivalences nat ( B Φ , B Ϙ ′ ◦ ( 𝐺𝑐𝑝, 𝐺𝑔 ′ ) op ) ≃ nat ( B Φ ◦ (id , 𝑓 ) op , B Ϙ ′ ◦ ( 𝐺𝑐𝑝, 𝐺 ) op ) ≃ nat ( (( 𝑐𝑝 ) op × id C op ) ! (B Φ ◦ (id , 𝑓 ) op ) , B Ϙ ′ ◦ ( 𝐺, 𝐺 ) op ) by adjunction. We now claim that already ( 𝑝 op × id C op ) ! (B Φ ◦ (id , 𝑓 ) op ≃ 0 : The left Kan extension is ob-tained by pullback along the right adjoint ( 𝑞, id C ) op of ( 𝑝, id C ) op and we precisely assumed that B Φ ◦ ( 𝑞, 𝑓 ) op ≃0 . (cid:3) Poincaré-Karoubi sequences.
In this section we study Poincaré-Karoubi sequences, the analogues ofPoincaré-Verdier sequences in the setting of idempotent complete Poincaré ∞ -categories. On the one hand,these are important in their own right when considering the hermitian analogue of non-connective K -theoryin §[IV].2.2, on the other it is often easier to establish Poincaré-Verdier sequences in a two-step process: Firstone constructs a Poincaré-Karoubi sequence using the Thomason-Neeman localisation theorem A.3.11,or in modern guise, the equivalence between small stable ∞ -categories, and compactly generated stable ∞ -categories, and then in a second step isolates subcategories forming Poincaré-Verdier sequences, seeProposition 1.4.5 for an example.We will, in fact, see that every Poincaré-Verdier sequence is a Poincaré-Karoubi sequence (Proposi-tion 1.3.8), and establish a simple criterion for a Poincaré-Karoubi sequence to be a Poincaré-Verdier se-quence (Corollary 1.3.10).Let us establish some terminology: We denote by C ♮ the idempotent completion of a small ∞ -category C and refer the reader to [Lur09a, §5.1.4] for its construction. The category C ♮ is stable if C is and the naturalfunctor 𝑖 ∶ C → C ♮ is fully faithful, exact and has dense essential image, where a full subcategory D ⊆ C is called dense if every object of C is a retract of one in D . Recall also that we call a functor a Karoubiequivalence if it is fully faithful with dense essential image, in other words if it induces an equivalence onthe idempotent completions (cf. Definition A.3.1).1.3.1.
Remark.
We avoid the common term Morita equivalence for what we call a Karoubi equivalence,since it conflicts with the notion of Morita equivalence of (discrete) rings: The very fact that invariants suchas K -, L - and Grothendieck-Witt spectra of a ring are defined via its (derived) module categories makes theminvariant under Morita equivalences in the latter sense, whereas invariance under Karoubi equivalences isan additional feature, that for example separates connective and non-connective K -theory. Definition.
A Poincaré ∞ -category is idempotent complete if its underlying stable ∞ -category is.We denote by Cat p∞ , idem ⊆ Cat p∞ the full subcategory spanned by the idempotent complete Poincaré ∞ -categories. A Poincaré functor ( 𝑓 , 𝜂 ) ∶ ( C , Ϙ ) → ( D , Φ) is a Karoubi equivalence if 𝑓 is a Karoubi equiva-lence and 𝜂 ∶ Ϙ → 𝑓 ∗ Φ is an equivalence.1.3.3. Proposition.
Let ( C , Ϙ ) be a Poincaré ∞ -category and 𝑖 ∶ C → C ♮ its idempotent completion. Thenthe left Kan extension 𝑖 ! Ϙ ∶ ( C ♮ ) op → S 𝑝 is a Poincaré functor on C ♮ and the canonical hermitian functor ( C , Ϙ ) → ( C ♮ , 𝑖 ! Ϙ ) is Poincaré, and a Karoubi equivalence.Moreover, for any idempotent-complete Poincaré ∞ -category ( D , Φ) the pullback functor Fun ex (( C ♮ , 𝑖 ! Ϙ ) , ( D , Φ)) → Fun ex (( C , Ϙ ) , ( D , Φ)) is an equivalence of Poincaré ∞ -categories. In particular the inclusion of Cat p∞ , idem ⊆ Cat p∞ of idempotent-complete Poincaré ∞ -categories has a left adjoint sending ( C , Ϙ ) to ( C ♮ , 𝑖 ! Ϙ ) . We will often write ( C , Ϙ ) ♮ for this left adjoint. Proof.
By Lemma [I].1.4.1 and Proposition [I].1.4.3 the functor 𝑖 ! Ϙ is quadratic with bilinear part ( 𝑖 × 𝑖 ) ! B Ϙ .To see that this is perfect, note first that it restricts back to B Ϙ since 𝑖 is fully faithful. Now, the idempotentcompletion of the equivalence D Ϙ ∶ C op → C is another equivalence D ∶ ( C ♮ ) op ≃ ( C op ) ♮ → C ♮ , and by theprevious observation, the functors Hom C ♮ (− , D−) and B 𝑖 ! Ϙ agree on C op × C op , and therefore on all of ( C ♮ ) op × ( C ♮ ) op by [Lur09a, Proposition 5.1.4.9]. This shows thatboth 𝑖 ! Ϙ and 𝑖 are Poincaré.Finally, let us fix ( D , Φ) an idempotent-complete Poincaré ∞ -category and consider the Poincaré functor 𝑖 ∗ ∶ Fun ex ( ( C ♮ , 𝑖 ! Ϙ ) , ( D , Φ) ) → Fun ex ( ( C , Ϙ ) , ( D , Φ) ) . By another application of [Lur09a, Proposition 5.1.4.9] this is an equivalence of the underlying stable ∞ -categories, so it suffices to show that it induces also an equivalence on the corresponding quadratic functors.But for an exact functor 𝑓 ∶ C ♮ → D this map is precisely the canonical equivalence nat( 𝑖 ! Ϙ , 𝑓 ∗ Φ) → nat( Ϙ , 𝑖 ∗ 𝑓 ∗ Φ) . (cid:3) Remark.
The adjunction 𝑖 ! ∶ Fun q ( C ) ⟂ Fun q ( C ♮ ) ∶ 𝑖 ∗ between hermitian structures on C and her-mitian structures on C ♮ is an equivalence, since 𝑖 ! is fully-faithful and 𝑖 ∗ is conservative by the density of 𝑖 . By Proposition 1.3.3 this equivalence restricts to an equivalence between Poincaré structures on C andPoincaré structures on C ♮ whose associated duality preserves C .1.3.5. Proposition.
The localisation of
Cat p∞ at the Karoubi equivalences admits both a left and a rightadjoint, the right adjoint is given by ( C , Ϙ ) ↦ ( C , Ϙ ) ♮ , and the left adjoint by ( C , Ϙ ) ↦ ( C min , 𝑗 ∗ Ϙ ) ,where C min is the full subcategory of C spanned the objects 𝑋 ∈ C with 𝑋 ] ∈ K ( C ) and 𝑗 is its inclusioninto C .In particular, the idempotent completion functor (−) ♮ ∶ Cat p∞ → Cat p∞ , idem preserves both limits andcolimits. The analogous statement for the underlying stable ∞ -categories is Proposition A.3.3. Proof.
We first note that C min ⊆ C is closed under the duality of C , since the duality acts by a grouphomomorphisms on K , and so ( C min , 𝑗 ∗ Ϙ ) is Poincaré by Observation [I].1.2.19.Now, according to Lemma A.2.1, we have to verify that Hom
Cat p∞ (( C min , 𝑗 ∗ Ϙ ) , −) and Hom
Cat p∞ (− , ( C ♮ , 𝑖 ! Ϙ )) invert Karoubi equivalences of Poincaré ∞ -categories. Both of these follow from their non-Poincaré coun-terparts established in Proposition A.3.3 by considering the induced maps on the cartesian squares Nat(Φ , 𝐹 ∗ Ψ) Hom
Cat p∞ (( D , Φ) , ( E , Ψ))Δ Hom
Cat ex∞ ( D , E ) . 𝐹 ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 27 For either ( D , Φ) = ( C min , 𝑗 ∗ Ϙ ) or ( E , Ψ) = ( C ♮ , 𝑖 ! Ϙ ) a Karoubi equivalence in the other variable inducesan equivalence by Lemma A.2.1 and Proposition A.3.3, and the induced map in the top left corner is anequivalence by [Lur09a, Proposition 5.1.4.9], since S 𝑝 is idempotent complete.The final clause follows since the adjoints are both automatically fully faithful by yet another applicationof Lemma A.2.1. (cid:3) Recall, that a sequence C 𝑓 ←←←←←←←→ D 𝑝 ←←←←←←→ E of exact functors with vanishing composite is a Karoubi sequence (Definition A.3.5) if the sequence C ♮ → D ♮ → E ♮ is both a fiber and a cofiber sequence in Cat ex∞ , idem . In this case we refer to 𝑓 as a Karoubi inclusion and to 𝑝 as a Karoubi projection .In the same spirit, we put:1.3.6.
Definition.
A sequence ( C , Ϙ ) ( 𝑓,𝜂 ) ←←←←←←←←←←←←←←←←←←←→ ( D , Φ) ( 𝑝,𝜃 ) ←←←←←←←←←←←←←←←←←→ ( E , Ψ) of Poincaré functors with vanishing composite is a Poincaré-Karoubi sequence if ( C , Ϙ ) ♮ ( 𝑓,𝜂 ) ♮ ←←←←←←←←←←←←←←←←←←←←←←→ ( D , Φ) ♮ ( 𝑝,𝜗 ) ♮ ←←←←←←←←←←←←←←←←←←←←←→ ( E , Ψ) ♮ is both a fibre sequence and a cofibre sequence in Cat p∞ , idem . We then call ( 𝑓 , 𝜂 ) a Poincaré-Karoubi inclu-sion and ( 𝑝, 𝜗 ) a Poincaré-Karoubi projection .We warn the reader that, contrary to the situation for (Poincaré-)Verdier sequences, a (Poincaré-)Karoubisequence is determined by its inclusion or its projection only up to idempotent completion of the third term.We record a few simple consequences of the definition.1.3.7.
Observation.
Since the forgetful and hyperbolic functors commute with idempotent completion byinspection, the sequence of stable ∞ -categories underlying a Poincaré-Karoubi sequence is a Karoubisequence and the hyperbolisation of a Karoubi sequence is a Poincaré-Karoubi sequence. By Proposition A.3.7, any Verdier sequence is a Karoubi sequence. Analogously:1.3.8.
Proposition.
Every Poincaré-Verdier sequence is a Poincaré-Karoubi sequence.Proof.
A fiber-cofiber sequence in
Cat p∞ remains so in Cat p∞ , idem after idempotent completion by Proposi-tion 1.3.5. (cid:3) Proposition.
Let ( C , Ϙ ) ( 𝑓,𝜂 ) ←←←←←←←←←←←←←←←←←←←→ ( D , Φ) ( 𝑝,𝜗 ) ←←←←←←←←←←←←←←←←←←→ ( E , Ψ) be a sequence of Poincaré functors with vanishing composite. Then:i) Its idempotent completion is a fibre sequence in Cat p∞ , idem if and only if the idempotent completion ofits underlying sequence is a fiber sequence in Cat ex∞ , idem and 𝜂 induces an equivalence Ϙ ⇒ 𝑓 ∗ Φ .ii) Its idempotent completion is a cofibre sequence in Cat p∞ , idem if and only if the idempotent completionof its underlying sequence is a cofiber sequence in Cat ex∞ , idem and 𝜗 exhibits Ψ as the left Kan extensionof Φ along 𝑝 .iii) It is a Poincaré-Karoubi sequence if and only if its underlying sequence is a Karoubi sequence andboth 𝜂 induces an equivalence Ϙ ⇒ 𝑓 ∗ Φ and 𝜗 exhibits Ψ as the left Kan extension of Φ along 𝑝 .Proof. By Proposition 1.3.5, fibers in
Cat p∞ , idem are computed in Cat p∞ while cofibers are computed asidempotent completions of cofibers in Cat p∞ . Thus, i) and ii) follow from Proposition 1.1.4 i) and ii), re-spectively, using the equivalence between quadratic functors on C and on C ♮ explained in Remark 1.3.4 (aswell as the ones for D and E ). Part iii) is i) and ii) put together. (cid:3) In particular, comparing Proposition 1.3.9 with Proposition 1.1.4 and investing Corollary A.1.10 for aconcrete description, we obtain:
Corollary.
A Poincaré-Karoubi sequence is a Poincaré-Verdier sequence if and only if its underlying(Karoubi) sequence is a Verdier sequence, i.e. concretely, the image of the inclusion is closed under retractsand the projection is essentially surjective.
Combining Proposition 1.3.9 with the concrete characterisation of Karoubi sequences given in Proposi-tion A.3.7, we also have:1.3.11.
Corollary.
A sequence of Poincaré functors ( C , Ϙ ) ( 𝑓,𝜂 ) ←←←←←←←←←←←←←←←←←←←→ ( D , Φ) ( 𝑝,𝜗 ) ←←←←←←←←←←←←←←←←←←→ ( E , Ψ) with vanishing composite is a Poincaré-Karoubi sequence if and only if bothi) 𝑓 is fully-faithful and the induced map D ∕ C → E is fully faithful with dense essential image andii) the map 𝜂 ∶ Ϙ ⇒ 𝑓 ∗ Φ is an equivalence, and the induced map 𝜗 ∶ Φ ⇒ 𝑝 ∗ Ψ exhibits Ψ as the left Kanextension of Φ along 𝑝 . Similarly, using Corollary A.3.8, we obtain:1.3.12.
Corollary. i) A Poincaré functor ( 𝑓 , 𝜂 ) ∶ ( C , Ϙ ) → ( D , Φ) is a Poincaré-Karoubi inclusion if and only if 𝑓 is fully-faithful and the map 𝜂 ∶ Ϙ ⇒ 𝑓 ∗ Φ is an equivalence.ii) A Poincaré functor ( 𝑝, 𝜗 ) ∶ ( D , Φ) → ( E , Ψ) is a Poincaré-Karoubi projection if and only if 𝑝 has denseessential image, the induced functor D → 𝑝 ( D ) is a localisation and 𝜗 ∶ Φ ⇒ 𝑝 ∗ Ψ exhibits Ψ as theleft Kan extension of Φ along 𝑝 . Finally, let us establish an analogue of the classification of Verdier and Karoubi projections from Propo-sition A.3.14, compare also [Nik20] for a slightly different treatment. To this end recall the category
Latt( C ) ⊂ Ar(Ind Pro( C )) spanned by the arrows from inductive to projective systems and the Verdierprojection cof ∶ Latt( C ) → Tate( C ) with Tate( C ) ⊆ Ind Pro( C ) the smallest stable subcategory containing Ind( C ) and Pro( C ) ; in the appendix we used Pro Ind instead of
Ind Pro to define
Latt and
Tate (which wasadvantageous in the proof of Proposition A.3.14) but evidently this makes no difference and the reverseorder will be more convenient here.Given a hermitian ∞ -category ( C , Ϙ ) we can endow both Ind( C ) and Pro( C ) with induced hermitianstructures since S 𝑝 is both complete and cocomplete, e.g. via Ind( C ) op ≃ Pro( C op ) Pro( Ϙ ) ←←←←←←←←←←←←←←←←←←←←←←←→ Pro( S 𝑝 ) lim ←←←←←←←←←←←←←→ S 𝑝. Even if Ϙ is Poincaré the same is, however, not usually true of these extensions. Instead the duality of Ϙ theninduces an equivalence Ind( C ) op ≃ Pro( C op ) Pro(D Ϙ ) ←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ Pro( C ) . From this statement it is readily checked that
Tate( C ) inherits a Poincaré structure by the above procedure(note, however, that it is not a small category in general), as does Latt( C ) from the hermitian structure on Ar(Ind Pro( C )) given by Ϙ ar ( 𝑥 → 𝑦 ) = Ϙ [1]met ( 𝑦 → cof( 𝑥 → 𝑦 )) , whose duality is given by D Ϙ ar ( 𝑥 → 𝑦 ) ≃ D Ϙ 𝑦 → D Ϙ 𝑥 ; see Definition [I].2.3.15 for a thorough discussionof this hermitian structure. By design there is generally an equivalence (Ar( D ) , Φ ar ) ≃ Met( D , Φ [1] ) bysending an arrow 𝑥 → 𝑦 to 𝑦 → cof ( 𝑥 → 𝑦 ) and this translates met ∶ Met( D , Φ [1] ) → ( D , Φ [1] ) to thefunctor cof ∶ (Ar( D ) , Φ ar ) → ( D , Φ [1] ) .1.3.13. Proposition.
For any Poincaré ∞ -category ( C , Ϙ ) the restriction cof ∶ (Latt( C ) , Ϙ ar ) ⟶ (Tate( C ) , Ϙ [1] ) of the hermitian functor just described is a Poincaré-Verdier projection (among large Poincaré ∞ -categories)with fibre ( C , Ϙ ) ♮ . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 29 Proof.
To see that it preserves the dualities simply note that cof D Ϙ ar ( 𝑥 → 𝑦 ) ≃ cof(D Ϙ 𝑦 → D Ϙ 𝑥 ) ≃ ΣD Ϙ cof( 𝑥 → 𝑦 ) ≃ D Ϙ [1] cof( 𝑥 → 𝑦 ) . Regarding the fibre we find that the kernel of cof is C ♮ by Proposition A.3.14 and that Ϙ ar restricts as desiredto C is immediate from its definition. But this determines the restriction to C ♮ as well since S 𝑝 is idempotentcomplete. Another application of Proposition A.3.14 also yields that cof is a Verdier projection, and tocheck that Ϙ [1] is the left Kan extension of Ϙ ar , we use Remark 1.1.8 to compute (cof ! Ϙ ar )(cof( 𝑖 → 𝑝 )) ≃ colim 𝑐 ∈ C op 𝑝 ∕ Ϙ ar (f ib( 𝑖 → 𝑐 ) → f ib( 𝑝 → 𝑐 ))≃ colim 𝑐 ∈ C op 𝑝 ∕ Ϙ [1]met (f ib( 𝑝 → 𝑐 ) → cof( 𝑖 → 𝑝 ))≃ colim 𝑐 ∈ C op 𝑝 ∕ f ib( Ϙ [1] (cof( 𝑖 → 𝑝 )) → Ϙ [1] (f ib( 𝑝 → 𝑐 ))) Now, the category C 𝑝 ∕ is cofiltered, so in particular contractible, whence the colimit can be moved throughthe first term, and we are left to show that colim ( 𝑐 ∈ C 𝑝 ∕ ) op Ϙ [1] (f ib( 𝑝 → 𝑐 )) ≃ 0 . But by construction the extension of Ϙ to projective systems commutes with filtered colimits, so colim 𝑐 ∈ C op 𝑝 ∕ Ϙ [1] (f ib( 𝑝 → 𝑐 )) ≃ Ϙ [1] ( lim 𝑐 ∈ C 𝑝 ∕ f ib( 𝑝 → 𝑐 ) ) ≃ Ϙ [1] (f ib( 𝑝 → 𝑝 ) ≃ 0 as desired. (cid:3) We claim that the Poincaré-Verdier projection just constructed is the universal example of such a pro-jection with fibre ( C , Ϙ ) ♮ , and cof ∶ (Latt( C ) , Ϙ ar ) → (Tate( C ) , Ϙ [1] ) ♮ is consequently the universal Poincaré-Karoubi projection. We construct the classifying morphism:1.3.14. Construction.
Given a Poincaré-Verdier sequence ( C , Ϙ ) 𝑓 ←←←←←←←→ ( D , Φ) 𝑝 ←←←←←←→ ( E , Ψ) consider the null-composite sequence Ind Pro( C ) → Ind Pro( D ) → Ind Pro( E ) . By Theorem A.3.11 andthe discussion thereafter it is a split Verdier sequence and endowing the middle term with the inducedhermitian structure, we claim it satisfies the assumption of Construction 1.2.8, i.e B Φ ( 𝑞 ( 𝑒 ) , Pro( 𝑓 )( 𝑐 )) ≃ 0 for all 𝑒 ∈ Ind Pro( E ) and 𝑐 ∈ Ind Pro( C ) . Before going to the proof, note that straight from the definitionthe restriction of this hermitian structure on Ind Pro( D ) to Ind Pro( C ) is just the extension of Ϙ . Now thevanishing of the bilinear part in fact holds before passing to inductive completions, and is preserved by thatprocess: For pro-objects 𝑑 = lim 𝑑 𝑖 and 𝑑 ′ = lim 𝑑 ′ 𝑗 one computes B Φ ( 𝑑, 𝑑 ′ ) ≃ colim 𝑖,𝑗 B Φ ( 𝑑 𝑖 , 𝑑 𝑗 ) ≃ colim 𝑖,𝑗 Hom D ( 𝑑 𝑖 , D Φ 𝑑 𝑗 ) ≃ colim 𝑗 Hom
Pro( D ) ( 𝑑, D Φ 𝑑 𝑗 ) ≃ Hom Ind Pro( D ) ( 𝑑, D Φ 𝑑 ) where the first equivalence follows straight from the definition of the extension of Φ to Pro( D ) op and D Φ in the final entry denotes the extension Pro( D ) op → Ind( D ) . With this in place we can compute B Φ ( 𝑞 ( 𝑒 ) , Pro( 𝑓 )( 𝑐 )) ≃ Hom Ind Pro( D ) ( 𝑞 ( 𝑒 ) , D Φ Pro( 𝑓 )( 𝑐 ))≃ Hom Ind Pro( D ) ( 𝑞 ( 𝑒 ) , Ind( 𝑓 )D Ϙ 𝑐 )≃ Hom Ind Pro( C ) ( 𝑒, Ind( 𝑝 ) Ind( 𝑓 )D Ϙ 𝑐 )≃ 0 for 𝑒 ∈ Pro( E ) and 𝑐 ∈ Ind( C ) . That the vanishing is still true after inductive completion follows from thegeneral formula B Φ ( 𝑑, 𝑑 ′ ) ≃ lim 𝑖,𝑗 B Φ ( 𝑑 𝑖 , 𝑑 ′ 𝑗 ) for inductive systems 𝑑 = colim 𝑖 𝑑 𝑖 and 𝑑 ′ = colim 𝑗 𝑑 ′ 𝑗 . We can therefore apply Construction 1.2.8 and the discussion immediately after, to obtain a cartesiandiagram (Ind Pro( D ) , Φ) Met(Ind Pro( C ) , Ϙ [1] )(Ind Pro( E ) , Ψ ′ ) (Ind Pro( C ) , Ϙ [1] ) 𝑔 ⇒ 𝑐𝑝 met 𝑐 of rather large hermitian ∞ -categories; note that we do not claim that the hermitian structure Ψ ′ in the lowerleft corner is the extension of Ψ . Using the equivalence between metabolic and arrow categories it can berewritten as (Ind Pro( D ) , Φ) Ar(Ind Pro( C ) , Ϙ ar )(Ind Pro( E ) , Ψ ′ ) (Ind Pro( C ) , Ϙ [1] ) 𝑔 ′ ⇒ 𝑔 cof 𝑐 and one readily checks that it restricts to a diagram ( D , Φ) (Latt( C ) , Ϙ ar )( E , Ψ) (Tate( C ) , Ϙ [1] ) , 𝑝 cof using Remark 1.1.8 to identify the hermitian structures in the lower left corner.1.3.15. Theorem.
For any Poincaré-Verdier sequence ( C , Ϙ ) 𝑓 ←←←←←←←→ ( D , Φ) 𝑝 ←←←←←←→ ( E , Ψ) the square constructed above consists of Poincaré functors and is cartesian.Proof. That the square of underlying ∞ -categories is cartesian is part of Proposition A.3.14 and that thehermitian structure on D is the pullback of the other three follows from the analogous statement in thepreceeding cartesian square involving the Ind Pro -categories. Since limits of Poincaré ∞ -categories aredetected among hermitian ∞ -categories, it only remains to check that the horizontal maps preserve thedualities. This is verified exactly as in Theorem 1.2.9. (cid:3) Finally, we again record the uniqueness of the classifying map. To state the result, we extend the notionof adjointability to non-split Verdier projections by requiring the diagrams
Ind( C ) Ind( C ′ ) Pro( C ) Pro( C ′ )Ind( D ) Ind( D ′ ) Pro( C ) Pro( C ′ ) 𝑖𝑝 𝑝 ′ 𝑖𝑝 𝑝 ′ 𝑗 𝑖 to be left and right adjointable, respectively.1.3.16. Corollary.
Given a Poincaré-Verdier sequence ( C , Ϙ ) → ( D , Φ) → ( E , Ψ) then for every Poincaré ∞ -category ( C ′ , Ϙ ′ ) the full subcategory of Fun ex (( D , Φ) , (Latt( C ′ ) , Ϙ ′ar )) spanned by the functors 𝜑 thatgive rise to adjointable squares D Latt( C ′ ) E Tate( C ′ ) 𝜑 cof 𝜑 is equivalent to Fun ex (( C , Ϙ ) , ( C ′ , Ϙ ′ )) via restriction to horizontal fibres.In particular, the classifying morphism in Theorem 1.3.15 is determined up to contractible choice byyielding a cartesian square and inducing the identity on vertical fibres.Proof. The first part follows from Proposition 1.2.11 by inspection, the second uses in addition that cartesiansquares with vertical Verdier-projections are adjointable, which is part of Proposition A.3.15. (cid:3)
ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 31 Examples of Poincaré-Verdier sequences.
In this section we consider various examples of interestof Poincaré-Verdier and Poincaré-Karoubi sequences. As one of the most important examples we alreadygave the metabolic fibre sequence in Example 1.2.5, which is at the core of our deduction of the main resultsfrom the additivity theorem in the next section. We repeat the statement for completeness’ sake: Given aPoincaré ∞ -category, the metabolic fibre sequence ( C , Ϙ [−1] ) ⟶ Met( C , Ϙ ) met ←←←←←←←←←←←←←←→ ( C , Ϙ ) is a split Poincaré-Verdier sequence; its left hand Poincaré functor is given by sending 𝑥 to 𝑥 → , togetherwith the identification Ω Ϙ ( 𝑋 ) ≃ f ib( Ϙ (0) → Ϙ ( 𝑥 )) .Next, we give a simple recognition criterion for Poincaré-Verdier sequences involving hyperbolic Poincaré ∞ -categories. Recall from Corollary [I].7.2.20 and Remark [I].7.2.21 that Hyp is both left and right adjointto the underlying category functor 𝑈 ∶ Cat p∞ → Cat ex∞ .1.4.1.
Lemma.
Let 𝑔 ∶ C → D be an exact functor. Then for a Poincaré structure Ϙ on C the functor 𝑔 hyp ∶ ( C , Ϙ ) ⟶ Hyp( D ) obtained by right adjointness of Hyp is a split Poincaré-Verdier projection if and only if 𝑔 is a split Verdierprojection and the restrictions of Ϙ to both the essential images of 𝑙 op and D op Ϙ ◦ 𝑟 vanish, where 𝑙 and 𝑟 denote the adjoints of 𝑔 .Similarly, for a Poincaré structure Φ on D the functor 𝑔 hyp ∶ Hyp( C ) ⟶ ( D , Φ) obtained by left adjointness of Hyp is a split Poincaré-Verdier inclusion if and only if 𝑔 is a split Verdierinclusion and the restrictions of Ϙ to both the essential images of 𝑔 op and D opΦ ◦ 𝑔 vanish.Proof. Let us prove the first statement, the second is entirely analogous. It is easy to check that the functor 𝑔 hyp , which is given by ( 𝑔, 𝑔 op ◦ D op Ϙ ) ∶ C ⟶ D ⊕ D op admits both adjoints if and only if 𝑔 does; in this case the left adjoint 𝑙 ′ to 𝑔 hyp is given by ( 𝑑, 𝑑 ′ ) ↦ 𝑙𝑑 ⊕ D Ϙ ( 𝑟𝑑 ′ ) , and the right adjoint by switching the roles of 𝑙 and 𝑟 .Similarly, one checks that the unit of the adjunction 𝑙 ′ ⊢ 𝑔 hyp is given by ( 𝑑, 𝑑 ′ ) (( 𝑢, , (0 ,𝑐 )) ←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ ( 𝑔𝑙𝑑 ⊕ 𝑔 D Ϙ ( 𝑟𝑑 ′ ) , 𝑔 D Ϙ ( 𝑙𝑥 ) ⊕ 𝑔𝑟𝑑 ′ ) , where 𝑢 is the unit of the adjunction 𝑙 ⊢ 𝑔 and 𝑐 the counit of 𝑔 ⊢ 𝑟 . If this unit is an equivalence then so are 𝑢 and 𝑐 making 𝑔 into a split Verdier projection by Corollary A.2.6. Conversely, if 𝑔 is a Verdier projectionboth 𝑢 and 𝑐 are equivalences and it remains to check that 𝑔 vanishes on the essential images of both D op Ϙ ◦ 𝑟 and D op Ϙ ◦ 𝑙 , but this is implied by Hom D ( 𝑔 D Ϙ ( 𝑟𝑑 ′ ) , 𝑑 ) ≃ Hom C (D Ϙ ( 𝑟𝑑 ′ ) , 𝑟𝑑 )≃ B Ϙ (D Ϙ ( 𝑟𝑑 ′ ) , D Ϙ ( 𝑟𝑑 ))Hom D ( 𝑑, 𝑔 D Ϙ ( 𝑙𝑑 ′ )) ≃ Hom C ( 𝑙𝑑, D Ϙ ( 𝑙𝑑 ′ ))≃ B Ϙ ( 𝑙𝑑, 𝑙𝑑 ′ ) both of which vanish by the assumption on Ϙ .Finally by Corollary 1.2.3, assuming the existence and full faithfulness of a left adjoint, 𝑔 hyp is a Poincaré-Verdier projection if and only if the map Ϙ ( 𝑙𝑑 ⊕ D Ϙ ( 𝑟𝑑 ′ )) ⟶ B Ϙ ( 𝑙𝑑 ⊕ D Ϙ ( 𝑟𝑑 ′ ) , 𝑙𝑑 ⊕ D Ϙ ( 𝑟𝑑 ′ )) ≃ Hom C ( 𝑙𝑑 ⊕ D Ϙ ( 𝑟𝑑 ′ ) , D Ϙ ( 𝑙𝑑 ) ⊕ 𝑟𝑑 ′ ) 𝑝 ←←←←←←→ Hom D ( 𝑑 ⊕ 𝑝 D Ϙ ( 𝑟𝑑 ′ ) , 𝑝 D Ϙ ( 𝑙𝑑 ) ⊕ 𝑑 ′ ) ((id 𝑑 , ∗ , (0 , id 𝑑 ′ ) ∗ ) ←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ Hom D ( 𝑑, 𝑑 ′ ) is an equivalence. Under the equivalence Ϙ ( 𝑙𝑑 ⊕ D Ϙ ( 𝑟𝑑 ′ )) ≃ Ϙ ( 𝑙𝑑 ) ⊕ Ϙ (D Ϙ ( 𝑟𝑑 ′ )) ⊕ Hom C ( 𝑙𝑑, 𝑟𝑑 ′ ) this map becomes the projection to the last summand followed by the natural equivalence Hom D ( 𝑙𝑑, 𝑟 ′ 𝑑 ) ≃ Hom C ( 𝑔𝑙𝑑, 𝑑 ′ ) ≃ Hom C ( 𝑑, 𝑑 ′ ) . Thus it is an equivalence if and only if Ϙ ( 𝑙𝑑 ) and Ϙ (D Ϙ ( 𝑟𝑑 ′ )) both vanish, which is precisely our assumption. (cid:3) We next work out the more substantial example of module ∞ -categories in detail, where the hermitianstructure is defined by means of a module with genuine involution, introduced in §[I].3.2 (compare also therecollection section). We do so first in the generality of a map of E -algebras 𝜙 ∶ 𝐴 → 𝐵 over some base E ∞ -ring 𝑘 together with a map 𝜂 of modules with genuine involution 𝜙 ! ( 𝑀, 𝑁, 𝛼 ) → ( 𝑀 ′ , 𝑁 ′ , 𝛽 ) over 𝐵 and eventually specialise to Ore localisations of discrete rings with anti-involution in Corollary 1.4.9. Thereader only interested in this case is invited to take 𝑘 the (Eilenberg-Mac Lane spectrum of the) integersand 𝐴 and 𝐵 (and even 𝑀 and 𝑀 ′ ) discrete from the start, though this does not simplify the discussion.Furthermore, it is important to allow 𝑁 and 𝑁 ′ to be non-discrete, so as to capture the genuine Poincaréstructures.Throughout, unmarked tensor products are always over 𝑘 , and in case 𝑘 = H 𝑅 will translate to thederived tensor product ⊗ 𝕃ℝ .We want to establish general conditions on 𝜙 under which the hermitian functor ( 𝜙 ! , 𝜂 ) becomes aPoincaré-Verdier or Poincaré-Karoubi projection. To obtain a Verdier sequence on the underlying stable ∞ -categories the following conditions are necessary and sufficient: A map 𝜙 ∶ 𝐴 → 𝐵 of E -ring spectrais said to be a localisation if the map 𝐵 ⊗ 𝐴 𝐵 → 𝐵, induced by the multiplication of 𝐵 is an equivalence of spectra. For such a localisation of ring spectradenote by 𝐼 ∈ Mod 𝐴 its fibre. Straight from the definition one finds that 𝐼 belongs to (Mod 𝐴 ) 𝐵 , the kernelof 𝜙 ! ∶ Mod 𝐴 → Mod 𝐵 . We say that 𝜙 has perfectly generated fibre if 𝐼 belongs to the smallest fullsubcategory of (Mod 𝐴 ) 𝐵 containing Mod 𝜔𝐴 ∩(Mod 𝐴 ) 𝐵 and closed under colimits.Summarising the discussion of Appendix A.4, we have by Proposition A.4.4 that if 𝜙 ∶ 𝐴 → 𝐵 is local-isation of E -rings with perfectly generated fibre, then for any subgroup c ⊆ K ( 𝐴 ) the induction functors 𝜙 𝜔 ! ∶ Mod 𝜔𝐴 → Mod 𝜔𝐵 and 𝜙 c! ∶ Mod c 𝐴 → Mod 𝜙 (c) 𝐵 are Karoubi and Verdier projections, respectively; here Mod c 𝐴 denotes the full subcategory of Mod 𝜔𝐴 spannedby all those 𝐴 -modules with [ 𝐴 ] ∈ c ⊆ K ( 𝐴 ) .We warn the reader explicitely, that when applied to the Eilenberg-Mac Lane spectra of discrete rings,this notion of localisation differs from that in ordinary algebra: If 𝐴 → 𝐵 is a localisation of discreterings, then H 𝐴 → H 𝐵 is a localisation in the sense above if and only if additionally Tor 𝐴𝑖 ( 𝐵, 𝐵 ) = 0 forall 𝑖 > . This is automatic for commutative rings, or more generally if the localisation satisfies an Orecondition, but not true in general. Moreover, there are quotient maps 𝐴 → 𝐴 ∕ 𝐼 of commutative rings suchthat H 𝐴 → H 𝐴 ∕ 𝐼 is a localisation. When specialising to the case of discrete rings we will therefore call amap 𝐴 → 𝐵 a derived localisation if H 𝐴 → H 𝐵 is a localisation in the sense above. We do not know of asimple ring theoretic characterisation of this condition; see §A.4 for a more thorough discussion.The following example will essentially cover all of our applications:1.4.2. Example. If 𝐴 is an E -ring spectrum and 𝑆 ⊆ 𝜋 ∗ 𝐴 is a multiplicatively closed subset of homoge-neous elements, which satisfies the left or right Ore condition, then the localisation map 𝜙 ∶ 𝐴 → 𝐴 [ 𝑆 −1 ] (see [Lur17, §7.2.3]) is a localisation by Lemma A.4.1, since the forgetful functor Mod 𝐴 [ 𝑆 −1 ] → Mod 𝐴 isfully faithful. In this case the modules 𝐴 ∕ 𝑠 = cof[ ⋅ 𝑠 ∶ 𝕊 𝑛 ⊗ 𝐴 → 𝐴 ] for 𝑠 ∈ 𝑆 and 𝑛 ∈ ℤ form a systemof generators for (Mod 𝐴 ) 𝐵 under shifts and colimits , see [Lur17, Lemma 7.2.3.13], so in particular 𝜙 hasperfectly generated fibre.Thus 𝜙 𝜔 ! ∶ Mod 𝜔𝐴 → Mod 𝜔𝐴 [ 𝑆 −1 ] is a Karoubi projection and 𝜙 c! ∶ Mod c 𝐴 → Mod im(c) 𝐴 [ 𝑆 −1 ] is a Verdierprojection for any c ⊆ K ( 𝐴 ) .Let us now introduce hermitian structures into the picture. As discussed in section §[I].3.2, an invertiblemodule with genuine involution ( 𝑀, 𝑁, 𝛼 ) over 𝐴 gives rise to a Poincaré structure Ϙ 𝛼𝑀 on Mod 𝜔𝐴 ; it restrictsto a Poincaré structure on Mod f 𝐴 provided that 𝑀 belongs to Mod c 𝐴 and provided c is closed under theinvolution on K ( 𝐴 ) induced by 𝑀 . For example, if c is the image of the canonical map ℤ → K ( 𝐴 ) , ↦ 𝐴 ,then Mod c 𝐴 = Mod f 𝐴 and this assumption is satisfied if also 𝑀 ∈ Mod f 𝐴 . We computed the left Kan ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 33 extension of this Poincaré structures along the functor 𝜙 𝜔 ! ∶ Mod 𝜔𝐴 → Mod 𝜔𝐵 in Corollary [I].3.3.1: It is thehermitian structure associated to the module with genuine involution(10) 𝜙 ! ( 𝑀, 𝑁, 𝛼 ) = ((
𝐵 ⊗ 𝐵 ) ⊗ 𝐴⊗𝐴
𝑀, 𝐵 ⊗ 𝐴 𝑁, 𝛽 ) , over 𝐵 ; here 𝛽 is the composition 𝐵 ⊗ 𝐴 𝑁 Δ ⊗𝛼 ←←←←←←←←←←←←←←←←←←←→ ( 𝐵 ⊗ 𝐵 ) tC ⊗ 𝐴 𝑀 tC ←←→ (( 𝐵 ⊗ 𝐵 ) ⊗ 𝐴⊗𝐴 𝑀 ) tC where Δ is the Tate diagonal. For example by Remark 1.1.8, the same formula then applies for the Kanextension along 𝜙 c! ∶ Mod c 𝐴 → Mod 𝜙 (c) 𝐵 .In order to obtain Poincaré-Karoubi projections, we need a compatibility condition between 𝜙 ∶ 𝐴 → 𝐵 and the module with involution 𝑀 over 𝐴 :1.4.3. Definition.
An invertible module with involution 𝑀 over 𝐴 is called compatible with a localisationof E -rings 𝐴 → 𝐵 if the composite 𝐵 ⊗ 𝐴 𝑀 ≃ ( 𝐵 ⊗ 𝐴 ) ⊗ 𝐴⊗𝐴 𝑀 ⟶ ( 𝐵 ⊗ 𝐵 ) ⊗ 𝐴⊗𝐴 𝑀 is an equivalence.1.4.4. Example. i) If 𝐴 is an E ∞ -ring and 𝑀 an invertible 𝐴 -module with 𝐴 -linear involution (regarded as an 𝐴 ⊗ 𝐴 -module via the multiplication map
𝐴 ⊗ 𝐴 → 𝐴 ), then compatibility is automatic, since in this case themap in question identifies with the evident one 𝐵 ⊗ 𝐴 𝑀 → 𝐵 ⊗ 𝐴 𝐵 ⊗ 𝐴 𝑀 which is an equivalenceby the assumption that 𝐴 → 𝐵 is a localisation.ii) If 𝑀 is the module with involution over 𝐴 associated to a Wall anti-structure ( 𝜖, 𝜎 ) on a discrete ring 𝐴 as in Example [I].3.1.10 (i.e. 𝑀 = 𝐴 regarded as an 𝐴 ⊗ 𝐴 -module using the involution 𝜎 , and thenequipped with the involution 𝜖𝜎 , where 𝜖 ∈ 𝐴 ∗ ) and 𝜙 ∶ ( 𝐴, 𝜖, 𝜎 ) ⟶ ( 𝐵, 𝛿, 𝜏 ) is a map of rings with anti-structure, then 𝑀 is also automatically compatible with 𝜙 if the latter is aderived localisation: For in this case it is readily checked that the maps 𝑏 ⊗ 𝑏 ′ ⊗ 𝑎 ⟼ 𝑏𝑎 ⊗ 𝜏 ( 𝑏 ′ ) and 𝑏 ⊗ 𝑏 ′ ⟼ 𝑏 ⊗ 𝑏 ′ ⊗ give inverse equivalences 𝐵 ⊗ 𝐵 ⊗
𝐴⊗𝐴 𝐴 ≃ 𝐵 ⊗ 𝐴 𝐵, which translates the map in Definition 1.4.3 to the unit map 𝐵 → 𝐵 ⊗ 𝐴 𝐵 which is an equivalencesince 𝜙 is a localisation.iii) If 𝜙 is an Ore localisation at the set 𝑆 ⊆ 𝜋 ∗ ( 𝐴 ) , and 𝑀 is an invertible module with involution over 𝐴 ,then 𝑀 is compatible with 𝜙 if after inverting the action of 𝑆 on 𝑀 using the first 𝐴 -module structure, 𝑆 operates invertibly through the second one.iv) Combining the two previous examples, if 𝑀 is the 𝐴 -module associated to a Wall anti-structure ( 𝜖, 𝜎 ) on 𝐴 , and 𝑆 ⊆ 𝐴 satisfies the Ore condition and is closed under the involution 𝜎 , then 𝑀 is compatiblewith the localisation map 𝐴 → 𝐴 [ 𝑆 −1 ] .1.4.5. Proposition.
Let 𝜙 ∶ 𝐴 → 𝐵 be a localisation of E -ring spectra, with perfectly generated fibre andlet ( 𝑀, 𝑁, 𝛼 ) be an invertible module with genuine involution over 𝐴 , such that 𝑀 is compatible with 𝜙 .Then 𝜙 ! ( 𝑀, 𝑁, 𝛼 ) is invertible and the associated functor 𝜙 𝜔 ! ∶ (Mod 𝜔𝐴 , Ϙ 𝛼𝑀 ) → (Mod 𝜔𝐵 , Ϙ 𝜙 ! 𝛼𝜙 ! 𝑀 ) is a Poincaré-Karoubi projection. It restricts to a Poincaré-Verdier projection 𝜙 c! ∶ (Mod c 𝐴 , Ϙ 𝛼𝑀 ) → (Mod 𝜙 (c) 𝐵 , Ϙ 𝜙 ! 𝛼𝜙 ! 𝑀 ) . if c ⊆ K ( 𝐴 ) is closed under the involution induced by 𝑀 . Proof.
The natural map
𝐵 ⊗ 𝐴 hom 𝐴 ( 𝑋, 𝑀 ) ⟶ hom 𝐵 ( 𝐵 ⊗ 𝐴 𝑋, 𝐵 ⊗ 𝐴 𝑀 ) is an equivalence for 𝑋 = 𝐴 and thus for every compact 𝐴 -module 𝑋 , in particular for 𝑋 = 𝑀 , whichshows that 𝐵 ⊗ 𝐴 𝑀 has 𝐵 as its 𝐵 -linear endomorphisms, and therefore by assumption ( 𝐵 ⊗𝐵 ) ⊗ 𝐴⊗𝐴 𝑀 isinvertible (or alternatively, one can apply Remark 1.1.7 together with Proposition [I].3.1.3). Both functors 𝜙 𝜔 ! and 𝜙 c! are then Poincaré by Lemma [I].3.3.3.By Corollary 1.1.6, the functor 𝜙 c! is a Poincaré-Verdier projection since the underlying map on modulecategories is a Verdier projection and by definition the Poincaré structure on the target is the left Kanextension of that on the source. Similarly, the functor 𝜙 𝜔 ! is a Poincaré-Karoubi projection by Corol-lary 1.3.12. (cid:3) Corollary.
Let 𝜙 ∶ 𝐴 → 𝐵 be a localisation of E -ring spectra, with perfectly generated fiber and let 𝑀 be an invertible module with involution over 𝐴 , that is compatible with 𝜙 .Then 𝜙 𝜔 ! ∶ (Mod 𝜔𝐴 , Ϙ q 𝑀 ) → (Mod 𝜔𝐵 , Ϙ q 𝜙 ! 𝑀 ) and 𝜙 c! ∶ (Mod c 𝐴 , Ϙ q 𝑀 ) → (Mod 𝜙 (c) 𝐵 , Ϙ q 𝜙 ! 𝑀 ) are a Poincaré-Karoubi and Poincaré-Verdier projection (for c ⊆ K ( 𝐴 ) closed under the duality), respec-tively. Symmetric Poincaré structures are not, however, generally preserved by left Kan extension:1.4.7.
Example.
The map 𝑝 ∶ 𝕊 → 𝕊 [ ] does not induce an equivalence 𝑝 ! Ϙ s 𝕊 ≃ Ϙ s 𝕊 [ ] and consequently thefunctor 𝑝 ! ∶ (Mod 𝜔 𝕊 , Ϙ s ) ⟶ (Mod 𝜔 𝕊 [ ] , Ϙ s ) , is not a Poincaré-Karoubi projection: By Lin’s theorem the linear part of 𝑝 ! Ϙ s is classified by 𝕊 [ ] ⊗ 𝕊 ∧2 ≃H ℚ , whereas 𝕊 [ ] tC ≃ 0 gives the linear part of the symmetric Poincaré structure on the target.In the discrete case, an additional flatness assumption excludes such examples, as we will see in the nextproposition.Recall that for discrete (or more generally connective) 𝐴 , and 𝑀 an invertible module with (non-genuine)involution over 𝐴 , we defined in §[I].3.2 the genuine family of Poincaré structures Ϙ ≥ 𝑚𝑀 for 𝑚 ∈ ℤ as thePoincaré structures associated to the modules with genuine involution ( 𝑀, 𝜏 ≥ 𝑚 𝑀 tC , 𝛼 ) where 𝛼 ∶ 𝜏 ≥ 𝑚 𝑀 tC → 𝑀 tC is the canonical map; the quadratic and symmetric Poincaré structures Ϙ q 𝑀 and Ϙ s 𝑀 are included inthe genuine family as 𝑚 = −∞ and 𝑚 = ∞ , respectively.1.4.8. Proposition.
Let 𝜙 ∶ 𝐴 → 𝐵 be a derived localisation between discrete rings with perfectly generatedfibre, that furthermore that makes 𝐵 into a flat right module over 𝐴 and let 𝑀 be a discrete invertible modulewith involution over 𝐴 that is compatible with 𝜙 . Then for arbitrary 𝑚 ∈ ℤ ∪ {±∞} the maps 𝜙 𝜔 ! ∶ ( D p ( 𝐴 ) , Ϙ ≥ 𝑚𝑀 ) → ( D p ( 𝐵 ) , Ϙ ≥ 𝑚𝜙 ! 𝑀 ) and 𝜙 c! ∶ ( D c ( 𝐴 ) , Ϙ ≥ 𝑚𝑀 ) → ( D 𝜙 (c) ( 𝐵 ) , Ϙ ≥ 𝑚𝜙 ! 𝑀 ) . are a Poincaré-Karoubi and a Poincaré-Verdier projection, respectively, for every c ⊆ K ( 𝐴 ) closed underthe duality.Proof. We use the following two inputs: firstly, 𝐵 being a flat 𝐴 op -module implies that it can be written asfiltered colimit of finitely generated free 𝐴 op -modules 𝐵 𝑖 and secondly, Tate cohomology commutes withfiltered colimits of discrete modules in the coefficients. The former statement is a classical theorem ofLazard, see e.g. [Laz69, Théorème 1.2] or [SP18, Tag 058G], and the second statement (for group coho-mology) was discovered by Brown in [Bro75, Theorem 3] for groups admitting a classifying space of finitetype; given the -periodicity of Tate cohomology for C the case at hand also follows immediately from thesame statement for group homology, which is obvious from the definitions.Now, recall the description of 𝜙 ! Ϙ ≥ 𝑚𝑀 via (10). We start by considering the case 𝑚 = ∞ ; in which casewe need to show that 𝐵 ⊗ 𝐴 𝑀 tC → (( 𝐵 ⊗ 𝐵 ) ⊗ 𝐴⊗𝐴 𝑀 ) tC is an equivalence. We can regard this as anatural transformation between spectrum valued functors 𝑋 ⟼ 𝑋 ⊗ 𝐴 𝑀 tC and 𝑋 ⟼ ( ( 𝑋 ⊗ 𝑋 ) ⊗ 𝐴⊗𝐴 𝑀 ) tC ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 35 on both the category of discrete 𝐴 op -modules and D ( 𝐴 op ) . From the latter case we obtain that it is anequivalence for every perfect 𝑋 , as both sides are exact functors and the claim is evidently true for 𝑋 = 𝐴 .In particular, the claim is true for all finitely generated projective 𝐴 op -modules since these are perfect whenregarded in D ( 𝐴 op ) . Since filtered colimits are in particular sifted (i.e. the diagonal of a filtered categoryis cofinal), regarding the two assignments as functors on the category of discrete 𝐴 op -modules the secondfact makes them commute with filtered colimits of finitely generated free 𝐴 op -modules (for 𝑋 a finitelygenerated free 𝐴 op -module, 𝑋 ⊗ 𝑋 is a finitely generated free ( 𝐴 ⊗ 𝐴 ) op -module, so ( 𝑋 ⊗ 𝑋 ) ⊗ 𝐴⊗𝐴 𝑀 remains discrete, despite 𝑋 ⊗ 𝑋 and
𝐴 ⊗ 𝐴 potentially having higher homotopy). Taken together, thetransformation is an equivalence for all flat 𝐴 -modules, so in particular for 𝑋 = 𝐵 as desired.To obtain the case of the genuine Poincaré structures, just observe that the flatness of 𝐵 also guaranteesthat the functor 𝐵 ⊗ 𝐴 − ∶ D ( 𝐴 ) → D ( 𝐵 ) commutes with the connective cover functors 𝜏 ≥ 𝑚 for all 𝑚 ∈ ℤ .The case of the quadratic Poincaré structure is trivial. (cid:3) As a special case we obtain:1.4.9.
Corollary.
Let ( 𝐴, 𝜖, 𝜎 ) a ring with Wall anti-structure, and 𝑆 ⊆ 𝐴 a multiplicative subset satisfyingthe left Ore condition and closed under the involution 𝜎 . Then if 𝑀 denotes the module with involutionover 𝐴 given by endowing 𝐴 with the 𝐴 ⊗ 𝐴 -module structure arising from 𝜎 and the involution 𝜖𝜎 we findfor all 𝑚 ∈ ℤ ∪ {±∞} a Poincaré-Karoubi sequence ( D p ( 𝐴 ) 𝑆 , Ϙ ≥ 𝑚𝑀 ) ⟶ ( D p ( 𝐴 ) , Ϙ ≥ 𝑚𝑀 ) ⟶ ( D p ( 𝐴 [ 𝑆 −1 ]) , Ϙ ≥ 𝑚𝑀 [ 𝑆 −1 ] ) and a Poincaré-Verdier sequence ( D c ( 𝐴 ) 𝑆 , Ϙ ≥ 𝑚𝑀 ) ⟶ ( D c ( 𝐴 ) , Ϙ ≥ 𝑚𝑀 ) ⟶ ( D im(c) ( 𝐴 [ 𝑆 −1 ]) , Ϙ ≥ 𝑚𝑀 [ 𝑆 −1 ] ) , where the subscript 𝑆 in the source denotes the full subcategory of complexes whose homology is 𝑆 -torsion. This example will serve as the main input to obtain localisation sequences of Grothendieck-Witt spectrain §4.4.
Proof.
Note only that 𝐴 [ 𝑆 −1 ] is flat thus a derived localisation on account of the Ore condition, as theconstruction of 𝐴 [ 𝑆 −1 ] as one-sided fractions displays it as a filtered colimit of free 𝐴 op -modules of rank , so that Proposition 1.4.8 applies. (cid:3) The Ore condition is in fact often necessary to achieve flatness of the localisation: In [Tei03, Main The-orem] Teichner shows that if 𝑆 is the set of elements that become invertible modulo a two-sided ideal 𝐼 ,then flatness of 𝐴 [ 𝑆 −1 ] as a right 𝐴 -module is equivalent to 𝑆 being left Ore.Let us finally consider examples involving diagram ∞ -categories as constructed in §[I].6.3; this will berequired for our analysis of the hermitian Q -construction in the next section. For simplicity let us restrictout attention to the case of finite posets, where we recall that our convention for interpreting a poset as acategory is that 𝑖 ≤ 𝑗 means a morphism from 𝑖 to 𝑗 . Given a finite poset J , a full subposet I ⊆ J is said tobe a upwards closed if 𝑖, 𝑗 ∈ J are such that 𝑖 ∈ I and 𝑖 ≤ 𝑗 then 𝑗 ∈ I . In particular, if 𝑟 ∶ I ↪ J is upwardsclosed then for every 𝑖 ∈ I the functor I 𝑖 ∕ → J 𝑟 ( 𝑖 )∕ is an isomorphism and hence 𝑟 satisfies the conditionof Proposition [I].6.3.18. Given a Poincaré ∞ -category ( C , Ϙ ) we then have that the functor 𝑟 ∗ ∶ C J → C I commutes with the respective (possibly non-perfect) dualities. Thus, in the case where both ( C I , Ϙ I ) and ( C J , Ϙ J ) are Poincaré the hermitian functor(11) ( 𝑟 ∗ , 𝜂 ) ∶ ( C J , Ϙ J ) → ( C I , Ϙ I ) is Poincaré as well.1.4.10. Proposition.
Let 𝑟 ∶ I ↪ J be an upwards closed inclusion between finite posets, and let ( C , Ϙ ) bea Poincaré ∞ -category such that the hermitian ∞ -categories ( C I , Ϙ I ) and ( C J , Ϙ J ) are Poincaré. Then thePoincaré functor (11) is a split Poincaré-Verdier projection.Proof. A fully-faithful left adjoint to 𝑟 is given by the exact functor 𝑟 ! ∶ C I → C J performing left Kanextension. In fact, since 𝑟 is upwards closed this left Kan extension admits a very explicit formula: for a diagram 𝜑 ∶ I → C the value of 𝑟 ! 𝜑 is given by 𝑟 ! 𝜑 ( 𝑗 ) = { 𝜑 ( 𝑗 ) 𝑗 ∈ I 𝑗 ∉ I Since Ϙ (0) ≃ 0 the spectrum valued diagram 𝑗 ↦ Ϙ ( 𝑟 ! 𝜑 ( 𝑗 )) is then, for a similar reason, a right Kanextension of its restriction to I op , and so the natural map Ϙ J ( 𝑟 ! 𝜑 ) ≃ lim 𝑗 ∈ J op Ϙ ( 𝑟 ! 𝜑 ( 𝑗 )) → lim 𝑖 ∈ I op Ϙ ( 𝜑 ( 𝑖 )) = Ϙ I ( 𝜑 ) is an equivalence. The Poincaré functor (11) is then Poincaré-Verdier projection by Corollary 1.2.3 ii). (cid:3) In this situation of Proposition 1.4.10, one may also consider instead the hermitian ∞ -categories ( C I , Ϙ I ) and ( C J , Ϙ J ) obtained by applying the tensor construction instead of the cotensor construction. When I isa finite poset, the underlying ∞ -category C I identifies with Fun( I op , C ) by Lemma [I].6.5.6, and Ϙ I sendssuch a diagram 𝜑 ∶ I op → C to colim 𝑖 ∈ I Ϙ ( 𝜑 ( 𝑖 )) ; of course the same holds for J . In the case where ( C I , Ϙ I ) and ( C J , Ϙ J ) are both Poincaré and 𝑟 ∶ I ↪ J is an upwards closed inclusion, the induced hermitian functor(12) ( 𝑟 ∗ , 𝜂 ) ∶ ( C I , Ϙ I ) → ( C J , Ϙ J ) refining the right Kan extension functor 𝑟 ∗ is Poincaré by Proposition [I].6.5.13; we shall use the symbol 𝑟 ∗ even though the right Kan extension is taken along the functor 𝑟 op ∶ I op → J op .1.4.11. Proposition.
Let 𝑟 ∶ I → J be an upwards closed inclusion of finite posets, and let ( C , Ϙ ) be aPoincaré ∞ -category such that the hermitian ∞ -categories ( C I , Ϙ I ) and ( C J , Ϙ J ) are Poincaré. Then thePoincaré functor (12) is a split Poincaré-Verdier inclusion.Proof. We first note that the right Kan extension 𝑟 ∗ is fully-faithful (since 𝑟 is) and admits a left adjointgiven by restriction. To finish the proof it will suffice by Corollary 1.2.3 i) to show that for every diagram 𝜑 ∶ I op → C the composed map colim 𝑖 ∈ I Ϙ ( 𝜑 ( 𝑖 )) → colim 𝑖 ∈ I Ϙ ( 𝑟 ∗ 𝑟 ∗ 𝜑 ( 𝑖 )) → colim 𝑗 ∈ J Ϙ ( 𝑟 ∗ 𝜑 ( 𝑗 )) is an equivalence. Here the first map is an equivalence since 𝑟 is fully-faithful. To see that the second mapis an equivalence we argue as in the proof of Proposition 1.4.10 and observe that 𝑟 ∗ 𝜑 ( 𝑗 ) = { 𝜑 ( 𝑗 ) 𝑗 ∈ I 𝑗 ∉ I The spectrum valued diagram 𝑗 ↦ Ϙ ( 𝑟 ∗ 𝜑 ( 𝑗 )) is then, for a similar reason, the left Kan extension of itsrestriction to I , and so the second map above is an equivalence as well. (cid:3) We next consider Poincaré-Verdier projections involving the exceptional functoriality from Construc-tion [I].6.5.14. To this end let 𝛼 ∶ I → J be a cofinal map between finite posets. In this case the restrictionand right Kan extension maps ( 𝛼 op ) ∗ ∶ Fun( J op , C ) ⟶ Fun( I op , C ) and 𝛼 ∗ ∶ Fun( I , C ) ⟶ Fun( J , C ) acquire canonical hermitian structure upgrading them to functors 𝛼 ∗ ∶ ( C , Ϙ ) J ⟶ ( C , Ϙ ) I and 𝛼 ∗ ∶ ( C , Ϙ ) I ⟶ ( C , Ϙ ) J . Proposition.
Suppose that ( C , Ϙ ) is a Poincaré ∞ -category and 𝛼 ∶ I ↪ J is a cofinal and fullyfaithful inclusion of finite posets such that ( C , Ϙ ) I and ( C , Ϙ ) J are Poincaré. Then 𝛼 ∗ ∶ ( C , Ϙ ) J → ( C , Ϙ ) I isa split Poincaré-Verdier projection.Proof. To prove the first claim note that 𝛼 ∗ admits fully faithful left and right adjoints given by left andright Kan extension. It follows direclty from the explicit formula [D I ( 𝜑 )]( 𝑗 ) = colim 𝑖 ∈ I D( 𝜑 ( 𝑖 )) hom I ( 𝑖,𝑗 ) of Proposition [I].6.5.8, that 𝛼 ∗ preserves the dualities.By Proposition 1.2.3 we are left to show that for 𝜑 ∈ Fun( J op , C ) the natural map Ϙ I ( 𝛼 op! 𝜑 ) → Ϙ J (( 𝛼 op ) ∗ 𝛼 op! 𝜑 ) → Ϙ J ( 𝜑 ) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 37 is an equivalence, where 𝛼 ! denotes the left Kan extension functor. Indeed Ϙ I ( 𝛼 op! 𝜑 ) ≃ colim 𝑖 ∈ I Ϙ ( 𝛼 op! 𝜑 ( 𝑖 )) ≃ colim 𝑗 ∈ J Ϙ ( 𝛼 op! 𝜑 ( 𝛼 ( 𝑗 ))) ≅ colim 𝑗 ∈ J Ϙ ( 𝜑 ( 𝑗 )) , where we have used that 𝛼 ∶ J → I is cofinal and that ( 𝛼 op ) ∗ 𝛼 op! 𝜑 ≅ 𝜑 as a consequence of 𝛼 being fullyfaithful. (cid:3) Proposition.
Suppose that ( C , Ϙ ) is a Poincaré ∞ -category and 𝛼 ∶ I ↪ J a localisation amongfinite posets such that ( C , Ϙ ) I and ( C , Ϙ ) J are Poincaré. Assume furthermore that the hermitian functor 𝛼 ∗ is duality preserving. Then 𝛼 ∗ ∶ ( C , Ϙ ) I → ( C , Ϙ ) J is a split Poincaré-Verdier projection. Note that by [Cis19, Proposition 7.1.10] localisations are cofinal, so that 𝛼 ∗ is well-defined. Proof.
First up, restriction along 𝛼 is left adjoint to 𝛼 ∗ and fully faithful since 𝛼 is a localisation. Thus by1.1.6 we are left to prove that the natural map Ϙ J ( 𝜑 ) ⟶ Ϙ I ( 𝜑 ◦ 𝛼 ) is an equivalence. Indeed, Ϙ J ( 𝜑 ) ≃ lim 𝑗 ∈ J op Ϙ ( 𝜑 ( 𝑗 )) ≃ lim 𝑖 ∈ I op Ϙ ( 𝜑𝛼 ( 𝑖 )) ≃ Ϙ I ( 𝜑 ◦ 𝛼 ) since 𝛼 op is final by [Cis19, Proposition 7.1.10]. (cid:3) Finally, let us also record:1.4.14.
Proposition.
Given a (split) Poincaré-Verdier sequence ( C , Ϙ ) → ( C ′ , Ϙ ′ ) → ( C ′′ , Ϙ ′′ ) and a finiteposet I such that (−) I ∶ Cat p∞ → Cat h∞ preserves Poincaré ∞ -categories. Then the induced sequences ( C , Ϙ ) I ⟶ ( C ′ , Ϙ ′ ) I ⟶ ( C ′′ , Ϙ ′′ ) I and ( C , Ϙ ) I ⟶ ( C ′ , Ϙ ′ ) I ⟶ ( C ′′ , Ϙ ′′ ) I are (split) Poincaré-Verdier sequence. Note that by [I].6.5.12, the functor (−) I ∶ Cat h∞ → Cat h∞ preserves Cat p∞ provided (−) I does, which inturn is equivalent to ( S 𝑝 𝜔 , Ϙ u ) I being Poincaré. Proof.
Let us treat the tensoring, the argument for the cotensoring being entirely dual. As a left adjoint,the tensoring construction generally preserves colimits, and by [I].6.5.10 tensoring with a finite poset alsopreserves limits. This gives the part of the statement disregarding splittings. But for example from [I].6.5.8we find that the operation (−) I ∶ Cat ex∞ → Cat ex∞ preserves adjoints, which implies the split case. (cid:3)
Additive and localising functors.
In this section we establish the basic notions of additive, Verdier-localising and Karoubi-localising functors. They are based on a mild generalization of Poincaré-Verdier andPonicaré-Karoubi sequences in the form of certain bicartesian squares. Sending these particular bicarte-sian squares to bicartesian squares isolates the localisation properties enjoyed by Grothendieck-Witt theoryaxiomatically.In the present paper we focus almost exclusively on Verdier-localising (or even additive) functors. To-gether with the principal example of the Karoubi-Grothendieck-Witt functor, Karoubi-localising functorsare studied thoroughly in Paper [IV] and we only briefly mention them here for completeness’ sake.1.5.1.
Definition. A (split) Verdier square is a commutative square(13) C DC ′ D ′ in Cat ex∞ which is cartesian and whose vertical maps are (split) Verdier projections. We say that a square asin (13) is a
Karoubi square if it becomes cartesian in
Cat ex∞ , idem after applying completion and its verticalmaps are Poincaré-Karoubi projections. A (split) Poincaré-Verdier square is a commutative square(14) ( C , Ϙ ) ( D , Φ)( C ′ , Ϙ ′ ) ( D ′ , Φ ′ ) in Cat p∞ which is cartesian and whose vertical maps are (split) Poincaré-Verdier projections. We say thata square as in (14) is a Poincaré-Karoubi square if it becomes cartesian after applying the idempotentcompletion functor of Proposition 1.3.3 and its vertical maps are Poincaré-Karoubi projections.1.5.2.
Remarks. i) A (split) Poincaré-Verdier square with lower left corner p∞ is exactly a (split) Poincaré-Verdiersequence. The same holds for Poincaré-Karoubi sequences.ii) The classifying squares of Theorem 1.2.9 and Proposition A.2.11 give examples of split (Poincaré-)Verdier squares.iii) Any Poincaré-Verdier square is also cocartesian in Cat p∞ : Indeed, extend (14) to a commutative rec-tangle(15) ( E , Ϙ | E ) ( C , Ϙ ) ( D , Φ)0 ( C ′ , Ϙ ′ ) ( D ′ , Φ ′ ) in which both squares are cartesian and the vertical maps are Poincaré-Verdier projections. Then theexternal rectangle is cartesian by the pasting lemma, and hence cocartesian since the right verticalmap is a Verdier projection. For the same reason the left square is cocartesian and so the right squareis cocartesian by the pasting lemma. Similarly, every Poincaré-Karoubi square becomes cocartesianin Cat p∞ , idem after applying idempotent completion.iv) By Corollary 1.2.6 and Corollary A.2.7 the collection of split (Poincaré-)Verdier projections is closedunder pullback. Therefore a cartesian square in Cat p∞ is a split Poincaré-Verdier square if only its rightvertical leg is a split-Verdier projection. The same statement holds for general Poincaré-Verdier squaresby Lemma A.1.11, Remark 1.1.8 and Proposition A.3.15. The case of Poincaré-Karoubi squares fol-lows from this.v) Proposition 1.3.5 implies that every Poincaré-Verdier square is a Poincaré-Karoubi square. Conversely,a Poincaré-Karoubi square involving idempotent complete Poincaré ∞ -categories is a Poincaré-Verdiersquare if and only if its vertical maps are essentially surjective.The following are useful recognition criteria for (Poincaré)-Verdier squares:1.5.3. Lemma.
Consider a diagram
C C ′ D D ′ 𝑖𝑝 𝑝 ′ 𝑗 in Cat ex∞ such that 𝑝 and 𝑝 ′ are (split) Verdier projections. Then the square is a Verdier square if and onlyif the induced map ker( 𝑝 ) → ker( 𝑝 ′ ) is an equivalence and the square is adjointable.The same statement holds for a diagram ( C , Ϙ ) ( C ′ , Ϙ ′ )( D , Φ) ( D ′ , Φ ′ ) 𝑖𝑝 𝑝 ′ 𝑗 in Cat p∞ whose vertical maps are (split) Poincaré-Verdier projections, i.e. it is cartesian if and only if the in-duced map (ker( 𝑝 ) , Ϙ ) → (ker( 𝑝 ′ ) , Ϙ ′ ) is an equivalence and the underlying diagram of stable ∞ -categoriesis adjointable.Furthermore, for a diagram in Cat p∞ as above, left adjointability and right adjointability are equivalent. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 39 The adjointablity condition is not automatic: Consider for example the shear map C → C , ( 𝑐, 𝑐 ′ ) ↦ ( 𝑐, 𝑐 ⊕ 𝑐 ′ ) as a self-map of the Verdier projection pr ∶ C → C . It is, however, easily checked in practise, especiallyin the Poincaré setting: For example, for a square 𝐴 𝐴 ′ 𝐵 𝐵 ′ of E -rings, left adjointability of the square formed by the extension-of-scalars functors on compact modulesis equivalent to the natural map 𝐴 ′ ⊗ 𝐴 𝐵 → 𝐵 ′ being an equivalence. Under the assumption that the map ker( 𝑝 ) → ker( 𝑝 ′ ) is an equivalence, it is also easily checked equivalent to the condition that 𝑖 inducesequivalences Hom C ( 𝑥, 𝑐 ) → Hom C ′ ( 𝑖 ( 𝑥 ) , 𝑖 ( 𝑐 )) and Hom C ( 𝑐, 𝑥 ) → Hom C ′ ( 𝑖 ( 𝑐 ) , 𝑖 ( 𝑥 )) for all 𝑥 ∈ ker( 𝑝 ) and 𝑐 ∈ C . In this guise the non-hermitian part of Lemma 1.5.3 is directly verified byKrause in [Kra20, Lemma 3.9], whereas we will simply appeal to our classification of (Poincaré-)Verdierprojections. Proof.
We explicitly checked that Verdier squares are adjointable as part of Proposition A.3.15; this containsthe much simpler case of split Verdier squares. As mentioned for the more interesting converse we appeal tothe classification results for Verdier sequences: In the split case Proposition A.2.11 implies that both 𝑝 and 𝑝 ′ are pulled back from the same split Verdier projection, and thus from one another, since the classifyingmap of 𝑝 is that of 𝑝 ′ composed with the arrow D → D ′ by Proposition A.2.13. In the case of a Karoubiprojection the same argument can be made using Proposition A.3.14 and Proposition A.3.15 instead. Forthe case of general Verdier sequences we then immediately obtain that C ♮ ( C ′ ) ♮ D ♮ ( D ′ ) ♮𝑖𝑝 𝑝 ′ 𝑗 is a cartesian square and deduce that the map from C to the pullback in the original square is fully faithful. Itremains to show that it is essentially surjective. This follows immediately from Thomasson’s classificationof dense subcategories Theorem A.3.2, since Verdier projections induce exact sequences on K .The Poincaré case follows by the exact same argument using Proposition 1.2.11 and Corollary 1.3.16instead of Proposition A.2.13 and Proposition A.3.15.Finally, to see that the two adjointability conditions are equivalent in the Poincaré case, simply note that D Ϙ ∶ C op → C induces an equivalence Pro( C ) op ≃ Ind( C op ) ⟶ Ind( C ) and similarly for C ′ , D and D ′ . Since all functors in sight commute with the dualities, conjugation with theabove equivalence exchanges left and right adjoints, which gives the claim. (cid:3) We now come to the main definition of this subsection.1.5.4.
Definition.
Let E be an ∞ -category which admits finite limits and F ∶ Cat p∞ → E a functor. Recallthat F is said to be reduced if F (0) is a terminal object in E . We say that a reduced functor F is addi-tive , Verdier-localising or Karoubi-localising , if it takes split Poincaré-Verdier squares, arbitrary Poincaré-Verdier squares or Poincaré-Karoubi squares to cartesian squares, respectively.We shall denote the ∞ -categories of these functors by Fun add (Cat p∞ , E ) , Fun vloc (Cat p∞ , E ) , and Fun kloc (Cat p∞ , E ) , respectively. It follows from Remark 1.5.2 that there are inclusions
Fun kloc (Cat p∞ , E ) ⊆ Fun vloc (Cat p∞ , E ) ⊆ Fun add (Cat p∞ , E ) as full subcategories. We note that additive, Verdier-localising and Karoubi-localising invariants are closedin Fun(Cat p∞ , E ) under limits (which are computed pointwise), such as taking loops. Colimits on the otherhand are generally not computed pointwise (unless E is stable), and we shall see in the next section that the Q -construction implements suspension in the category Fun add (Cat p∞ , S ) , which is ultimately the reason forthe universal property of Grothendieck-Witt theory. Warning.
Here we follow the convention of the fifth author and Tamme to divorce the preservation of fil-tered colimits from the preservation of certain fibre sequences and squares. As a result, the ∞ -categoriesappearing in the end of Definition 1.5.4 are not locally small. The reader who is adverse to non-locally small ∞ -categories is invited to restrict attention only to accessible additive/Verdier-localising/Karoubi-localisingfunctors; this will not affect any of the statements in this paper, nor their proofs.We also note that if one fixes a regular cardinal 𝜅 and restricts attention only to those additive/Verdier-localising/Karoubi-localising functors that preserve 𝜅 -filtered colimits then the corresponding variants of Fun add (Cat p∞ , E ) , Fun vloc (Cat p∞ , E ) and Fun kloc (Cat p∞ , E ) become presentable, and a reader who so prefersmay fix at this moment once and for all a sufficiently large such 𝜅 . At any rate, the most interesting examplesof such functors that appear in this paper, such as the Grothendieck-Witt, K - and L -theory spectra, evenpreserve 𝜔 -filtered colimits.Any additive, Verdier-localising or Karoubi-localising functor sends split Poincaré-Verdier, Poincaré-Verdier or Poincaré-Karoubi sequences, respectively, to fiber sequences. If E is stable, the converse holdsas well:1.5.5. Proposition.
A reduced functor F ∶ Cat p∞ → E with E stable is additive, Verdier-localising orKaroubi-localising if and only if it takes split Poincaré-Verdier, Poincaré-Verdier or all Poincaré-Karoubisequences to exact sequences in E .Proof. Apply F to the rectangle in Remark 1.5.2 and use the pasting lemma. (cid:3) For non-stable E we expect, however, that the condition of being additive or Verdier-localising is strictlystronger than sending split Poincaré-Verdier or Poincaré-Verdier sequences to fiber sequences, and similarlyfor the condition of being Karoubi-localising. We will need the stronger variant in §2.5 with target S , whenwe discuss the additivity theorem for cobordism categories.1.5.6. Proposition.
A functor F ∶ Cat p∞ → E is Karoubi-localising if and only if it is Verdier-localising andinvariant under Karoubi equivalences.Proof. The “only if” part follows from Remark 1.5.2 and the fact that ( C , Ϙ ) 0( C , Ϙ ) ♮ forms a Poincaré-Karoubi square. The other direction follows from the fact that every Poincaré-Karoubisquare is Karoubi equivalent to a Poincaré-Verdier square: Assume that F is Verdier-localising and sendsKaroubi equivalences to equivalences. By definition of Karoubi squares it suffices to consider squares(16) ( C , Ϙ ) ( D , Φ)( C ′ , Ϙ ′ ) ( D ′ , Φ ′ ) , all of whose corners are idempotent complete and whose vertical legs are Poincaré-Karoubi projections.Letthen A ⊆ C ′ and B ⊆ D ′ be the essential images of the left and right vertical arrows, respectively, whichare invariant under the respective dualities since these vertical arrows are Poincaré. Furthermore, their ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 41 inclusions are Karoubi equivalences by Corollary 1.3.12. Since (16) is cartesian the full subcategory A coincides with the inverse image of B ⊆ D ′ and the square(17) ( C , Ϙ ) ( D , Φ)( A , Ϙ ′ | A ) ( B , Φ ′ | B ) is again cartesian. Finally, it follows from Corollary 1.3.10 that the vertical maps in (17) are Poincaré-Verdier projections, which gives the claim. (cid:3) Lemma.
The categories
Fun add (Cat p∞ , E ) , Fun vloc (Cat p∞ , E ) and Fun kloc (Cat p∞ , E ) are semi-additive and the forgetful functor Fun add (Cat p∞ , Mon E ∞ ( E )) ⟶ Fun add (Cat p∞ , E ) and its localising analogues are equivalences.Proof. Since the ∞ -category Cat p∞ is semi-additive the category of product preserving functors Cat p∞ → E is also semi-additive by [GGN15, Corollary 2.4]. But products of additive, Verdier or Karoubi localisingfunctors are again such, which implies the first statement. The second follows from [GGN15, Corollary 2.5iii)]. (cid:3) This allows us to set:1.5.8.
Definition.
An additive functor F ∶ Cat p∞ → E is called grouplike if the canonical lift of F to Mon E ∞ ( E ) actually takes values in the full subcategory Grp E ∞ ( E ) .Equivalently, this is the same as saying that for every Poincaré ∞ -category ( C , Ϙ ) it takes the shear map ( C , Ϙ ) × ( C , Ϙ ) → ( C , Ϙ ) × ( C , Ϙ ) (given at the level of objects by ( 𝑥, 𝑦 ) ↦ ( 𝑥, 𝑥 ⊕ 𝑦 ) ) to an equivalence in E .1.5.9. Remark. If E is additive then any additive functor F ∶ Cat p∞ → E is group-like, because both forgetfulfunctors Mon E ∞ ( E ) → E and Grp E ∞ ( E ) → E are equivalences.1.5.10. Examples.
The functors Cr and Pn ∶ Cat p∞ → S are Verdier-localising since, by virtue of being rep-resentable, they preserve all limits. They are not grouplike. The functors K ∶ Cat p∞ → S and K ∶ Cat p∞ → S 𝑝 , which associate to a Poincaré ∞ -category the algebraic K -theory space or spectrum of its underlyingstable ∞ -category are Verdier-localising and group-like; this essentially follows from Waldhausen’s addi-tivity and fibration theorems, as implemented in the setting of stable ∞ -categories by Blumberg-Gepner-Tabuada [BGT13], we will review the situation in §2.6. Similarly, the functor 𝕂 ∶ Cat p∞ → S 𝑝 which asso-ciates to a Poincaré ∞ -category the nonconnective K -theory spectrum of its underlying stable ∞ -categoryis localizing by [BGT13]. The functor K ◦ (−) ♮ (where (−) ♮ is the idempotent completion functor of Propo-sition 1.3.3) is an example of an additive, but non-Verdier-localising, functor.Finally, we record the following simple consequence of the splitting lemma:1.5.11. Proposition.
Let (18) ( C , Ϙ ) 𝑖 ⟶ ( C ′ , Ϙ ′ ) 𝑝 ⟶ ( C ′′ , Ϙ ′′ ) be a (split) Poincaré-Verdier sequence and let F ∶ Cat p∞ ⟶ E be a grouplike (additive or) Verdier-localising functor. Assume that the Verdier projection ( C ′ , Ϙ ′ ) ⟶ ( C ′′ , Ϙ ′′ ) admits a section 𝑠 ∶ ( C ′′ , Ϙ ′′ ) ⟶ ( C ′ , Ϙ ′ ) in Cat p∞ . Then 𝑖 and 𝑠 together induce an equivalence (19) F ( C , Ϙ ) ⊕ F ( C ′′ , Ϙ ′′ ) ⟶ F ( C ′ , Ϙ ′ ) . If, in addition, the Poincaré functor 𝑖 admits a retraction 𝑟 ∶ ( C ′ , Ϙ ′ ) ⟶ ( C , Ϙ ) in Cat p∞ then 𝑝 and 𝑟 togetherinduce an equivalence (20) F ( C ′ , Ϙ ′ ) ⟶ F ( C , Ϙ ) ⊕ F ( C ′′ , Ϙ ′′ ) . This equivalence is inverse to (19) when 𝑟 ◦ 𝑠 is the zero Poincaré functor. Note, that since
Cat p∞ is only semi-additive, but not additive, the middle term in a Poincaré-Verdiersequence admitting a Poincaré split as above, need not split as a direct sum before applying F .The proof of Proposition 1.5.11 relies on the following version of the classical splitting lemma [Mac67,Proposition I.4.3] from homological algebra (it should be considered standard, but we were not able tolocate a reference).1.5.12. Lemma.
Let A be an additive ∞ -category which admits fibers and cofibers and let 𝑥 𝑖 ⟶ 𝑦 𝑟 ⟶ 𝑥 be a retract diagram. Then the following statement hold:i) The maps 𝑖 ∶ 𝑥 ⟶ 𝑦 and f ib( 𝑟 ) ⟶ 𝑦 induce an equivalence 𝑥 ⊕ f ib( 𝑟 ) ⟶ 𝑦 .ii) The maps 𝑟 ∶ 𝑦 ⟶ 𝑥 and 𝑦 ⟶ cof( 𝑖 ) induce an equivalence 𝑦 ⟶ 𝑥 ⊕ cof ( 𝑖 ) .iii) The fiber sequence f ib( 𝑟 ) ⟶ 𝑦 ⟶ 𝑥 is also a cofiber sequence.iv) The cofiber sequence 𝑥 ⟶ 𝑦 ⟶ cof( 𝑖 ) is also a fiber sequence.v) The composite map f ib( 𝑟 ) ⟶ 𝑦 ⟶ cof ( 𝑖 ) is an equivalence.Proof. We first note that ii) and iv) follow from i) and iii), respectively, applied to the additive ∞ -category A op . To prove i), it is actually enough to argue at the level of the homotopy category. To see this, observethat for every 𝑧 we have a fiber sequence of spaces Map A ( 𝑧, f ib( 𝑟 )) ⟶ Map A ( 𝑧, 𝑦 ) ⟶ Map A ( 𝑧, 𝑥 ) . Since 𝑟 admits a section the map 𝜋 Map E ( 𝑧, 𝑦 ) ⟶ 𝜋 Map E ( 𝑧, 𝑥 ) is surjective and hence the long exactsequence in homotopy groups ends with a fiber sequence 𝜋 Map A ( 𝑧, f ib( 𝑟 )) ⟶ 𝜋 Map A ( 𝑧, 𝑦 ) ⟶ 𝜋 Map A ( 𝑧, 𝑥 ) . of sets. This means that f ib( 𝑟 ) is also the fiber of 𝑟 in the homotopy category Ho( A ) . Now since productsand coproducts descend to Ho( A ) we have that Ho( A ) is additive and the functor A ⟶ Ho( A ) preservesdirect sums. It will hence suffice to show that i) holds for Ho( A ) , which is the classical splitting lemma(see, e.g., [Bor94, Proposition 1.8.7]); the splitting lemma is usually phrased for abelian categories only,but the proof from loc.cit. works verbatim in the additive case. Alternatively, it can be deduced from theablian case by embedding Ho( A ) into its abelian envelope.Let us prove iii). Note that i) provides us in particular with a retraction 𝑦 ⟶ f ib( 𝑟 ) which vanisheswhen restricted to 𝑥 . We may then consider the resulting commutative diagram 𝑟 ) 0 𝑥 𝑦 𝑥 𝑟 ) 0 𝑖 𝑟 in which the middle row and middle column are retract diagrams. By i) the top left square is cocartesianand hence by the pasting lemma the top right square is cocartesian as well. This gives iii).To obtain v) use the pasting lemma to deduce that the bottom left square is cocartesian, which induces anequivalence f ib( 𝑟 ) ⟶ cof( 𝑖 ) . But this map is the same as the one obtained from the composition f ib( 𝑟 ) ⟶ 𝑦 ⟶ cof( 𝑖 ) because the f ib( 𝑟 ) ⟶ 𝑦 ⟶ f ib( 𝑟 ) is a retract diagram. (cid:3) Proof of Proposition 1.5.11.
To obtain the first equivalence (19) we apply part i) of the splitting lemma1.5.12 to the retract diagram F ( C ′′ , Ϙ ′′ ) 𝑠 ∗ ←←←←←←←←←→ F ( C ′ , Ϙ ′ ) 𝑝 ∗ ←←←←←←←←←←→ F ( C ′′ , Ϙ ′′ ) in the additive ∞ -category Grp E ∞ ( E ) and identify the fiber of 𝑝 ∗ with F ( C , Ϙ ) . By Part iii) of the samelemma it follows that the fiber sequence F ( C , Ϙ ) 𝑖 ∗ ⟶ F ( C ′ , Ϙ ′ ) 𝑝 ∗ ⟶ F ( C ′′ , Ϙ ′′ ) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 43 is also a cofiber sequence in A . The second equivalence (20) then follows from Part ii) of the splittinglemma applied the retract diagram F ( C , Ϙ ) 𝑖 ∗ ←←←←←←←←→ F ( C ′ , Ϙ ′ ) 𝑟 ∗ ←←←←←←←←←→ F ( C , Ϙ ) , after identifying F ( C , Ϙ ) with the cofiber of 𝑖 ∗ with F ( C ′′ , Ϙ ′′ ) using the above. To see the final statementnote that two equivalences are inverse to each other if and only if they are one-sided inverses. Composingin one direction we get the functor(21) F ( C , Ϙ ) ⊕ F ( C ′′ , Ϙ ′′ ) → F ( C , Ϙ ) ⊕ F ( C ′′ , Ϙ ′′ ) whose “matrix components” are ( id 𝑟 ∗ 𝑠 ∗ ) , and so (21) is homotopic to the identity as soon as 𝑟 ◦ 𝑠 is thezero Poincaré functor. (cid:3)
2. T
HE HERMITIAN Q - CONSTRUCTION AND ALGEBRAIC COBORDISM CATEGORIES
In this section we introduce the main objects of study, namely the cobordism category constructed froma Poincaré ∞ -category. To motivate our perspective let ( C , Ϙ ) be a Poincaré ∞ -category and ( 𝑥, 𝑞 ) , ( 𝑥 ′ , 𝑞 ′ ) be two Poincaré objects in C . A cobordism from ( 𝑥, 𝑞 ) to ( 𝑥 ′ , 𝑞 ′ ) is a span of the form 𝑥 𝛼 ←←←←←←← 𝑤 𝛽 ←←←←←←→ 𝑥 ′ together with a path 𝜂 ∶ 𝛼 ∗ 𝑞 → 𝛽 ∗ 𝑞 ′ in the space Ω ∞ Ϙ ( 𝑤 ) of hermitian structures on 𝑤 , such that 𝑤 satisfiesthe Poincaré-Lefschetz condition with respect to 𝑥 and 𝑥 ′ , i.e. that the canonical map(22) f ib( 𝑤 → 𝑥 ) ≃ f ib( 𝑥 ′ → 𝑥 ∪ 𝑤 𝑥 ′ ) → f ib( 𝑥 ′ → D Ϙ 𝑤 ) ≃ ΩD Ϙ (f ib( 𝑤 → 𝑥 ′ )) , is an equivalence; here the middle map is induced by the map 𝑤 → D Ϙ 𝑥 × D Ϙ 𝑤 D Ϙ 𝑥 ′ provided by 𝜂 and the condition above can also be phrased as asking this map to be an equivalence.For example, if 𝑊 is an oriented cobordism between two 𝑑 -manifolds 𝑀 and 𝑁 we obtain a span of theform 𝐶 ∗ ( 𝑀 ) ← 𝐶 ∗ ( 𝑊 ) → 𝐶 ∗ ( 𝑁 ) and the fundamental class [ 𝑊 ] determines a path relating the pullbacks of the two symmetric Poincaréstructures 𝑞 𝑀 and 𝑞 𝑁 on 𝐶 ∗ ( 𝑀 ) and 𝐶 ∗ ( 𝑁 ) , respectively. Lefschetz duality for oriented manifolds preciselyimplies that this path exhibits the span as a cobordism between the Poincaré objects ( 𝐶 ∗ ( 𝑀 ) , 𝑞 𝑀 ) and ( 𝐶 ∗ ( 𝑁 ) , 𝑞 𝑁 ) of ( D p ( ℤ ) , Ϙ 𝑠 ℤ [− 𝑑 ] ) in the sense above.Now, cobordisms can be composed in a natural way, by first forming the corresponding composition atthe level of spans and then at the level of the paths between hermitian structures. This will allow us to definean ∞ -category Cob( C , Ϙ ) whose objects are the Poincaré objects of ( C , Ϙ [1] ) and whose morphisms are givenby cobordisms; the choice in shifts adheres to the usual convention from manifold theory that the category Cob 𝑑 have ( 𝑑 − 1) -dimensional closed manifolds as objects and 𝑑 -dimensional cobordisms as morphisms.To make this idea precise, we interpret a cobordism in ( C , Ϙ ) as a Poincaré object in the diagram category (Fun( 𝑃 , C ) , Ϙ 𝑃 ) , where 𝑃 is the category ∙ ← ∙ → ∙ , and Ϙ 𝑃 is the Poincaré structure on the diagram categorygiven by the limit of the values of Ϙ on the diagram. This construction turns out to be the degree partof a simplicial Poincaré ∞ -category Q( C , Ϙ ) , whose Poincaré objects in degree 𝑛 may be interpreted as thedatum of 𝑛 composable tuples of cobordisms. Varying ( C , Ϙ ) this construction gives rise to a functor Q ∶ Cat p∞ → sCat p∞ , our implementation of the hermitian Q -construction, see §2.1. By considering the spaces of Poincaré objectsof these diagram categories we will therefore obtain a complete Segal space and then extract Cob( C , Ϙ ) ∈Cat ∞ as the associated category in §2.2.Then we develop the two main tools that will allow us to analyse this cobordism category and its homo-topy type. First, we show how to describe the cobordism category using Ranicki’s algebraic surgery tech-niques from [Ran80], adapted to the setting of Poincaré ∞ -categories by Lurie in [Lur11]. Beside its uses in the present paper, this serves as a fundamental tool in [HS20] to compare our definition of Grothendieck-Witt theory with the classical L -, Witt and Grothendieck-Witt groups, and is also used extensively in Paper[III]. The second topic, in §2.4 and §2.5, is the additivity theorem, which says that the functor | Cob − | = | Pn Q(−) | ∶ Cat p∞ → S ; is additive. This will be the basis for most of the structural results we prove about Grothendieck-Witt theory.As far as the additivity theorem is concerned, the only property of the functor Pn that enters the proof,is that is itself is additive. In fact, we will show that the functor | F Q(−) | ∶ Cat p∞ → S is additive whenever F ∶ Cat p∞ → S is additive. This added layer of generality will be used to establish theadditivity of the Grothendieck-Witt functor, defined via iteration of the hermitian Q -construction, and alsoenters into the proof of its universal property.Finally, in §2.6 we explain how our methods give rise to a new proof of the more classical additivitytheorem for the algebraic 𝐾 -theory of stable ∞ -categories.2.1. The hermitian Q -construction. Let 𝐾 be a poset and ( C , Ϙ ) a hermitian ∞ -category.2.1.1. Definition.
Let Q 𝐾 ( C , Ϙ ) denote the following hermitian ∞ -category: The underlying stable ∞ -category is given as the full subcategory Q 𝐾 ( C ) of Fun(TwAr( 𝐾 ) , C ) spanned by those functors 𝐹 suchthat for every 𝑖 ≤ 𝑗 ≤ 𝑘 ≤ 𝑙 ∈ 𝐾 the square 𝐹 ( 𝑖 ≤ 𝑙 ) 𝐹 ( 𝑗 ≤ 𝑙 ) 𝐹 ( 𝑖 ≤ 𝑘 ) 𝐹 ( 𝑗 ≤ 𝑘 ) is bicartesian. The hermitian structure is given by restricting the quadratic functor Ϙ TwAr( 𝐾 ) ( 𝐹 ) = lim TwAr( 𝐾 ) op Ϙ ◦ 𝐹 op from Proposition [I].6.3.2.When 𝐾 = Δ 𝑛 we will shorten notation and denote Q 𝐾 ( C , Ϙ ) by Q 𝑛 ( C , Ϙ ) and Ϙ Δ 𝑛 by Ϙ 𝑛 . Also bydefinition the hermitian ∞ -category (Fun(TwAr( 𝐾 ) , C ) , Ϙ TwAr( 𝐾 ) ) is the cotensor ( C , Ϙ ) TwAr( 𝐾 ) , in the senseof §[I].6.3. It is usually not Poincaré, while Q 𝑛 ( C , Ϙ ) is, as we will see below.2.1.2. Remark.
By the pasting lemma for cartesian squares, see [Lur09a, Lemma 4.4.2.1], we find that inorder to establish the condition in Definition 2.1.1 for all 𝑖 ≤ 𝑗 ≤ 𝑘 ≤ 𝑙 to suffices to check the case 𝑗 = 𝑘 .2.1.3. Examples. i) We have
TwAr(Δ ) = ∙ ← ∙ → ∙ , so Q ( C ) is simply the category of spans in C , with no conditionimposed. Using Proposition [I].6.3.2, the duality on Q ( C , Ϙ ) is given by the rule ( 𝑋 ← 𝑌 → 𝑍 ) ↦ ( D Ϙ 𝑋 ⟵ D Ϙ 𝑋 × D Ϙ 𝑌 D Ϙ 𝑍 ⟶ D Ϙ 𝑍 ) . Following our explanation above, we interpret Q ( C , Ϙ ) as the category of cobordisms in ( C , Ϙ ) .ii) Q ( C ) consists of those diagrams 𝐹 (0 ≤ 𝐹 (0 ≤ 𝐹 (1 ≤ 𝐹 (0 ≤ 𝐹 (1 ≤ 𝐹 (2 ≤ in which the top square is bicartesian. It is therefore reasonable to think of Q ( C , Ϙ ) as the category oftwo composable cobordisms equipped with a chosen composite. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 45 iii) By i), the functor 𝑑 ∶ Q ( C , Ϙ ) → Q ( C , Ϙ ) = ( C , Ϙ ) , ( 𝑋 ← 𝑌 → 𝑍 ) ↦ 𝑋 is duality-preserving so that its kernel is closed under the duality of Q ( C , Ϙ ) , and therefore a Poincaré ∞ -category with the restricted Poincaré structure. In fact, there is a canonical equivalence of Poincaré ∞ -categories ker( 𝑑 ) ≃ Met( C , Ϙ ) that sends ← 𝑤 → 𝑐 to 𝑤 → 𝑐 .iv) We note that for the category I 𝑛 ⊆ TwAr(Δ 𝑛 ) spanned by the pairs ( 𝑖, 𝑗 ) with 𝑗 ≤ 𝑖 + 1 (the zig-zagalong the bottom) the restriction functor Q 𝑛 ( C , Ϙ ) → (Fun( I 𝑛 , C ) , Ϙ I 𝑛 ) = ( C , Ϙ ) I 𝑛 is an equivalence of hermitian ∞ -categories: On underlying categories, it follows from [Lur09a,Proposition 4.3.2.15], that the right Kan extension functor Fun( I 𝑛 , C ) → Fun(TwAr(Δ 𝑛 ) , C ) is bothfully faithful and a left inverse to restriction. For 𝑋 ∈ Fun(TwAr(Δ 𝑛 ) , C ) it is then readily checkedfrom the pointwise formulae [Lur09a, Lemma 4.3.2.13] that being in Q 𝑛 ( C ) is equivalent to being rightKan extended from I 𝑛 . For the quadratic functor it follows since the inclusion I op 𝑛 ⊆ TwAr(Δ 𝑛 ) op is final. By Remark [I].6.5.18 the arising hermitian structure on the right Kan extension functor Fun( I 𝑛 , C ) → Fun(TwAr(Δ 𝑛 ) , C ) is an instance of the exceptional functoriality of Construction [I].6.5.14.This description justifies us in thinking of Q 𝑛 ( C , Ϙ ) as the category of 𝑛 composable cobordisms in ( C , Ϙ ) also for larger 𝑛 .v) There is another description of the category underlying Q 𝑛 ( C , Ϙ ) : Letting J 𝑛 ⊆ TwAr(Δ 𝑛 ) denote thesubset of those ( 𝑖, 𝑗 ) with either 𝑖 = 0 or 𝑗 = 𝑛 (the arch along the top), the restriction functor Q 𝑛 ( C ) → Fun( J 𝑛 , C ) is also an equivalence: A functor 𝐹 ∶ TwAr(Δ 𝑛 ) → C is in Q 𝑛 ( C ) if and only if it is left Kan extendedfrom J 𝑛 . However, this equivalence does not translate the quadratic functor Ϙ 𝑛 into Ϙ J 𝑛 , once 𝑛 ≥ .For example, for the element 𝑋 id 𝑋 ←←←←←←←←←←←←←←← 𝑋 ← → 𝑌 id 𝑌 ←←←←←←←←←←←←←→ 𝑌 in Fun( J , C ) we find Ϙ J 𝑛 given by Ϙ ( 𝑋 ) ⊕ Ϙ ( 𝑌 ) , where as Ϙ yields Ϙ ( 𝑋 ⊕ 𝑌 ) , and these two termsdiffer by B Ϙ ( 𝑋, 𝑌 ) . In fact, (Fun( J 𝑛 , C ) , Ϙ J 𝑛 ) is not Poincaré, whereas we will next establish this for Q 𝑛 ( C , Ϙ ) .Denoting the category of finite posets by Posets we thus obtain a functorPosets op × Cat h∞ → Cat h∞ , ( 𝐾, C , Ϙ ) ↦ Q 𝐾 ( C , Ϙ ) , from Proposition [I].6.3.11, since clearly induced maps perserve the cartesianness condition of Defini-tion 2.1.1. Restricting along the inclusion Δ ⊆ Posets and adjoining the construction above we thus obtaina simplicial object Q( C , Ϙ ) ∈ sCat h∞ .2.1.4. Definition.
We call the functor
Q ∶ Cat h∞ → sCat h∞ just described the hermitian Q -construction .We immediately note that the underlying category of Q 𝑛 ( C , Ϙ ) only depends on C , and agrees withBarwick-Rognes’ implementation Q 𝑛 ( C ) of the Q -construction, see [BR13, §3] upon restricting their setupto stable ∞ -categories.The following is at the heart of the present section:2.1.5. Lemma.
For every hermitian ∞ -category ( C , Ϙ ) the simplical hermitian ∞ -category Q( C , Ϙ ) is aSegal object of Cat h∞ . Furthermore, it is complete in the sense that the diagram Q ( C , Ϙ ) Q ( C , Ϙ )Q ( C , Ϙ ) Q ( C , Ϙ ) 𝑠 Δ ( 𝑑 ,𝑑 )( 𝑠,𝑠 ) is cartesian in Cat h∞ , with horizontal maps given by total degeneracies. Proof.
We need to show that for every ≤ 𝑖 ≤ 𝑛 the square Q 𝑛 C Q [0 ,𝑖 ] C Q [ 𝑖,𝑛 ] C Q [ 𝑖,𝑖 ] C is a pullback square of Poincaré ∞ -categories. This will follow readily from Example 2.1.3 iv). To this end,note that the inclusions TwAr(Δ 𝑖 ) → TwAr(Δ 𝑛 ) and TwAr(Δ { 𝑖, … ,𝑛 } ) → TwAr(Δ 𝑛 ) take the subcategories I 𝑖 and I [ 𝑖, … ,𝑛 ] to I 𝑛 , and in fact the induced diagram I [ 𝑖,𝑖 ] I [ 𝑖,𝑛 ] I 𝑖 I 𝑛 is readily checked to be cocartesian in Cat ∞ , thus cartesian in Cat op∞ . But the functor
Cat op∞ → Cat p∞ , 𝐼 ↦ (Fun( 𝐼, C ) , Ϙ 𝐼 ) being a right adjoint preserves limits, whence we obtain the first claim.To see that Q( C , Ϙ ) is complete, we first show the claim on underlying stable ∞ -categories as follows:Limits in Cat ex∞ may be computed in
Cat ∞ (as limits in Cat ∞ of diagrams of stable ∞ -categories, are easilychecked to be stable again), so the map 𝑃 → Q ( C ) from the pullback 𝑃 of the diagram Q ( C ) → Q ( C ) ← Q ( C ) is fully faithful, since the degeneracy Q ( C ) → Q ( C ) is, and fully faithful functors are stable under pullback.Its essential image is given by the diagrams consisting entirely of equivalences, as one can check directlyusing the defining property of the Q -construction, and these are precisely the constant diagrams, i.e., thetotally degenerate ones.The claim for the hermitian structure is immediate from Remark [I].6.1.3, since the diagram Ϙ ( 𝑋 ) → Ϙ ( 𝑠𝑋 ) ← Ϙ ( 𝑠𝑋 ) , whose pullback defines the hermitian structure on 𝑃 , evaluates to Ϙ ( 𝑋 ) ←←←←←←←←→ Ϙ ( 𝑋 ) ←←←←←←←←← Ϙ ( 𝑋 ) , so has pullback Ϙ ( 𝑋 ) . (cid:3) We next show:2.1.6.
Lemma.
The functor
Q ∶ Cat h∞ → sCat h∞ restricts to a functor Cat p∞ → sCat p∞ . In particular, Q( C , Ϙ ) is a complete Segal object of Cat p∞ , whenever ( C , Ϙ ) is Poincaré. Lemma.
For ( C , Ϙ ) a Poincaré ∞ -category all face maps in Q( C , Ϙ ) , and more generally all mapsinduced by injections in Δ , are split Poincaré-Verdier projections.Proof of Lemmata 2.1.6 & 2.1.7. There are two good approaches to the statements. Either, one directlyattacks them using the machinery developed in §[I].6.6, or one reduces the statement to explicit checks forsmall values of 𝑛 using the Segal condition. At the cost of being less elementary, we will here use the formerroute as it leads to shorter proofs.That the categories Q 𝑛 ( C , Ϙ ) are Poincaré follows immediately from Proposition [I].6.6.1 and Exam-ples 2.1.3, since I 𝑛 is the poset of faces for the triangulation of the interval using 𝑛 + 1 -vertices.To see that the induced hermitian functors 𝛼 ∗ ∶ Q 𝑛 ( C , Ϙ ) → Q 𝑚 ( C , Ϙ ) for 𝛼 ∶ Δ 𝑚 → Δ 𝑛 preserve thedualities, we distinguish two cases, namely the inner face maps on the one hand, and the outer face mapsand degeneracies on the other. Since every morphism in Δ can be written as a composition of such, thiswill suffice for the claim.The latter maps all take the subset I 𝑚 ⊆ TwAr(Δ 𝑚 ) into I 𝑛 , and the restriction is induced by a map of thesimplicial complexes giving rise to I 𝑚 and I 𝑛 . Thus Proposition [I].6.6.2 gives the claim. The interior faces ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 47 do not preserves the subsets I 𝑚 , however. Instead, we claim that they are instances of the exceptional func-toriality of Construction [I].6.5.14 associated to a refinement among triangulations. Namely, one readilychecks that 𝑑 𝑖 ∶ TwAr(Δ 𝑛 ) → TwAr(Δ 𝑛 +1 ) admits a right adjoint 𝑟 𝑖 ∶ TwAr(Δ 𝑛 +1 ) → TwAr(Δ 𝑛 ) explicitlygiven by ( 𝑘 ≤ 𝑙 ) ⟼ ⎧⎪⎪⎨⎪⎪⎩ ( 𝑘 ≤ 𝑙 ) 𝑙 < 𝑖 or 𝑘 < 𝑙 = 𝑖 ( 𝑘 − 1 ≤ 𝑙 ) 𝑘 = 𝑙 = 𝑖 ( 𝑘 ≤ 𝑙 − 1) 𝑘 < 𝑖 < 𝑙 ( 𝑘 − 1 ≤ 𝑙 − 1) 𝑖 ≤ 𝑘 < 𝑙 or 𝑖 < 𝑘 = 𝑙 As a right adjoint 𝑟 𝑖 is cofinal, so by Example [I].6.5.15 the pullback functor ( 𝑑 𝑖 ) ∗ ∶ ( C , Ϙ ) TwAr(Δ 𝑛 +1 ) ⟶ ( C , Ϙ ) TwAr(Δ 𝑛 ) agrees with the exceptional functoriality along 𝑟 𝑖 . From the explicit formula it is clear that 𝑟 𝑖 takes I 𝑛 +1 into I 𝑛 , so we find a commutative square ( C , Ϙ ) I 𝑛 +1 ( C , Ϙ ) I 𝑛 ( C , Ϙ ) TwAr(Δ 𝑛 +1 ) ( C , Ϙ ) TwAr(Δ 𝑛 )( 𝑟 𝑖 ) ∗ ( 𝑟 𝑖 ) ∗ where vertical maps are the exceptional functorialities associated to the inclusions I 𝑛 ⊆ TwAr(Δ 𝑛 ) whichare also cofinal (the diagram commutes since exceptional functorialities compose by Remark [I].6.5.17).But the vertical maps are equivalences onto Q 𝑛 ( C , Ϙ ) by Example 2.1.3 iv). The claim now follows fromProposition [I].6.6.2, since the restriction of 𝑟 𝑖 to I 𝑛 +1 → I 𝑛 comes from the refinement of triangulation ofthe interval that adds a new 𝑖 th vertex.This completes the proof of Lemma 2.1.6. To show Lemma 2.1.7, we only need to consider face maps,since split Poincaré-Verdier projections are stable under composition by the characterisation in Corol-lary 1.1.6. For the inner faces this is immediate from Proposition 1.4.13, since 𝑟 𝑖 ∶ I 𝑛 +1 → I 𝑛 is evidently alocalisation at the edges ( 𝑖 − 1 ≤ 𝑖 ) → ( 𝑖 ≤ 𝑖 ) and ( 𝑖 ≤ 𝑖 + 1) → ( 𝑖 ≤ 𝑖 ) . For the outer faces it is an instanceof Proposition 1.4.10. (cid:3) Remark. If ( C , Ϙ ) is a commutative algebra in Cat p∞ with respect to the symmetric monoidal structureconstructed in §[I].5.2, then each Q 𝑛 ( C , Ϙ ) inherits such a structure again; however these structures are not compatible with the simplicial structure.2.2. The cobordism category of a Poincaré ∞ -category. We now proceed to extract the cobordism cat-egory from the hermitian Q -construction. As mentioned in the introduction it will be useful to do this inthe generality of an arbitrary additive F ∶ Cat p∞ → S , but the reader is encouraged to envision F = Pn throughout.2.2.1. Proposition.
Let ( C , Ϙ ) be a Poincaré ∞ -category and F ∶ Cat p∞ → S an additive functor. Then F Q( C , Ϙ ) is a Segal space and if, furthermore, F preserves arbitrary pullbacks, it is complete. When F is the functor Cr ∶ Cat p∞ → S completeness was established in [BR13, 3.4 Proposition] bydifferent means. For arbitrary additive F , the Segal space F Q( C , Ϙ ) is in general not complete. For example,if F is grouplike, then F Q( C , Ϙ ) is complete if and only if F Hyp( C ) ≃ 0 , see Remark 3.2.17. Proof.
For the first part we need to show that F Q 𝑛 ( C , Ϙ ) F Q [0 ,𝑖 ] ( C , Ϙ ) F Q [ 𝑖,𝑛 ] ( C , Ϙ ) F Q [ 𝑖,𝑖 ] ( C , Ϙ ) is cartesian for every ≤ 𝑖 ≤ 𝑛 . But before applying F the square is a Poincaré-Verdier square by Lem-mas 2.1.7 and 2.1.5, and by assumption F preserves the cartesianness of such squares. The assertion on completeness is immediate from the final part of Lemma 2.1.5 (see [Lur09b, Proposition1.1.13] for the characterisation of completeness used here). (cid:3)
The results of Rezk [Rez01] (suitably reformulated in [Lur09b, §1]) therefore allow us to extract an ∞ -category from F Q( C , Ϙ ) . Let us briefly recall the relevant facts.Rezk constructed an adjoint pair of functors asscat ∶ s S ⟂ Cat ∞ ∶ N with the Rezk nerve as right adjoint, given by the simplicial space N( C ) 𝑛 = 𝜄 Fun(Δ 𝑛 , C ) , and left adjoint given by left Kan extending the cosimplicial category Δ − ∶ Δ ⟶ Cat ∞ along the Yoneda embedding Δ → s S . By [Lur09b, Corollary 4.3.16], the nerve is fully faithful withessential image the complete Segal spaces cSS ⊆ s S , in particular making Cat ∞ a left Bousfield localisationof s S . Consequently, there is also a left adjoint comp∶ s S → cSS to the inclusion, often referred to ascompletion, and composing adjoints we find asscat ◦ comp = asscat .Furthermore, the restriction of the nerve functor to S ⊂ Cat ∞ is given by the inclusion of constantdiagrams S → s S and passing to adjoints again shows that | asscat 𝑋 | ≃ | 𝑋 | for every simplicial space 𝑋 .In particular, 𝜋 | asscat 𝑋 | is always the coequaliser of the two boundary maps 𝜋 𝑋 → 𝜋 𝑋 .Furthermore, Rezk showed in [Rez01, §14], see also [Lur09b, Proposition 1.2.27], that for any Segalspace 𝑋 the natural map 𝑋 → comp 𝑋 induces equivalences from the fibres of ( 𝑑 , 𝑑 ) ∶ 𝑋 → 𝑋 × 𝑋 to the same expression for comp 𝑋 . For 𝑋 = N C this fibre, say over ( 𝑥, 𝑦 ) , is given by Hom C ( 𝑥, 𝑦 ) . Wetherefore find that for any Segal space 𝑋 and any 𝑥, 𝑦 ∈ 𝑋 we have a canonical equivalence Hom asscat 𝑋 ( 𝑥, 𝑦 ) ≃ f ib ( 𝑥,𝑦 ) ( 𝑋 → 𝑋 × 𝑋 ) . We will also have to use that the completion functor commutes with finite products, when restricted toSegal spaces. This follows immediately from [Lur09b, Proposition 1.2.27] since Segal equivalences areevidently closed under finite products.Note finally that, for every simplicial space 𝑋 , the inclusion of -simplices induces a natural map 𝑋 → 𝜄 (asscat 𝑋 ) on associated categories, which is a surjection on 𝜋 by [Lur09b, Remark 1.2.17]. For 𝑋 a Segal space, itis an equivalence if and only if 𝑋 is complete.2.2.2. Definition.
Let
Cob F ( C , Ϙ ) denote the category associated to the Segal space F Q( C , Ϙ [1] ) . We shallwrite Cob( C , Ϙ ) for Cob Pn ( C , Ϙ ) and call it the cobordism category of ( C , Ϙ ) . Furthermore, we set Cob 𝜕 ( C , Ϙ ) =Cob(Met( C , Ϙ [1] )) , the cobordism category with boundaries .We shall refer to Cob F ( C , Ϙ ) as the F -based cobordism category and hope the two possible superscripts( F and 𝜕 ) will not lead to confusion. By the functoriality of the Q -construction and the previous discussionthe construction of these categories assemble into a functor Fun add (Cat p∞ , S ) × Cat p∞ → Cat ∞ . An entirely analogous definition can be made for additive functors F ∶ Cat ex∞ → S (i.e. reduced and sendingVerdier squares to cartesian squares), resulting a category Span F ( C ) , with F = Cr giving rise to the usualspan category considered in [BR13].2.2.3. Example. i) Straight from the definition we have
Cob Cr ( C , Ϙ ) ≃ Span( C ) for every small stable ∞ -category C .ii) Similarly one obtains an equivalence Cob F (Hyp( C )) ≃ Span F ◦ Hyp ( C ) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 49 by commuting the hyperbolic and Q -constructions: From the natural equivalences of Remarks [I].6.4.6and [I].7.2.23, we find Fun ex (( E , Ϙ ) , Q 𝑛 Hyp( C )) ≃ Fun ex (( E , Ϙ ) , Hyp( C ) I 𝑛 )≃ Fun ex (( E , Ϙ ) I 𝑛 , Hyp( C ))≃ Hyp(Fun ex ( E I 𝑛 , C ))≃ Hyp(Fun ex ( E , C I 𝑛 ))≃ Fun ex (( E , Ϙ ) , Hyp Q 𝑛 ( C )) . so the natural map Q Hyp( C ) ⇒ Hyp Q( C ) in sCat p∞ is an equivalence.iii) In particular, Pn Hyp( C ) ≃ 𝜄 ( C ) gives Cob(Hyp( C )) ≃ Span( C ) for every stable ∞ -category, see Proposition [I].2.2.5.iv) There are canonical equivalences Cob( C , Ϙ s ) ≃ Span( C ) hC ∶ By Remark [I].2.2.8, a Poincaré structure on an ∞ -category D induces a natural C -action on 𝜄 D . Inparticular, we Ϙ s induces a C -action on the simplicial space 𝜄 Q C and therefore a C -action on theassociated category Span( C ) . By Proposition [I].6.2.2, the Poincaré structure ( Ϙ s ) TwAr[ 𝑛 ] is symmetricso that by Proposition [I].2.2.11 Pn Q 𝑛 ( C , Ϙ s ) ≃ 𝜄 Q 𝑛 ( C ) hC . As 𝜄 Q C is a complete Segal space, thisimplies the claim.v) There is a canonical equivalence Cob F ( C , Ϙ ) ≃ Cob F ( C , Ϙ ) op natural in the Poincaré ∞ -category ( C , Ϙ ) , since Q( C , Ϙ ) is naturally identfied with Q( C , Ϙ ) op (the rever-sal of the simplicial object) via the canonical identification TwAr(Δ 𝑛 ) ≅ TwAr((Δ 𝑛 ) op ) of cosimplicialobjects.We will now collect a few basic properties of such cobordism categories. Note that the inclusion of -simplices of F Q( C , Ϙ [1] ) gives a natural map F ( C , Ϙ [1] ) ⟶ 𝜄 Cob F ( C , Ϙ ) that is surjective on 𝜋 . Informally, for F = Pn this map takes any Poincaré object to itself and an equiva-lence 𝑓 ∶ 𝑥 → 𝑥 ′ to the cobordism 𝑥 id 𝑥 ←←←←←←←←←←←←← 𝑥 𝑓 ←←←←←←←→ 𝑥 ′ . Proposition 2.2.1 implies:2.2.4. Corollary.
The natural map F ( C , Ϙ [1] ) → 𝜄 Cob F ( C , Ϙ ) is an equivalence, whenever F preserves pullbacks. In particular, a Poincaré cobordism ( 𝑥, 𝑞 ) ← ( 𝑤, 𝑝 ) → ( 𝑥 ′ , 𝑞 ′ ) considered as a morphism in Cob( C , Ϙ ) is invertible if and only if both underlying maps 𝑤 → 𝑥 and 𝑤 → 𝑥 ′ are equivalences in C . Remark.
In the geometric cobordism category
Cob 𝑑 , one can perform a similar analysis: If a mor-phism 𝑊 in Cob 𝑑 is invertible, then it is an ℎ -cobordism and the converse is true if 𝑑 ≠ , the inverse of 𝑊 given by the ℎ -cobordism with Whitehead torsion − 𝜏 ( 𝑊 ) ∈ Wh( 𝜋 ( 𝜕 𝑊 )) .Furthermore, the homotopy type of 𝜄 Cob 𝑑 is closely related to the classifying space for ℎ -cobordisms[RS19].Since the association C ↦ Q 𝑛 C preserves products, as does completion of Segal spaces, it follows that thefunctor Cob F ∶ Cat p∞ ⟶ Cat ∞ preserves products. Since Cat p∞ is pre-additive (see Proposition [I].6.1.7)the categories Cob F ( C , Ϙ ) acquire natural symmetric monoidal structures induced by the direct sum op-eration in C . In particular, 𝜋 | Cob F ( C , Ϙ ) | is naturally a commutative monoid; explicitly when F = Pn , 𝜋 | Cob( C , Ϙ ) | is the monoid of cobordism classes of Poincaré objects in ( C , Ϙ ) under orthogonal sum. Now, 𝜋 | Cob F ( C , Ϙ ) | is in fact a group by the following result: Proposition.
The Poincaré functor (id C , −id Ϙ ) ∶ ( C , Ϙ ) → ( C , Ϙ ) induces an inversion map on 𝜋 | Cob F ( C , Ϙ ) | for every additive F ∶ Cat p∞ → S and every Poincaré ∞ -category ( C , Ϙ ) . In particular, | Cob F ( C , Ϙ ) | is al-ways an E ∞ -group in its canonical E ∞ -structure. Remark.
Let us warn the reader, that the Poincaré functor (id C , −id Ϙ ) does not generally induce theinversion on the entirety of | Cob F ( C , Ϙ ) | , the difference between the two maps merely vanishes on 𝜋 . Wewill give a formula for the inversion map at the space level in Corollary 3.1.8 below.For the proof we need a construction which will reappear later:2.2.8. Construction.
Consider the hermitian functor bcyl∶ ( C , Ϙ ) → Q ( C , Ϙ ) , representing a bent cylinder ,which consists of the functor 𝑋 ↦ [ 𝑋 ⊕ 𝑋 Δ 𝑋 ←←←←←←←←←←←←←←← 𝑋 → and the map of quadratic functors induced by the commutative diagram Ϙ 𝑋 Ϙ (bcyl 𝑋 ) Ϙ ( 𝑋 ⊕ 𝑋 ) Ϙ 𝑋 ⊕ Ϙ 𝑋 ⊕ B Ϙ ( 𝑋, 𝑋 )∗ Ϙ 𝑋 (id , −id , ∗ pr +pr whose left hand square is cartesian by definition of Ϙ , and whose right most horizontal map is an equiv-alence by definition of B Ϙ . The construction is readily checked to give a Poincaré functor by unwindingdefinitions.Informally, the bent cylinder provides a nullbordism of the sum of any Poincaré object with its reversedhermitian form. Proof of Proposition 2.2.6.
Recall from the discussion of Segal spaces that 𝜋 | Cob F ( C , Ϙ ) | is the coequaliserof the two boundary maps 𝜋 F Q ( C , Ϙ [1] ) → 𝜋 F ( C , Ϙ [1] ) . By construction then the element bcyl ∗ 𝑥 ∈ 𝜋 F Q ( C , Ϙ [1] ) witnesses 𝑥 + (id C , −id Ϙ ) ∗ 𝑥 ∈ 𝜋 F ( C , Ϙ [1] ) for every 𝑥 ∈ 𝜋 F ( C , Ϙ [1] ) . The claim follows. (cid:3) Corollary.
For any additive functor F ∶ Cat p∞ → S , the natural map 𝜋 F ( C , Ϙ [1] ) → 𝜋 | Cob F ( C , Ϙ ) | fits into a cocartesian square 𝜋 F (Met( C , Ϙ [1] )) 𝜋 F ( C , Ϙ [1] )0 𝜋 | Cob F ( C , Ϙ ) | met of commutative monoids. In other words, 𝜋 | Cob F ( C , Ϙ ) | is the quotient of 𝜋 F ( C , Ϙ [1] ) by the congruence relation identifying 𝑥 and 𝑥 ′ if there exist 𝑦, 𝑦 ′ ∈ 𝜋 F Met( C , Ϙ [1] ) such that 𝑥 + met( 𝑦 ) = 𝑥 ′ + met( 𝑦 ′ ) . In particular, for F = Pn we obtain an isomorphism 𝜋 | Cob( C , Ϙ ) | ≅ L −1 ( C , Ϙ ) with the L -groups from §[I].2.3. We will further explain the relation in §4.4 below. Proof.
The two formulations are equivalent by the description of cokernels in the category of commutativemonoids. Now recall that 𝜋 | Cob F ( C , Ϙ ) | is the coequaliser of the two boundary maps 𝑑 , 𝑑 ∶ 𝜋 F Q ( C , Ϙ [1] ) → 𝜋 F ( C , Ϙ [1] ) . Using Example 2.1.3 iii), we conclude that the diagram is indeed commutative, and the rightvertical map surjective. Therefore, we obtain an induced surjective map on the cokernel of met and we claimthat this map has trivial kernel. To see this, we note that the image of ( 𝑑 , 𝑑 ) is an equivalence relation in ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 51 𝜋 F ( C , Ϙ [1] ) : It is clearly reflexive and transitive, and symmetry follows from the evident automorphism of Q ( C , Ϙ ) swapping source and target. Thus if 𝑥 ∈ 𝜋 F ( C , Ϙ [1] ) vanishes in 𝜋 | F Q( C , Ϙ [1] ) | then there existsa 𝑤 ∈ 𝜋 F Q ( C , Ϙ [1] ) with 𝑑 𝑤 = 𝑥 and 𝑑 𝑤 = 0 . But since F (Met( C , Ϙ [1] )) → F Q ( C , Ϙ [1] ) → F ( C , Ϙ [1] ) is a fibre sequence by Lemma 2.1.7, we conclude that 𝑤 lifts to 𝜋 F (Met( C , Ϙ [1] )) and therefore vanishes inthe cokernel of met .Now the map from 𝜋 F ( C , Ϙ [1] ) into the cokernel is surjective (by the description of cokernels), and bythe same argument as in the proof of Proposition 2.2.6, we see that the cokernel of met is a group, just as 𝜋 | Cob F ( C , Ϙ ) | , so that the vanishing of the kernel implies injectivity. (cid:3) As the maps met ∶ Met(Met( C , Ϙ )) → Met( C , Ϙ ) and met ∶ Met(Hyp( C )) → Hyp( C ) are split by Remark [I].7.3.23 and Corollary [I].2.3.23, we obtain:2.2.10. Corollary.
For any Poincaré ∞ -category ( C , Ϙ ) , any small stable ∞ -category D and any additivefunctor F ∶ Cat p∞ → S the categories Cob F (Met( C , Ϙ )) and Cob F (Hyp( D )) are connected. Let us have a closer look at these two cobordism categories. We recorded in Example 2.2.3 that theforgetful functor
Cob(Hyp( C )) → Span( C ) is an equivalence, so in particular we find:2.2.11. Observation.
For every small stable ∞ -category C there is a canonical equivalence | Cob(Hyp( C )) | ≃ Ω ∞−1 K( C ) . Here, K( C ) denotes the connective algebraic 𝐾 -theory spectrum of C , defined for instance through theiterated Q -construction for stable ∞ -categories. In the case of Met( C ) , we have:2.2.12. Proposition.
There is a natural equivalence of ∞ -categories Cob(Met( C , Ϙ [1] )) → Span(He( C , Ϙ )) . Furthermore, the forgetful functor
Span(He( C , Ϙ )) → Span( C ) induces an equivalence on realisations. Thus, | Cob(Met( C , Ϙ )) | ≃ Ω ∞−1 K( C ) . The resulting equivalence | Cob(Met( C , Ϙ )) | ≃ | Cob(Hyp( C )) | in fact holds more generally for the F -based cobordism categories as a formal consequence | Cob F − | beingadditive and group-like, see Corollary 3.1.5. Proof.
Commuting diagram categories we find
Q(Met( C , Ϙ )) ≃ Met Q( C , Ϙ ) so that Proposition [I].2.3.20 implies Pn Q(Met( C , Ϙ [1] )) ≃ Fm Q( C , Ϙ ) . But without a non-degeneracy condition hermitian objects in a diagram category are just diagrams of her-mitian objects, see Corollary [I].6.3.15. So the right hand side is equivalent to 𝜄 Q(He( C , Ϙ )) . Passing toassociated categories gives the first claim.For the second claim we will show that 𝜋 ∶ Span(He( C , Ϙ )) ⟶ Span( C ) is cofinal and appeal to [Lur09a, Theorem Corollary 4.1.1.12]. By [Lur09a, Theorem 4.1.3.1], it suffices toshow that for every 𝑥 ∈ Span( C ) the under category Span(He( C )) 𝑥 ∕ is contractible. Since 𝜄 Span( C ) ≃ 𝜄 C (which is immediate from our discussion of Segal spaces) we may naturally interpret 𝑥 as an object of C and hence consider the comparison map(23) (He( C , Ϙ ) ∕ 𝑥 ) op ≃ (He( C , Ϙ ) op ) 𝑥 ∕ ⟶ Span(He( C , Ϙ )) 𝑥 ∕ induced by the following functor He( C , Ϙ ) op → Span(He( C , Ϙ )) : It is given by the identity on objects andtakes a morphism 𝑓 ∶ 𝑥 ′ → 𝑥 ′′ to the span 𝑥 ′′ 𝑓 ←←←←←←←← 𝑥 ′ id ←←←←←←←←→ 𝑥 ′ ; more formally the target functor TwAr(Δ 𝑛 ) → (Δ 𝑛 ) op gives a natural transformation of complete Segal spaces 𝜄 Fun(Δ , − op ) ≃ 𝜄 Fun(Δ op , −) → 𝜄 Q(−) , which has the desired behaviour on associated categories. But the functor (23) admits a right adjoint:Using the fibre sequence relating mapping spaces in under-categories with those in the original categoryone readily checks that 𝑤 → 𝑥 is right adjoint to 𝑥 ′ ← 𝑤 → 𝑥 , and thus Yoneda’s lemma assembles thisassignment into a right adjoint functor. We conclude that (23) induces an equivalence on realisations. Butthe category He( C , Ϙ ) ∕ 𝑥 has an initial object (the zero object of C with the trivial hermitian structure) andis hence contractible. (cid:3) Algebraic surgery.
In this subsection we translate Ranicki’s algebraic surgery to our set-up. Thisprovides a useful way of producing cobordisms, that we will heavily exploit in §[III].1 of Paper [III] and[HS20], and gives a description of slice categories of
Cob( C , Ϙ ) . We will approach these statements bytranslating them into assertions about certain Segal spaces derived from the Q -construction, and for thepresent paper it is, in fact, the analysis thereof that will play the largest role. We will follow the basicdescription of algebraic surgery given by Lurie in [Lur11, Lecture 11].Let ( C , Ϙ ) be a Poincaré ∞ -category, and ( 𝑋, 𝑞 ) be a Poincaré object therein. A surgery datum on ( 𝑋, 𝑞 ) consists of a map 𝑟 ∶ 𝑇 → 𝑋 and a nullhomotopy of 𝑓 ∗ 𝑞 ∈ Ω ∞ Ϙ ( 𝑇 ) . In other words, this is the extension of ( 𝑋, 𝑞 ) to a hermitian (but not necessarily Poincaré) nullbordism, i.e. to an object of He Met( C , Ϙ ) . Surgerydata organize into a space, and, more generally, into a category:2.3.1. Definition.
The category of surgery data in ( C , Ϙ ) is given by Surg( C , Ϙ ) = Pn( C , Ϙ ) × He( C , Ϙ ) He(Met( C , Ϙ )) , where the right hand map in the pullback is induced by met ∶ Met( C , Ϙ ) → ( C , Ϙ ) . The fibre of Surg( C , Ϙ ) over some ( 𝑋, 𝑞 ) ∈ Pn( C , Ϙ ) is called the category of surgery data on ( 𝑋, 𝑞 ) and denoted by Surg ( 𝑋,𝑞 ) ( C , Ϙ ) .We shall refer to the groupoid cores of these categories as the spaces of surgery data .2.3.2. Remark.
In geometric topology, a surgery datum on a closed oriented 𝑑 -dimensional manifold 𝑀 is afinite collection of disjointly embedded spheres ⨿ 𝑖 𝑆 𝑘 with trivialised normal bundles. The induced map onsingular chains inherits the structure of an algebraic surgery datum in ( D p ( ℤ ) , Ϙ s[− 𝑑 ] ) after applying chains(the Poincaré form on the target arises via its identification with C ∗ ( 𝑀 ; ℤ ) trough Poincaré duality), for ex-ample by feeding the trace of the geometric surgery datum into the surgery equivalence of Proposition 2.3.3below.Let us warn the reader that our presentation of algebraic surgery does not follow the overall conventionof creating Poincaré chain complexes from manifolds via their cochains; that convention would require usto describe an algebraic surgery datum in a more cumbersome fashion via the map 𝑋 → 𝑆 = D Ϙ 𝑇 , togetherwith a null-homotopy of the form after pull-back along D Ϙ 𝑆 ⟶ D Ϙ 𝑋 .Like in the geometric setting, surgery data can be used to produce cobordisms: Given a surgery datum ( 𝑓 ∶ 𝑇 → 𝑋, ℎ ∶ 𝑓 ∗ 𝑞 ≃ 0) , the composition 𝑇 𝑓 ←←←←←←←→ 𝑋 𝑞 ♯ ←←←←←←←←←→ D Ϙ 𝑋 D Ϙ 𝑓 ←←←←←←←←←←←←←←←←←←→ D Ϙ 𝑇 is identified with ( 𝑓 ∗ 𝑞 ) ♯ and therefore null via ℎ . Therefore one can form the following diagram 𝑇 𝑇 𝜒 ( 𝑓 ) 𝑋 D Ϙ 𝑇𝑋 𝑓 𝑋 ∕ 𝑇 D Ϙ 𝑇 𝑓 ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 53 with exact rows and columns: Here 𝜒 ( 𝑓 ) is the fibre of the composition 𝑋 ≃ D Ϙ 𝑋 D Ϙ 𝑓 ←←←←←←←←←←←←←←←←←←→ D Ϙ 𝑇 and 𝑋 𝑓 isdefined to be the cofibre of 𝑇 → 𝜒 ( 𝑓 ) .The resulting span [ 𝑋 ← 𝜒 ( 𝑓 ) → 𝑋 𝑓 ] ∈ Q ( C ) will then be the underlying object of the desiredcobordism:2.3.3. Proposition (Surgery equivalence) . The association 𝜒 upgrades to an equivalence 𝜒 ∶ 𝜄 Surg( C , Ϙ ) → Pn Q ( C , Ϙ ) , such that the diagram 𝜄 Surg( C , Ϙ ) Pn Q ( C , Ϙ )Pn( C , Ϙ ) . 𝜒 pr 𝑑 commutes, naturally in the Poincaré ∞ -category ( C , Ϙ ) . The image of a surgery datum under this equivalence is called the trace of the surgery. By the commu-tativity of the diagram above, the trace of a surgery on ( 𝑋, 𝑞 ) starts at ( 𝑋, 𝑞 ) , and the other end of the trace,that is 𝑋 𝑓 , is called the result of surgery . As already done here, we will use 𝜒 ( 𝑓 ) for both the trace and itstotal object. Proof.
We identify the Q ( C , Ϙ ) with the full subcategory of Met(Met( C , Ϙ [1] )) on those objects whose“boundary of the boundary” is zero, i.e., with the fibre of Met(Met( C , Ϙ [1] )) met ←←←←←←←←←←←←←←→ Met( C , Ϙ [1] ) met ←←←←←←←←←←←←←←→ ( C , Ϙ [1] ) . One readily checks that this yields an equivalence Q ( C , Ϙ ) ≃ ( C , Ϙ ) × Met( C , Ϙ [1] ) Met Met( C , Ϙ [1] )) in Cat p∞ , where the maps in the pull-back are given by taking boundaries on the right, and including objectswith boundary zero on the left. We obtain an equivalence Pn(Q ( C , Ϙ )) ≃ Pn( C , Ϙ ) × Pn(Met( C , Ϙ [1] )) Pn(Met Met( C , Ϙ [1] ))≃ Pn( C , Ϙ ) × Fm( C , Ϙ ) Fm(Met( C , Ϙ ))= 𝜄 Surg( C , Ϙ ) as desired from the algebraic Thom isomorphism (see Corollary [I].2.3.20). (cid:3) Remark.
From the proof one also obtains the following explicit description of the inverse equivalenceon objects. Given a Poincaré cobordism with underlying object 𝑋 ← 𝑊 → 𝑌 , its associated surgery datum has as underlying object the canonical map f ib( 𝑊 → 𝑌 ) → 𝑋. The form on f ib( 𝑊 → 𝑌 ) is the pull-back of the form on 𝑊 to the fibre, which comes with a canonicalnullhomotopy, since the form pulls back from 𝑌 .Now, by construction of Cob( C , Ϙ ) there is a cartesian square Hom
Cob( C , Ϙ ) ( 𝑋, 𝑌 ) Pn Q ( C , Ϙ [1] )Δ Pn( C , Ϙ [1] ) , ( 𝑑 ,𝑑 )( 𝑋,𝑌 ) so from a surgery datum 𝑇 on ( 𝑋, 𝑞 ) ∈ Pn( C , Ϙ −1 ) , we obtain an element Hom
Cob( C , Ϙ ) ( 𝑋, 𝑋 𝑇 ) . As men-tioned we will make extensive use of this construction in §[III].1. Due to the inherently asymmetrical natureof the surgery process, it is, however, not particularly convenient to describe the spaces Hom
Cob( C , Ϙ ) ( 𝑋, 𝑌 ) themselves (with prescribed 𝑌 ) in terms of surgery data on ( 𝑋, 𝑞 ) . The entire process does, however, gen-eralise very well to describe the slice categories Cob( C , Ϙ ) 𝑋 ∕ and more generally the comma category of 𝜄 Cob( C , Ϙ ) over Cob( C , Ϙ ) . Let us denote the latter category by dec(Cob( C , Ϙ )) , so that there is a pullbackdiagram dec(Cob( C , Ϙ )) Ar(Cob( C , Ϙ ))Pn( C , Ϙ [−1] ) Cob( C , Ϙ ) 𝑠 with 𝑠 the source map. The terminology dec is issued from the word decalage, see Lemma 2.3.7 below.2.3.5. Theorem.
The surgery process results in an equivalence 𝜒 Surg( C , Ϙ [−1] ) dec(Cob( C , Ϙ ))Pn( C , Ϙ [−1] ) 𝜒 pr 𝑠 natural in the Poincaré ∞ -category ( C , Ϙ ) . In particular, there result equivalences Surg 𝑋 ( C , Ϙ [−1] ) ≃ Cob( C , Ϙ ) 𝑋 ∕ for all 𝑋 ∈ Pn( C , Ϙ [1] ) . Remark.
We will not exploit this description of
Cob( C , Ϙ ) 𝑋 ∕ in the present paper as we are forcedto consider Cob F ( C , Ϙ ) for arbitrary additive F ∶ Cat p∞ → S in the sequel and do not know a similarly nicedescription in that generality (see Remark 2.3.11 below for a discussion of this point). The descriptionfeatures very prominently in [HS20].The proof of Theorem 2.3.5 will occupy the remainder of this section. The construction of the equiva-lence will proceed by first translating the assertion to the language of Segal spaces, see Proposition 2.3.10below. To this end, let us first recall the following well-known construction of slice categories in Segalspaces (for which we could not find a reference). We denote by dec∶ s S → s S the shifting or décalagefunctor induced by the endofunctor [0] ∗ − ∶ Δ op → Δ op , and similarly for simplicial objects in othercategories. Recall also that we set dec( C ) ≃ 𝜄 ( C ) × C Ar( C ) for any ∞ -category C .2.3.7. Lemma.
There are canonical equivalences
N(dec( C )) ≃ dec N( C ) natural in the category C , under which the nerve of the source and target functors dec( C ) → 𝜄 C and dec( C ) ⟶ C correspond to the maps N 𝑛 ( C ) ⟶ N ( C )N 𝑛 ( C ) ⟶ N 𝑛 ( C ) induced by +1 ∶ [ 𝑛 ] → [1 + 𝑛 ] and the inclusion [0] → [1 + 𝑛 ] , respectively. In particular, there resultequivalences N( C 𝑋 ∕ ) ≃ f ib( 𝑡 ∶ dec(N( C )) → N ( C )) , naturally in 𝑛 and ( C , 𝑋 ) , where the fibre is taken over 𝑋 ∈ 𝜄 C = N ( C ) .Proof. Note that the statement is entirely analogous to the comparison of thin and fat slices in the theoryof quasicategories and the proof is conceptually similar as well. Unwinding the definitions the claim isequivalent to there being a cocartesian square Δ 𝑛 Δ 𝑛 × Δ Δ Δ 𝑛𝑑 𝑑 ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 55 in Cat ∞ , that is natural in 𝑛 ; the contraction of Δ 𝑛 onto its first vertex produces the map Δ 𝑛 × Δ → Δ 𝑛 appearing on the right. Explicitely, it is given by ( 𝑘, ⟼ and ( 𝑘, ⟼ 𝑘 + 1 . That this diagram is cocartesian can be deduced from [Lur09a, Proposition 4.2.1.2], together with the factthat homotopy cartesian diagrams in Joyal’s model structure on simplicial sets give cocartesian diagramsin
Cat ∞ .But one can also give an internal argument: The contraction admits an explicit degreewise right inverse(which is not natural in 𝑛 ) as follows: Simply include Δ 𝑛 +1 into Δ 𝑛 × Δ as by sending to (0 , and 𝑘 to ( 𝑘 − 1 , for all < 𝑘 ≤ 𝑛 + 1 . Then the composition Δ 𝑛 +1 ⟶ Δ 𝑛 × Δ ⟶ Δ 𝑛 +1 is the identity, and conversely the composition Δ 𝑛 × Δ ⟶ Δ 𝑛 +1 ⟶ Δ 𝑛 × Δ comes with a unique natural transformation Δ 𝑛 × Δ × Δ → Δ 𝑛 × Δ to the identity. Given now a category E against which to test the cocartesianness of the square above, this transformation preserves Δ 𝑛 × {0} soadjoins to a transformation Δ × Fun(Δ 𝑛 × Δ , E ) × Fun(Δ 𝑛 , E ) E ⟶ Fun(Δ 𝑛 × Δ , E ) × Fun(Δ 𝑛 , E ) E from the composition in question to the identity. This is readily checked to be a pointwise equivalence. (cid:3) It follows conversely that for a complete Segal space C ∈ cSS and 𝑋 ∈ C we find asscat( C ) 𝑋 ∕ ≃ asscat(f ib( 𝑠 ∶ dec( C ) ⟶ C )) where the fibres are taken over 𝑋 . It is also easy to see that the right hand side is not affected by completion,so this formula is valid for all Segal spaces C .In particular, the ∞ -category Cob( C , Ϙ ) is modelled by the following Segal object:2.3.8. Definition.
Let ( C , Ϙ ) be a Poincaré ∞ -category. We define the simplicial object Null( C , Ϙ ) in Cat p∞ as the fibre of the simplicial map dec(Q( C , Ϙ )) → Q ( C , Ϙ ) = ( C , Ϙ ) .Explicitly, Null 𝑛 ( C , Ϙ ) consists of those diagrams 𝜑 ∶ TwAr[1 + 𝑛 ] ⟶ C in Q 𝑛 ( C , Ϙ ) such that 𝜑 (0 ≤
0) = 0 , with the Poincaré structure restricted from Q 𝑛 ( C , Ϙ ) . In particular, Null ( C , Ϙ ) ≅ Met( C , Ϙ ) .In fact, the Poincaré ∞ -category Null 𝑛 ( C , Ϙ ) is metabolic in the sense of Definition [I].7.3.10: Let L − 𝑛 ⊆ Null 𝑛 ( C , Ϙ ) be the full subcategory spanned by those diagrams 𝜑 ∶ TwAr[1 + 𝑛 ] op ⟶ C with 𝜑 (0 ≤ 𝑖 ) ≃ 0 for all 𝑖 ∈ [1 + 𝑛 ] . Then, since 𝜑 is left Kan extended from J 𝑛 ⊆ TwAr(Δ 𝑛 ) by Examples 2.1.3, therestriction to the subposet of TwAr(Δ 𝑛 ) spanned by all ( 𝑗 ≤ 𝑛 ) , 𝑗 ≠ gives an equivalence 𝑝 𝑛 ∶ L − 𝑛 → Fun(Δ 𝑛 , C ) and furthermore one readly checks that the restriction of the hermitian structure of Null 𝑛 ( C , Ϙ ) correspondsprecisely to Ϙ Δ 𝑛 under 𝑝 𝑛 .2.3.9. Proposition.
We have a natural equivalence
Pn(Null 𝑛 ( C , Ϙ [1] )) ≃ Fm(Fun(Δ 𝑛 , C ) , Ϙ Δ 𝑛 ) . for Poincaré ∞ -categories ( C , Ϙ ) .Proof. We will show more generally that the full subcategory L + 𝑛 ⊆ Null 𝑛 ( C ) formed by the duals of theobjects in L − 𝑛 is a Lagrangian in the sense of Definition [I].7.3.10; so that 𝑝 𝑛 induces an equivalence Pair ( Fun(Δ 𝑛 , C ) , Ϙ Δ 𝑛 ) → Null 𝑛 ( C , Ϙ [1] ) by the recognition principle for pairing categories, Proposition [I].7.3.11, from which the claim followsfrom the generalised algebraic Thom isomorphism, Proposition [I].7.3.5.To see this, we observe that L + 𝑛 consists of all those 𝜑 ∶ TwAr[1 + 𝑛 ] op ⟶ C that are left Kan-extendedfrom the subposet 𝐵 𝑛 of TwAr(Δ 𝑛 ) spanned by all (0 ≤ 𝑗 ) , or equivalently for which 𝜑 (0 ≤ 𝑗 ) ⟶ 𝜑 ( 𝑖 ≤ 𝑗 ) is an equivalence for 𝑖 ≤ 𝑗 ∈ [1 + 𝑛 ] (in addition to 𝜑 (0 ≤
0) = 0 ). The second description immediatelyimplies that the restriction of Ϙ 𝑛 indeed vanishes, while the first exhibits left Kan extension from 𝐵 𝑛 as a right adjoint 𝑅 to the inclusion L + 𝑛 ⊆ Null 𝑛 ( C , Ϙ ) . Since also, by definition, L − 𝑛 = ker( 𝑅 ) , the subcategory L + 𝑛 is indeed a Lagrangian. (cid:3) Now under the equivalences of Lemma 2.3.7, Theorem 2.3.5 translates to the following generalisationof Proposition 2.3.3:2.3.10.
Proposition.
The algebraic surgery construction extends to a cartesian diagram
Pn Q 𝑛 ( C , Ϙ ) Pn( C , Ϙ ) 𝜄 Fun(Δ 𝑛 , He(Met( C , Ϙ ))) 𝜄 Fun(Δ 𝑛 , He( C , Ϙ )) 𝑑 𝐷𝑒𝑙𝑡𝑎 met of functors
Cat p∞ × Δ op → S .Proof. Identify the Poincaré ∞ -category Q 𝑛 ( C , Ϙ ) with the full Poincaré subcategory of Null 𝑛 Met( C , Ϙ [1] ) on all objects whose boundary in Null 𝑛 ( C , Ϙ [1] ) is of the form 𝑏 … 𝑏 , that is with the fibre of the composition(24) Null 𝑛 Met( C , Ϙ [1] ) met ←←←←←←←←←←←←←←→ Null 𝑛 ( C , Ϙ [1] ) 𝑑 ←←←←←←←←←←→ Q 𝑛 ( C , Ϙ [1] ); this is achieved by the equivalences Q 𝑛 ( C , Ϙ ) ≃ Q ( C , Ϙ ) × ( C , Ϙ ) Q 𝑛 ( C , Ϙ )≃ Met Met( C , Ϙ [1] ) × Met( C , Ϙ [1] ) Q 𝑛 ( C , Ϙ )≃ Null 𝑛 Met( C , Ϙ [1] ) × Q 𝑛 Met( C , Ϙ [1] ) Q 𝑛 ( C , Ϙ ) , where the third identification is obtained from the pullback Met( C , Ϙ ) ( C , Ϙ )Null 𝑛 ( C , Ϙ ) Q 𝑛 ( C , Ϙ ) met 𝑑 (straight from the Segal condition Lemma 2.1.5) by exchanging the order of pullbacks. Since the right handmap in the last description is fully faithful, this embeds Q 𝑛 ( C , Ϙ ) fully faithfully into Null 𝑛 Met( C , Ϙ [1] ) ,and it is clear that the essential image is as desired. But invoking the displayed pullback again, we find thatthe fibre of (24) is equivalent to f ib(Met( C , Ϙ [1] ) → ( C , Ϙ [1] )) ≃ ( C , Ϙ ) , the latter by the metabolic fibre sequence of Example 1.2.5. In total, we obtain an equivalence Q 𝑛 ( C , Ϙ ) ≃ ( C , Ϙ ) × Null 𝑛 ( C , Ϙ [1] ) Null 𝑛 Met( C , Ϙ [1] ) in Cat p∞ . But the functor Pn preserves limits, so Pn Q 𝑛 ( C , Ϙ ) ≃ PnNull 𝑛 Met( C , Ϙ [1] ) × PnNull 𝑛 ( C , Ϙ [1] ) Pn( C , Ϙ )≃ Fm Fun(Δ 𝑛 , Met( C , Ϙ )) × Fm Fun(Δ 𝑛 , ( C , Ϙ )) Pn( C , Ϙ )≃ 𝜄 Fun(Δ 𝑛 , He Met( C , Ϙ )) × 𝜄 Fun(Δ 𝑛 , He( C , Ϙ )) Pn( C , Ϙ ) the second equivalence by Proposition 2.3.9 and the third from Corollary [I].6.3.15. (cid:3) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 57 Proof of 2.3.5.
Since the inclusion of constant diagrams induces an equivalence
Pn( C , Ϙ ) ⟶ 𝜄 Fun(Δ 𝑛 , Pn( C , Q)) by the contractibility of Δ 𝑛 , Proposition 2.3.10 can be restated as an equivalence dec(Pn Q( C , Ϙ )) ≃ N(Surg( C , Ϙ )) . The claim thus follows from Lemma 2.3.7. (cid:3)
Remark.
Finally, let us explain the reason for sticking to the functor
Pn ∶ Cat p∞ → S in this section:For an arbitrary additive functor F ∶ Cat p∞ → S one can produce a functor c F ∶ Cat p∞ → Cat ∞ by setting c F ( C , Ϙ ) = asscat F (Null( C , Ϙ [1] )) . For example, cPn = He by Proposition 2.3.9. One can then set
Surg F ( C , Ϙ ) = F ( C , Q) × c F ( C , Ϙ ) c F (Met( C , Ϙ )) and attempt to obtain generalisations of Proposition 2.3.3 and Theorem 2.3.5 for arbitrary additive F . Thecrucial (and in fact necessary) ingredient for these statement is, however, that the tautological map F Met( C , Ϙ ) → 𝜄 (c F ( C , Ϙ [−1] )) is an equivalence, which unwinds exactly to the completeness of the Segal space F (Null( C , Ϙ [1] )) . As alreadymentioned after Proposition 2.2.1, this generally fails unless F preserves arbitrary pullbacks.2.4. The additivity theorem.
As we will see, the decisive step towards understanding the homotopy typeof the cobordism categories
Cob( C , Ϙ ) consists in analysing their behaviour under split Poincaré-Verdiersequences. To this end we show:2.4.1. Theorem (Additivity) . Let F ∶ Cat p∞ → S be additive. Then the functor | Cob F | is also additive. Inparticular, a split Poincaré-Verdier sequence ( C , Ϙ ) → ( C ′ , Ϙ ′ ) → ( C ′′ , Ϙ ′′ ) induces a fibre sequence | Cob F ( C , Ϙ ) | → | Cob F ( C ′ , Ϙ ′ ) | → | Cob F ( C ′′ , Ϙ ′′ ) | of E ∞ -groups. We heavily exploit the result in §3 below. In particular, we use it to compute 𝜋 | Cob F ( C , Ϙ ) | , producedeloopings of | Cob F ( C , Ϙ ) | via the iterated Q -construction and it also serves as the basis for Grothendieck-Witt theory in §4. It contains Waldhausens’s additivity theorem for K -theory as a special case, as we willdetail in §2.6 below.On the other hand, Theorem 2.4.1 yields an algebraic analogue of Genauer’s fibre sequence from geo-metric topology. To explain this analogy recall that there exists a fibre sequence | Cob 𝑑 +1 | → | Cob 𝜕𝑑 +1 | → | Cob 𝑑 | relating cobordism categories of manifolds of different dimension (with the middle term allowing objectsto have boundary). As mentioned in the introduction this was originally proven by identifying the sequenceterm by term with the infinite loop spaces of certain Thom spectra, together with a direct verification thatthese Thom-spectra form a fibre sequence, see [Gen12, Proposition 6.2] and the main result of [GTMW09].Applying Theorem 2.4.1 for F = Pn to the metabolic Poincaré-Verdier sequence ( C , Ϙ [−1] ) → Met( C , Ϙ [−1] ) → ( C , Ϙ ) from Example 1.2.5, we obtain the following algebraic analogue of the Genauer fibre sequence:2.4.2. Corollary.
For every Poincaré ∞ -category ( C , Ϙ ) there is a fibre sequence | Cob( C , Ϙ [−1] ) | → | Cob 𝜕 ( C , Ϙ [−1] ) | → | Cob( C , Ϙ ) | of E ∞ -groups. Even more, our proof of the Additivity theorem will follow the strategy developed in [Ste18] by the 9’thauthor in his approach to Genauer’s fibre sequence. It is based on a recognition criterion for realisationfibrations, whose assumption we verify with the following result:2.4.3.
Theorem.
Let F ∶ Cat p∞ → S be additive and ( 𝑝, 𝜂 ) ∶ ( C , Ϙ ) → ( C ′ , Ϙ ′ ) a split Poincaré-Verdierprojection. Then the induced map ( 𝑝, 𝜂 ) ∗ ∶ F Q( C , Ϙ ) ⟶ F Q( C , Ϙ ) is a bicartesian fibration of Segal spaces and in particular ( 𝑝, 𝜂 ) ∗ ∶ Cob F ( C , Ϙ ) ⟶ Cob F ( C ′ , Ϙ ′ ) a bicartesian fibration of ∞ -categories. We refer to [Ste18, section 2] for the definition of (co-)cartesian fibration between Segal spaces. Theproof of Theorem 2.4.3 will indeed show that an edge in F Q( C , Ϙ ) is F Q( 𝑝 ) -cocartesian if and only if it liesin the image of F Q( E , Ϙ ) where E ⊆ Q ( C ) is the subcategory spanned by those diagrams 𝑥 ← 𝑤 → 𝑦 with left hand map 𝑝 -cartesian and right hand map 𝑝 -cocartesian; the roles are reversed for Q( 𝑝 ) -cartesianedges.2.4.4. Remark. i) A similar result in the context of ∞ -categories of spans was given by Barwick as partof his unfurling construction in [Bar17, Theorem 12.2]. While the main motivation for that construc-tion is also K -theoretic in nature, its use does not seem at all related to additivity in Barwick’s work.Our proof, furthermore, proceeds rather differently than Barwick’s combinatorial approach.ii) Neither Theorem 2.4.1 nor Theorem 2.4.3 remain true upon assuming F Verdier-localising and theinput Poincaré-Verdier, but not necessarily split. For example, with F = K ◦ (−) ♮ , which is Karoubi-localising and grouplike, Corollary 2.2.9 and Theorem 3.3.4 below in combination show that | Cob F − | ≃ B K ◦ (−) ♮ is the connectied delooping, which is famously not (Poincaré-)Verdier-localising, sinceVerdier projections need not induce surjections on K ◦ (−) ♮ . Proof of the Additivity theorem, assuming Theorem 2.4.3.
Suppose given a Poincaré-Verdier square ( C , Ϙ ) ( D , Φ)( C ′ , Ϙ ′ ) ( D ′ , Φ ′ ) . Since
Q ∶ Cat p∞ → sCat p∞ preserves limits, we find an associated cartesian square of Segal spaces, whence[Ste18, Theorem 2.11] together with the equivalence | Cob F − | ≃ | F Q − | shows that | Cob F ( C , Ϙ ) | | Cob F ( D , Φ) || Cob F ( C ′ , Ϙ ′ ) | | Cob F ( D ′ , Φ ′ ) | is cartesian as desired. (cid:3) Remark.
We do not know, whether in general the analogous square involving the ∞ -categories Cob F (i.e. the completion of the square obtained by applying Q ), is cartesian if F does not preserve arbitrary pull-backs, since completion and the extraction of associated ∞ -categories do not generally preserve pullbacks.2.5. Fibrations between cobordism categories.
The present section is devoted to the proof of Theo-rem 2.4.3.We will throughout write ( C , Ϙ ) J = (Fun( J , C ) , Ϙ J ) for the cotensoring of a hermitian ∞ -category ( C , Ϙ ) with an ∞ -category J .The strategy of proof is as follows: After recording that a split Verdier projection (of stable ∞ -categories)is a bicartesian fibration, we improve on this by showing that the maps 𝑝 ∗ ∶ ( C , Ϙ ) Δ 𝑛 → ( C ′ , Ϙ ′ ) Δ 𝑛 ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 59 behave like a bicartesian fibration between Segal objects in Cat h∞ ; we will not give a formal definition ofthis term, but instead formulate the relevant statements directly in Lemmas 2.5.3 and 2.5.4. We then usethis to show that the map Q( 𝑝 ) ∶ Q( C , Ϙ ) → Q( C ′ , Ϙ ′ ) also behaves like such a bicartesian fibration; the cocartesian part is formulated in Lemmas 2.5.7 and 2.5.8and the cartesian one follows by invariance of the Q -construction under taking opposites. From there wewill deduce the theorem by observing that any additive functor F can be used as a ‘cut-off’ to obtain abicartesian fibration F Q( C , Ϙ ) → F Q( C ′ , Ϙ ′ ) of Segal objects in S , which implies the result.To get started we need:2.5.1. Lemma.
Let 𝑝 ∶ C → C ′ be a functor with left adjoint 𝑔 . Then:i) A morphism 𝛼 ∶ 𝑥 → 𝑦 in C is 𝑝 -cocartesian if and only if the square 𝑔𝑝 ( 𝑥 ) 𝑔𝑝 ( 𝑦 ) 𝑥 𝑦, 𝑔𝑝 ( 𝛼 )c 𝑥 c 𝑦 𝛼 obtained by applying the counit transformation to 𝛼 , is a pushout square.ii) If C admits pushouts which 𝑝 preserves and 𝑔 is fully faithful, then 𝑝 is a cocartesian fibration.Proof. The first statement is immediate from the mapping space criterion for cocartesian morphisms [Lur09a,Proposition 2.4.4.3]. For the second one readily checks that for 𝑐 ∈ C and a map 𝑝 ( 𝑐 ) → 𝑑 in C ′ the edge 𝑐 → 𝑐 ∪ 𝑔𝑝 ( 𝑐 ) 𝑔 ( 𝑑 ) is a 𝑝 -cocartesian lift; here the pushout is formed using the counit 𝑔𝑝 ( 𝑐 ) → 𝑐 of theadjunction. (cid:3) Applying the previous corollary also to the opposite category we find:2.5.2.
Corollary.
Any split Verdier projection 𝑝 ∶ C → C ′ of stable ∞ -categories is a bicartesian fibration. Now denote by
Cart( 𝑝 ) , Cocart( 𝑝 ) ⊆ Ar( C ) the full subcategories on 𝑝 -cartesian, resp. 𝑝 -cocartesian morphisms. These are stable subcategories asa consequence of Lemma 2.5.1, and the hermitian structure Ϙ Δ endows Cart( C ) and Cocart( C ) with thestructure of hermitian ∞ -categories (we warn the reader that Ϙ Δ ( 𝑥 → 𝑦 ) ≃ Ϙ ( 𝑦 ) is distinct from thePoincaré structure Ϙ ar from §[I].2.3). Finally, we denote by 𝑠 and 𝑡 ∶ Ar( C ) → C source and target functor,respectively.2.5.3. Lemma.
Let 𝑝 ∶ ( C , Ϙ ) → ( C ′ , Ϙ ′ ) be a Poincaré-Verdier projection. Then the diagrams (Cart( 𝑝 ) , Ϙ Δ ) ( C , Ϙ )( C ′ , Ϙ ′ ) Δ ( C ′ , Ϙ ′ ) 𝑡𝑝 𝑝𝑡 (Cocart( 𝑝 ) , Ϙ Δ ) ( C , Ϙ )( C ′ , Ϙ ′ ) Δ ( C ′ , Ϙ ′ ) 𝑠𝑝 𝑝𝑠 in Cat h∞ are cartesian. Lemma.
Let 𝑝 ∶ ( C , Ϙ ) → ( C ′ , Ϙ ′ ) be a Poincaré-Verdier projection. Then the square (Cart( C ) , Ϙ Δ ) × ( C , Ϙ ) Δ1 ( C , Ϙ ) Δ (Cart( C ) , Ϙ Δ ) × ( C , Ϙ ) ( C , Ϙ ) Δ ( C ′ , Ϙ ′ ) Δ ( C ′ , Ϙ ′ ) Δ × ( C ′ , Ϙ ′ ) ( C ′ , Ϙ ′ ) Δ , (id ,𝑑 ) 𝑝 𝑝 ( 𝑑 ,𝑑 ) where the pullback in the top left corner is formed using 𝑑 Δ → Δ and those on the right using the targetfunctor is cartesian in Cat h∞ . Similarly, (Cocart( C ) , Ϙ Δ ) × ( C , Ϙ ) Δ1 ( C , Ϙ ) Δ (Cocart( C ) , Ϙ Δ ) × ( C , Ϙ ) ( C , Ϙ ) Δ ( C ′ , Ϙ ′ ) Δ ( C ′ , Ϙ ′ ) Δ × ( C ′ , Ϙ ′ ) ( C ′ , Ϙ ′ ) Δ . (id ,𝑑 ) 𝑝 𝑝 ( 𝑑 ,𝑑 ) with top left corner formed using 𝑑 ∶ Δ → Δ and right hand using the source functor, is cartesian in Cat h∞ .Proof of Lemma 2.5.3. One readily checks straight from the definitions and the mapping space criterion forcartesian edges [Lur09a, Proposition 2.4.4.3] that the map
Cart( 𝑝 ) → Ar( C ′ ) × C ′ C is essentially surjectiveand fully faithful for any cartesian fibration 𝑝 .To see that this map is an equivalence Cat h∞ , note first that by the discussion in §[I].6.1 it is enough toshow that for a cartesian morphism 𝑓 ∶ Δ → C the square lim( Ϙ ◦ 𝑓 op ) Ϙ ( 𝑓 (1))lim( Ϙ ′ ◦ ( 𝑝𝑓 ) op ) Ϙ ′ ( 𝑝𝑓 (1)) is a pullback of spectra. But this is clear since the horizontal maps are equivalences, as is initial in (Δ ) op .Now we deal with the second square. That the underlying square of ∞ -categories is cartesian is againeasy (or indeed follows from the cartesian case applied to 𝑝 op ). For the hermitian structure we need to showthat(25) Ϙ ( 𝑓 (1)) Ϙ ( 𝑓 (0)) Ϙ ′ ( 𝑝𝑓 (1)) Ϙ ′ ( 𝑝𝑓 (0)) . is a pullback for every 𝑝 -cocartesian morphism 𝑓 .To see this, recall from Lemma 2.5.1 that 𝑓 (1) ≃ 𝑓 (0) ∪ 𝑙𝑝𝑓 (0) 𝑙𝑝𝑓 (1) , where 𝑙 is the left adjoint to 𝑝 .Furthermore, the canonical map Ϙ ◦ 𝑙 → Ϙ ′ is an equivalence, since 𝑝 is a split Poincaré-Verdier projection,see Corollary 1.2.3. Thus, the square in question is equivalent to Ϙ ( 𝑓 (0) ∪ 𝑙𝑝𝑓 (0) 𝑙𝑝𝑓 (1)) Ϙ ( 𝑓 (0)) Ϙ ( 𝑙𝑝𝑓 (1)) Ϙ ( 𝑙𝑝𝑓 (0)) By Lemma [I].1.1.19 it is therefore enough to show that B Ϙ (cof( 𝑙𝑝𝑓 ) , cof( 𝑐 )) ≃ 0 , where 𝑐 ∶ 𝑙𝑝𝑓 (0) → 𝑓 (0) is the counit of the adjunction We compute B Ϙ (cof( 𝑙𝑝𝑓 ) , cof( 𝑐 )) ≃ Hom C ( 𝑙 cof( 𝑝𝑓 ) , D Ϙ cof ( 𝑐 ))≃ Hom C ′ (cof( 𝑝𝑓 ) , 𝑝 D Ϙ cof ( 𝑐 ))≃ Hom C ′ (cof( 𝑝𝑓 ) , D Ϙ ′ cof( 𝑝𝑐 )) but 𝑝𝑐 is an equivalence so this term vanishes as desired. (cid:3) For the proof of Lemma 2.5.4 we will use the following observation, compare [Lur09a, Corollary 2.4.2.5]:2.5.5.
Observation.
Let 𝑝 ∶ C → C ′ be a cartesian fibration, and C ⊆ C be a full subcategory that containsall 𝑝 -cartesian morphisms whose target lies in C . Then the restricted functor 𝑝 ∶ C → C ′ is also a cartesianfibration. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 61 Proof of Lemma 2.5.4.
We again start by showing that the upper square is a pullback of ∞ -categories. Wefirst claim that both vertical maps are cartesian fibrations. By [Lur09a, 3.1.2.1] the functors 𝑝 ∗ ∶ Fun( 𝐾, C ) → Fun( 𝐾, C ′ ) are again cartesian fibrations, with cartesian edges detected pointwise. Applying this with 𝐾 = Δ and Λ , the claim easily follows from Observation 2.5.5 and the cancellability of cartesian edges[Lur09a, Proposition 2.4.1.7]. The pointwise nature of cartesian edges also implies that the top horizontalmap perserves cartesian edges, so to check that the underlying diagram is cartesian in Cat ∞ it suffices tocheck that the induced map on vertical fibres are equivalences by [Lur09a, Corollary 2.4.4.4].But here again, one readily checks that the induced functors are fully faithful and essentially surjectivestraight from the mapping space criterion for cartesian edges [Lur09a, Proposition 2.4.4.3] together withthe description of spaces of natural transformation as iterated pullbacks arising from [GHN17, Proposition5.1].This concludes the proof that the underlying square on the left is a pullback in Cat ∞ , and the argumentfor the right one is entirely analogous. To make the left square a pullback in Cat h∞ , we need to show thatfor each 𝑓 ∶ Δ → C , the square of spectra lim (Δ ) op Ϙ ◦ 𝑓 op lim (Λ ) op Ϙ ◦ 𝑓 op lim (Δ ) op Ϙ ′ ◦ 𝑝𝑓 op lim (Λ ) op Ϙ ′ ◦ 𝑝𝑓 op is a pullback. But this is clear since is terminal in both Δ and Λ so the inclusion (Λ ) op ⊂ (Δ ) op is finaland the horizontal maps are equivalences.To see that the second square is a pullback in Cat h∞ , we have to show that the following square is apull-back: lim (Δ ) op Ϙ ◦ 𝑓 op lim (Λ ) op Ϙ ◦ 𝑓 op lim (Δ ) op Ϙ ′ ◦ 𝑝𝑓 op lim (Λ ) op Ϙ ′ ◦ 𝑝𝑓 op Since is terminal in Δ this reads Ϙ ( 𝑓 (2)) Ϙ ( 𝑓 (1)) × Ϙ ( 𝑓 (0)) Ϙ ( 𝑓 (2)) Ϙ ′ ( 𝑝𝑓 (2)) Ϙ ′ ( 𝑝𝑓 (1)) × Ϙ ′ ( 𝑝𝑓 (0)) Ϙ ′ ( 𝑝𝑓 (2)) . But either by decoding the statement of Lemma 2.5.3 or more directly from (25), we find Ϙ ( 𝑓 (1)) ≃ Ϙ ′ ( 𝑝𝑓 (1)) × Ϙ ′ ( 𝑝𝑓 (0)) Ϙ ( 𝑓 (0)) , since 𝑓 (0) → 𝑓 (1) is 𝑝 -cocartesian by assumption. (cid:3) Let E ⊂ Q ( C ) denote the full subcategory on objects of the form 𝑐 ← 𝑤 → 𝑑 where the left arrow is 𝑝 -cartesian, and the right arrow is 𝑝 -cocartesian. This is a stable subcategory which inherits a hermitianstructure from Q ( C ) .2.5.6. Lemma. E ⊂ Q ( C ) is closed under the duality D Ϙ . Therefore, ( E , Ϙ ) is a Poincaré ∞ -category and the inclusion functor E → Q ( C ) tautologically refinesto a Poincaré functor. Proof.
Let 𝑐 𝑓 ←←←←←←←← 𝑤 𝑔 ←←←←←←→ 𝑑 be an object of E , so that 𝑓 is 𝑝 -cartesian and 𝑔 is 𝑝 -cocartesian. The dual arrowis obtained by first completing the diagram to a pushout square; then applying D Ϙ termwise, and deletingthe value at the terminal object of the square, see Proposition [I].6.3.2. The claim now follows from the fact that 𝑝 -(co-)cartesian morphisms are stable under (co-)base change, and that the dualities interchange 𝑝 -cartesian with 𝑝 -cocartesian morphisms since the diagram C op CC ′op C ′D Ϙ 𝑝 op 𝑝 D Ϙ ′ commutes as 𝑝 is Poincaré. (cid:3) We are now ready to state the main technical results of this section, namely that Q( 𝑝 ) ∶ Q( C , Ϙ ) → Q( C ′ , Ϙ ′ ) behaves like a cocartesian fibration of Segal objects in Cat h∞ , with cocartesian lifts given by ( E , Ϙ ) ⊂ Q ( C , Ϙ ) . Since the Q -construction is invariant under taking the opposite simplicial object, itfollows that it also behaves like a cartesian fibration, see Example 2.2.3.2.5.7. Lemma.
The diagram ( E , Ϙ ) 𝑝 ( C , Ϙ )Q ( C ′ , Ϙ ′ ) ( C ′ , Ϙ ′ ) 𝑑 𝑝𝑑 is a split Poincaré-Verdier square. Lemma.
The diagram ( E , Ϙ ) × Q ( C , Ϙ ) Q ( C , Ϙ ) ( E , Ϙ ) × ( C , Ϙ ) Q ( C , Ϙ )Q ( C ′ , Ϙ ′ ) Q ( C ′ , Ϙ ′ ) × ( C ′ , Ϙ ′ ) Q ( C ′ , Ϙ ′ ) . (id ,𝑑 ) 𝑝 𝑝 ( 𝑑 ,𝑑 ) where the upper left pullback is formed using 𝑑 ∶ Q ( C , Ϙ ) → Q ( C , Ϙ ) and the right hand ones using 𝑑 ,is a split Poincaré-Verdier square. In particular, both diagrams are cartesian in
Cat p∞ . Proof of Lemma 2.5.7.
We factor the square in question as ( E , Ϙ ) (Cart( C ) , Ϙ Δ ) ( C , Ϙ )Q ( C ′ , Ϙ ′ ) ( C ′ , Ϙ ′ ) Δ ( C ′ , Ϙ ′ ) . 𝑝 𝑡𝑝 𝑝𝑡 Here the left horizontal maps are given by including Δ into TwAr Δ as the morphism (0 ≤ → (0 ≤ .The right square is a pullback by Lemma 2.5.3. Now Q ( C , Ϙ ) ≃ ( C , Ϙ ) Λ ≃ ( C , Ϙ ) Δ × ( C , Ϙ ) ( C , Ϙ ) Δ using the source and target arrows for the pullback, and this equivalence restricts to an equivalence E ≃ Cart( C ) × C Cocart( C ) by construction. So, the left square is obtained by pullback from the right hand square of Lemma 2.5.3 andtherefore cartesian as well (in Cat h∞ , and hence in Cat p∞ ).Since 𝑝 is a split Poincaré-Verdier projection by assumption this implies the claim by Corollary 1.2.6. (cid:3) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 63 Proof of Lemma 2.5.8.
The ∞ -category in the upper left corner is equivalent (as a hermitian ∞ -category)to the full subcategory of Q ( C ) on those diagrams 𝐹 ∶ TwAr Δ → C ,(26) 𝐹 (0 ≤ 𝐹 (0 ≤ 𝐹 (1 ≤ 𝐹 (0 ≤ 𝐹 (1 ≤ 𝐹 (2 ≤ (III)(I) (II) such that (i) the map labelled by (I) is 𝑝 -cartesian, (ii) the map labelled by (II) is 𝑝 -cocartesian, and (iii) themiddle square is exact. In view of Lemma 2.5.1, one easily checks by pasting exact squares that condition(iii) is equivalent to the following two conditions: (iii’) the map labelled by (III) is 𝑝 -cocartesian, and (iii”)the image of the middle square in C ′ is exact. In other words, if we denote by ( C , Ϙ ) TwAr(Δ ) 𝑝 ⊂ ( C , Ϙ ) TwAr(Δ ) the full subcategory on diagrams satisfying (i), (ii), and (iii’), then the diagram(27) ( E , Ϙ ) × Q ( C , Ϙ ) Q ( C , Ϙ ) ( C , Ϙ ) TwAr(Δ ) 𝑝 Q ( C ′ , Ϙ ′ ) ( C ′ , Ϙ ′ ) TwAr(Δ ) 𝑝 𝑝 is a pullback in Cat h∞ , since it is one in Cat ∞ and the hermitian structures on the left are the restrictions ofthose on the right.Now consider the following filtration 𝐼 → 𝐼 → … 𝐼 = TwAr(Δ ) through (non-full) subposets, starting with 𝐼 = 𝑑 (TwAr Δ ) ∪ 𝑑 (TwAr Δ ) . The remaining 𝐼 𝑖 are obtained by adding relations in the order indicated in the following picture: (0 ≤ ≤
1) (1 ≤ ≤
0) (1 ≤
1) (2 ≤ ○ ○ ○ ○ Now one readily checks that each 𝐼 𝑖 → 𝐼 𝑖 +1 is obtained from an outer horn inclusion by cobase change(namely using Λ , Λ and then Λ twice) in Cat ∞ : This either follows from a simple direct argument bywriting the posets involved as iterated pushouts of simplices, or from the corresponding statement at thelevel of simplicial sets using that homotopy pushouts in the Joyal model structure model pushouts in Cat ∞ ,or For 𝑖 ∈ {0 , … , , let ( C , Ϙ ) 𝐼 𝑖 𝑝 ⊂ ( C , Ϙ ) 𝐼 𝑖 denote the full subcategory on functors that satisfy whicheverof condition (i), (ii), and (iii’) apply. Then for 𝑖 = 0 the map ( C , Ϙ ) 𝐼 𝑖 𝑝 → ( C ′ , Ϙ ′ ) 𝐼 𝑖 induced by 𝑝 is equivalentto that in the right hand column of the statement of the Lemma, and for 𝑖 = 4 it is the right hand map in(27).We then claim that the diagram(28) ( C , Ϙ ) 𝐼 𝑖 𝑝 ( C , Ϙ ) 𝐼 𝑖 −1 𝑝 ( C ′ , Ϙ ′ ) 𝐼 𝑖 ( C ′ , Ϙ ′ ) 𝐼 𝑖 −1 , 𝑝 𝑝 with horizontal maps given by restriction, is a pullback in Cat h∞ . This establishes the lemma by pastingpullbacks.Indeed, 𝐼 is obtained from 𝐼 by filling the -horn Λ ⊂ Δ with a cocartesian edge, so that the restrictionmap ( C , Ϙ ) 𝐼 → ( C , Ϙ ) 𝐼 is pulled back from the restriction map 𝑠 ∶ ( C , Ϙ ) Δ → ( C , Ϙ ) . It follows that thediagram in question is obtained from the second diagram of Lemma 2.5.3 by base changes, and therefore isa pullback.Similarly, we see that the diagrams for 𝑖 = 1 , , are obtained by base-changes from the diagrams ofLemma 2.5.4 and therefore pullbacks.We are left to show that the right vertical map in the statement of the lemma, namely ( E , Ϙ ) × ( C , Ϙ ) Q ( C , Ϙ ) ⟶ Q ( C ′ , Ϙ ′ ) × ( C ′ , Ϙ ′ ) Q ( C ′ , Ϙ ′ ) , is a split Poincaré-Verdier projection. But Lemma 2.5.7 identifies this map as a base change of Q ( 𝑝 ) ∶ Q ( C , Ϙ ) → Q ( C ′ , Ϙ ′ ) , which is a Poincaré-Verdier projection by Proposition 1.4.14. The claim thus follows from Corol-lary 1.2.6. (cid:3) Proof of Theorem 2.4.3.
Applying F to the squares of Lemmas 2.5.7 and 2.5.8, and using additivity, wededuce that the following squares are also pullbacks: F ( E , Ϙ ) F ( C , Ϙ ) F (Q ( C ′ , Ϙ ′ )) F ( C ′ , Ϙ ′ ) 𝑑 𝑝 𝑝𝑑 F ( E , Ϙ ) × F (Q ( C , Ϙ )) F (Q ( C , Ϙ )) F ( E , Ϙ ) × F ( C , Ϙ ) F (Q ( C , Ϙ )) F (Q ( C ′ , Ϙ ′ )) F (Q ( C ′ , Ϙ ′ )) × F ( C ′ , Ϙ ′ ) F (Q ( C ′ , Ϙ ′ )); (id ,𝑑 ) 𝑝 𝑝 ( 𝑑 ,𝑑 ) here the pullback in the right hand square is formed using 𝑑 on the left and 𝑑 on the right. Now the righthand square tells us that the image of 𝜋 F ( E , Ϙ ) → 𝜋 F (Q ( C , Ϙ )) consists of F Q( 𝑝 ) -cocartesian arrows,whence the left hand square provides sufficiently many F Q( 𝑝 ) -cocartesian lifts to make F Q( 𝑝 ) ∶ F Q( C , Ϙ ) → F Q( C ′ , Ϙ ′ ) into a cocartesian fibration of Segal spaces: To see the former claim map the right square to F ( E , Ϙ ) F ( E , Ϙ ) F Q ( C ′ , Ϙ ′ ) F Q ( C ′ , Ϙ ′ ) in the evident fashion and take fibres of a given point ̂𝑓 ∈ F ( E , Ϙ ) and its images. The resulting fibresquare is precisely the necessary square making its image 𝑓 ∈ F (Q ( C , Ϙ )) a F Q( 𝑝 ) -cocartesian morphism,see [Ste18, Definition 2.6]. Mapping instead to the square F ( E , Ϙ ) × F ( C , Ϙ ) F ( E , Ϙ ) × F ( C , Ϙ ) F Q ( C ′ , Ϙ ′ ) × F ( C ′ , Ϙ ′ ) F Q ( C ′ , Ϙ ′ ) × F ( C ′ , Ϙ ′ ) by addtionally extracting the last vertex and passing to fibres over ( 𝑓 , 𝑡 ) ∈ F ( E , Ϙ ) × F ( C , Ϙ ) , we obtain thecartesian square Hom
Cob F ( C , Ϙ ) ( 𝑡 ( 𝑓 ) , 𝑥 ) Hom Cob F ( C , Ϙ ) ( 𝑠 ( 𝑓 ) , 𝑡 )Hom Cob F ( C ′ , Ϙ ′ ) ( 𝑡 ( 𝑝𝑓 ) , 𝑝𝑥 ) Hom Cob F ( C ′ , Ϙ ′ ) ( 𝑠 ( 𝑝𝑓 ) , 𝑝𝑥 ) − ◦ 𝑓 − ◦ 𝑝𝑓 via the equivalence Hom
Cob F ( C , Ϙ ) ( 𝑐, 𝑑 ) ≃ f ib ( 𝑐,𝑑 ) ( F (Q ( C , Ϙ )) ( 𝑑 ,𝑑 ) ←←←←←←←←←←←←←←←←←←←←←←←←←→ F ( C , Ϙ ) ) . Thus, any 𝑓 ∈ F ( E , Ϙ ) also defines a Cob F ( 𝑝 ) -cocartesian morphism making Cob F ( 𝑝 ) a cocartesian fibra-tion as well. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 65 Since Q( C ) is naturally identfied with Q( C ) op through the canonical identification TwAr(Δ 𝑛 ) ≅ TwAr((Δ 𝑛 ) op ) ,see Example 2.2.3, we conclude that both F Q( 𝑝 ) and Cob F ( 𝑝 ) are also cartesian fibrations. (cid:3) Additivity in K -Theory. The arguments presented in the previous section work verbatim upon drop-ping hermitian structures and working with additive functors
Cat ex∞ → S . In the present section we brieflyrecord the statements that are obtained this way.Let us first formally set terminology obviously analogous to that of Definition 1.5.4.2.6.1. Definition.
Let E be an ∞ -category with finite limits and F ∶ Cat ex∞ → E a reduced functor. Wesay that F is additive , Verdier-localising or Karoubi-localising if it sends split Verdier squares, arbitraryVerdier squares or Karoubi squares to cartesian squares, respectively.The gist of the following result also appears in [BR13], though in incommensurable generality.2.6.2.
Proposition.
For a stable ∞ -category C the simplicial category Q( C ) is a Segal object in Cat ∞ ,whose boundary maps are split Verdier projections. For an additive functor F ∶ Cat ex∞ → S , and F Q( C ) isa Segal space, which is complete if F preserves pullbacks.Proof. This first two statements are obtained during the proofs of Lemmas 2.1.5 and 2.1.7. The latter twostatement are proven just as Proposition 2.2.1. (cid:3)
In particular, we can extract a category
Span F ( C ) from F Q( C ) , and it inherits a symmetric monoidalstructure since C ↦ Span F ( C ) preserves products. The proof of Corollary 2.2.9 gives the statement that 𝜋 F (Ar( C )) 𝜋 F ( C )0 𝜋 | Span F ( C ) | 𝑡 is a pushout. In the non-hermitian situations, the top horizontal map is, however, surjective: It is split forexample by the exact functor 𝑥 ↦ (0 → 𝑥 ) . We obtain:2.6.3. Proposition.
The category
Span F ( C ) is connected for any stable C and additive F ∶ Cat ex∞ → S . In particular, | Span F ( C ) | is always an E ∞ -group. Furthermore, replacing He ∶ Cat p∞ → Cat ∞ by Ar ∶ Cat ex∞ → Cat ∞ , the first statement of Proposition 2.2.12 becomes tautological, and the second partbecomes Waldhausen’s additivity theorem that | Span( C ) | ≃ | Span(Ar( C )) | . The simple proof of Propo-sition 2.2.12, however, uses the identification Cob(Hyp( C )) ≃ Span( C ) , which has no analogue in the nonhermitian set-up. Waldhausen’s additivity theorem instead follows from the following analogue of the ad-ditivity theorem 2.4.1:2.6.4. Theorem (Additivity) . If F ∶ Cat ex∞ → S is additive, then so is | Span F − | ≃ | F Q − | . As above, this theorem is deduced from the following statement:2.6.5.
Theorem.
Let F ∶ Cat ex∞ → S be additive and 𝑝 ∶ C → C ′ a split Verdier projection, then 𝑝 ∶ F Q( C ) → F Q( C ′ ) is a bicartesian fibration of Segal spaces and thus a realisation fibration. The proof of Theorem 2.4.3 in §2.5, in particular, verifies Theorem 2.6.5 upon dropping all mention ofPoincaré structures (which in fact made up the bulk of the work).Waldhausen’s additivity theorem now follows, by inserting the analogue of the metabolic sequence, i.e.the split Verdier sequence C → Ar( C ) 𝑡 ←←←←→ C into the corollary (whence our terminology), and noting that either adjoint of 𝑡 give rise to splittings of thesequence | Span( C ) | → | Span(Ar( C )) | 𝑡 ←←←←→ | Span( C ) | . This runs contrary to the situation of the metabolic sequence, where the adjoints of met ∶ Met( C , Ϙ ) → ( C , Ϙ ) are not compatible with the Poincaré structures. The splitting lemma then gives the equivalence | Span( C ) | ≃ | Span( C ) | ≃ | Span(Ar( C )) | and, since Ω | Span( C ) | = K ( C ) taking loop spaces gives K ( C ) ≃ K ( C ) ≃ K (Ar( C )) as desired.In summary, the metabolic fibre sequence is not just an algebraic analogue of Genauer’s fibre sequenceregarding geometric cobordism categories, but also of Waldhausen’s additivity, the connection betweenwhich was first realised by the ninth author in [Ste18].Finally, as we will have to make use of this result in the next section, let us also record the computationof 𝜋 K( C ) = 𝜋 | Span( C ) | in the generality of an arbitrary additive F ∶ Cat ex∞ → S : The natural equivalence Hom
Span F C (0 ,
0) ≃ F ( C ) provides maps 𝜋 F ( C ) ( 𝑠, cof) 𝜋 F (Ar C ) 𝜋 F ( C ) 𝜋 | Span F ( C ) | 𝜋 | Span F (Ar C ) | 𝜋 | Span F ( C ) | , t where 𝑠, 𝑡 and cof take the source, target, and cofibre of a morphism. The additivity theorem implies thatthe lower left horizontal map is an isomorphism. Inverting it produces a commutative diagram 𝜋 F (Ar C ) ( 𝑠, cof) 𝜋 F ( C ) 𝜋 F ( C ) 𝜋 | Span F ( C ) | 𝑡 of abelian monoids natural in both C and F .2.6.6. Proposition.
This square is cocartesian for every stable C and every additive F ∶ Cat ex∞ → S . In particular, for F = Cr we recover the standard fact that K ( C ) is given by 𝜋 𝜄 ( C ) modulo extensions. Proof.
While a proof internal to the Q -construction is certainly possible, the quickest route is throughthe well-known subdivision equivalence | F Q( C ) | ≃ | F S( C ) | with the Segal construction, as employed byWaldhausen. In S( C ) the 0-,1- and 2-simplices are given by ∗ , C and Cof( C ) , respectively, where Cof ( C ) denotes the category of cofibre sequences in C . This is equivalent to Ar( C ) and under this identification theboundary maps of S( C ) are given by source, target and cofibre. Thus we find 𝜋 | F S( C ) | given by 𝜋 F ( C ) modulo the relation 𝑠 ( 𝑓 )+cof( 𝑓 ) = 𝑡 ( 𝑓 ) for every 𝑓 ∈ 𝜋 F (Ar( C )) , which is exactly the pushout above. (cid:3)
3. S
TRUCTURE THEORY FOR ADDITIVE FUNCTORS
The objective of this section is to derive the fundamental theorems of Grothendieck-Witt theory from theadditivity theorem. We will, however, do so in the generality of arbitrary additive functors
Cat p∞ → S . Evenwhen only interested in Grothendieck-Witt spectra this additional layer of generality is useful, for exampleit enters our proof of the universal property of GW ∶ Cat p∞ → S 𝑝 . The reader is encouraged to keep the twofundamental examples Pn and | Cob(−) | ∶ Cat p∞ → S in mind throughout. In §4 below, we will specialise the results of this section to define Grothendieck-Witttheory and conclude the main theorems of this paper.We begin by introducing the notion of a cobordism between Poincaré functors, and use this to establishsome fundamental results for group-like additive functors. Chief among these is the agreeance of theirvalues on hyperbolic and metabolic categories. In the case of | Cob(−) | we already proved this claim inProposition 2.2.12 by explicit identification of both sides. Using the general statement as a base case,we develop a general theory of isotropic decompositions of Poincaré ∞ -categories, which allows for thecomputations of the values of a group-like additive functor F applied to many Poincaré ∞ -categories ( C , Ϙ ) of interest, e.g. Q 𝑛 ( C , Ϙ ) for all 𝑛 , in terms of hyperbolic pieces and parts that are often simpler then theoriginal ( C , Ϙ ) . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 67 We then use this machinery to establish precise relationships between additive functors taking valuesin the categories of E ∞ -monoids, E ∞ -groups and spectra, in particular constructing left adjoints to theevident forgetful functors. The adjoint passing from E ∞ -monoid- to E ∞ -group-valued functors, the group-completion , is given by F ⟶ Ω | Cob F (−) | ∶= Ω | F Q(− [1] ) | , using the F -based cobordism categoryfrom section §2, and the adjoint from E ∞ -group-valued to spectrum-valued functors, the spectrification ,is given by iterating the Q -construction on F . This generalises the work of Blumberg-Gepner-Tabuada onthe universality of algebraic K -theory [BGT13]. Many of our constructions also have geometric precursorsin the work of Bökstedt-Madsen on the connection between iterated cobordism categories and algebraicK-theory [BM14]. We will expand on these analogies in §4.We then turn to a more detailed analysis of spectrum-valued additive functors. To this end we introducethe notion of a bordism-invariant functor (i.e. one that vanishes on metabolic categories), the principalexample being L ∶ Cat p∞ → S 𝑝 , the L -theory functor of Ranicki and Lurie. We show that the inclusion ofbordism invariant functors into all additive functors also admits a left adjoint bord . It will then follow forrather formal reasons that there always is a natural bicartesian square F ( C , Ϙ ) F bord ( C , Ϙ ) F (Hyp( C )) hC F (Hyp( C )) tC of spectra, which can in principle be used to compute F from its hyperbolisation F hyp = F ◦ Hyp andits bordification F bord , each of which may be easier to understand than F . We also provide two directformulas for F bord , which again have precursors in manifold theory. We will use these in §4 to identify thebordification of Grothendieck-Witt theory with L-theory, completing the proof of the main theorem.3.1. Cobordisms of Poincaré functors.
In the previous section we introduced the concept of cobordism ina Poincaré ∞ -category. When applied to the Poincaré ∞ -category of exact functors between two Poincaré ∞ -categories this yields a natural notion of a cobordism between functors:3.1.1. Definition.
Let ( C , Ϙ ) and ( D , Φ) be two Poincaré ∞ -categories and let 𝑓 , 𝑔 ∶ ( C , Ϙ ) → ( D , Φ) be twoPoincaré functors. By a cobordism from 𝑓 to 𝑔 we shall mean a cobordism in the Poincaré ∞ -category Fun ex (( C , Ϙ ) , ( D , Φ)) between the Poincaré objects corresponding to 𝑓 and 𝑔 .We note that the data of such a cobordism can equivalently be encoded by a Poincaré functor 𝜙 ∶ ( C , Ϙ ) → Q ( D , Φ) such that 𝑑 𝜙 = 𝑓 and 𝑑 𝜙 = 𝑔 .Our first goal is to describe the behaviour of group-like additive functors under such cobordisms. Recallfrom Definition 1.5.8 that an additive functor F ∶ Cat p∞ → E into a category admitting finite products iscalled group-like if its canonical lift Cat p∞ → Mon E ∞ ( E ) arising from the semi-addivity of Cat p∞ actuallytakes values in the full subcategory Grp E ∞ ( E ) ⊆ Mon E ∞ ( E ) . Regarded as functor Cat p∞ → Grp E ∞ ( E ) , F isthen again additive, since limits in Grp E ∞ ( E ) are computed in E , and group-like, since Grp E ∞ ( E ) is additive.We start by analysing the universal case of a Poincaré cobordism between functors with target ( C , Ϙ ) . Itis given by the two Poincaré functors 𝑑 , 𝑑 ∶ Q ( C , Ϙ ) → Q ( C , Ϙ ) = ( C , Ϙ ) , which are equipped with atautological cobordism between them.To this end, consider the functor(29) 𝑖 ∶ C ⟶ Q ( C ) , 𝑥 ⟼ [ ← 𝑥 → 𝑥 ] and its right adjoint 𝑝 ∶ Q ( C ) ⟶ C , [ 𝑥 ← 𝑤 → 𝑦 ] ⟼ f ib( 𝑤 → 𝑥 ) . Note that the unit transformation id ⇒ 𝑝𝑖 is an equivalence, so 𝑖 is fully-faithful. By the universal propertyof the hyperbolic construction, Corollary [I].7.2.20, we obtain a pair of Poincaré functors(30) Hyp( C ) Q ( C , Ϙ ) Hyp( C ) 𝑖 hyp 𝑝 hyp which is a retract diagram in Cat p∞ . We also note that 𝑖 hyp ∶ Hyp( C ) → Q ( C , Ϙ ) factors through Met( C , Ϙ ) ⊆ Q ( C , Ϙ ) ; the corresponding restriction of 𝑖 hyp agrees with can ∶ Hyp( C ) → Met( C , Ϙ ) (compare the recol-lection section for a review of notation). Similary, the restriction of 𝑝 hyp to Met( C , Ϙ ) ⊆ Q ( C , Ϙ ) is exactly lag ∶ Met( C , Ϙ ) → Hyp( C ) . In particular, we obtain the commutative diagram(31) ( C , Ϙ )Met( C , Ϙ ) Q ( C , Ϙ ) ( C , Ϙ )Hyp( C ) cyl idlag 𝑝 hyp 𝑑 with cyl the inclusion of constant functors and 𝑝 hyp split (as a Poincaré functor) by 𝑖 hyp .3.1.2. Lemma.
For ( C , Ϙ ) a Poincaré ∞ -category both the horizontal and vertical sequence of (31) are splitPoincaré-Verdier sequences.Proof. For the horizontal sequence this is immediate from Lemma 2.1.7. For the vertical sequence weshall check that 𝑝 satisfies the assumptions of Lemma 1.4.1 to conclude that 𝑝 hyp is a split Poincaré-Verdierprojection; the kernel of 𝑝 hyp is evidently given by the diagrams TwAr(Δ ) → Cr C , and since | TwAr(Δ ) | iscontractible these are exactly the constant diagrams, which embed C fully faithfully into Q ( C ) and evidently cyl ∗ Ϙ ≃ Ϙ .We already recorded above that 𝑝 admits a fully faithful left adjoint 𝑖 taking 𝑥 to ← 𝑥 → 𝑥 , and Ϙ (0 ← 𝑥 → 𝑥 ) ≃ Ϙ (0) ≃ 0 . A right adjoint 𝑟 to 𝑝 is readily checked to be given by the formula 𝑥 ⟼ [Σ 𝑥 ← → and since D Ϙ ( [Σ 𝑥 ← → ) ≃ [ΩD Ϙ 𝑥 ← ΩD Ϙ 𝑥 → we also find Ϙ (D Ϙ ( 𝑟𝑥 )) ≃ 0 for all 𝑥 ∈ C as desired. (cid:3) Applying Proposition 1.5.11 to (31) we thus obtain:3.1.3.
Corollary.
Let F ∶ Cat p∞ → E be a group-like additive functor. Then the following holds:i) The Poincaré functor cyl ∶ ( C , Ϙ ) → Q ( C , Ϙ ) and the inclusion Met( C , Ϙ ) → Q ( C , Ϙ ) induce anequivalence F ( C , Ϙ ) × F (Met( C , Ϙ )) ⟶ F (Q ( C , Ϙ )) , and F sends the horizontal sequence of (31) to a bifibre sequence in Grp E ∞ ( E ) .ii) The functors cyl ∶ ( C , Ϙ ) → Q ( C , Ϙ ) and 𝑖 hyp ∶ Hyp( C ) → Q ( C , Ϙ ) induce an equivalence F ( C , Ϙ ) × F (Hyp( C )) ⟶ F (Q ( C , Ϙ )) , and F sends the vertical sequence of (31) to a bifibre sequence in Grp E ∞ ( E ) .iii) The functors 𝑑 ∶ Q ( C , Ϙ ) → ( C , Ϙ ) and 𝑝 hyp ∶ Q ( C , Ϙ ) → Hyp( C ) induce an equivalence F (Q ( C , Ϙ )) ⟶ F ( C , Ϙ ) × F (Hyp( C )) . As a consequence of the above we obtain the following corollary, which will play a fundamental rolethroughout this paper.3.1.4.
Corollary.
Let F ∶ Cat p∞ → E be a group-like additive functor. Then the functors lag ∶ Met( C , Ϙ ) → Hyp( C ) and can ∶ Hyp( C ) → Met( C , Ϙ ) induce inverse equivalences F (Met( C , Ϙ )) ≃ F (Hyp( C )) . Proof.
The composite ( C , Ϙ ) × Met( C , Ϙ ) (cyl , inc) ←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ Q ( C , Ϙ ) ( 𝑑 ,𝑝 hyp ) ←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ ( C , Ϙ ) × Hyp( C ) is equivalent to the map id ( C , Ϙ ) × lag . Since both constituents of this composite become equivalences afterapplying F by the previous corollary, F (lag) is a retract of an equivalence and therefore an equivalence itself.Since the functor can is a one-sided inverse to lag at the level of Poincaré ∞ -categories it must induce theinverse equivalence after applying F . (cid:3) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 69 Applying Corollary 3.1.4 to the group-like additive functor ( C , Ϙ ) ↦ | Cob F ( C , Ϙ ) | for F a not necessarilygroup-like additive functor, we deduce immediately:3.1.5. Corollary.
The functors lag and can induce inverse equivalences | Cob F (Met( C , Ϙ )) | ≃ | Cob F (Hyp( C )) | , for every additive functor F ∶ Cat p∞ → S . This in particular gives an alternative proof of the second half of Proposition 2.2.12 that does not usethe algebraic Thom construction. As explained in §2.6, it is furthermore a direct analogue to Waldhausen’sadditivity theorem in the non-hermitian setting.To exploit Corollary 3.1.3 further we need:3.1.6.
Construction.
Given two Poincaré ∞ -categories ( C , Ϙ ) , ( D , Φ) and an exact functor 𝑓 ∶ C → D between the underlying categories, we obtain a Poincaré functor N 𝑓 ∶ ( C , Ϙ ) → ( D , Φ) by forming thecomposition ( C , Ϙ ) Hyp( D ) ( D , Φ) 𝑓 hyp id hyp using that the hyperbolic construction is both a left and a right adjoint to the forgetful functor Cat p∞ → Cat ex∞ .We will refer to N 𝑓 as the norm of 𝑓 .Unwinding this construction, we find (N 𝑓 )( 𝑥 ) ≃ 𝑓 ( 𝑥 ) ⊕ D Φ 𝑓 op (D Ϙ 𝑥 ) . Applying Corollary 3.1.3 to ageneral bordism between Poincaré functors we then obtain:3.1.7. Proposition.
Let F ∶ Cat p∞ → E be a group-like additive functor. Let ( C , Ϙ ) and ( D , Φ) be Poincaré ∞ -categories and let (32) 𝑓 ℎ 𝑔 be a cobordism between two Poincaré functors 𝑓 , 𝑔 ∶ ( C , Ϙ ) → ( D , Φ) . Let 𝑘 ∶ C → D be the exact functorgiven by the formula 𝑘 ( 𝑥 ) = f ib( ℎ ( 𝑥 ) → 𝑓 ( 𝑥 )) and let N 𝑘 ∶ ( C , Ϙ ) → ( D , Φ) be its norm. Then there is acanonical homotopy F ( 𝑔 ) − F ( 𝑓 ) ∼ F (N 𝑘 ) ∶ F ( C , Ϙ ) ⟶ F ( D , Φ) of maps F ( C , Ϙ ) → F ( D , Φ) .Proof. By corollary 3.1.3 we have a pair of equivalences(33) F ( D , Φ) ⊕ F (Hyp( D )) F (Q ( D , Φ)) F ( D , Φ) ⊕ F (Hyp( D )) . F ( 𝑠 ) ⊕ F ( 𝑖 hyp ) ( F ( 𝑑 ) , F ( 𝑝 hyp )) These equivalences are inverse to each other: indeed, the composite equivalence F ( D , Φ) ⊕ F (Hyp( D )) ≃ ⟶ F ( D , Φ) ⊕ F (Hyp( D )) is equivalent to the identity since 𝑝 hyp 𝑖 hyp and 𝑑 𝑠 are equivalent to the respective identity functors while 𝑑 𝑖 hyp and 𝑝 hyp 𝑠 are equivalent to the respective zero functors. The equivalences (33) then determine ahomotopy between the identity map id ∶ F (Q ( D , Φ)) → F (Q ( D , Φ)) and the sum F ( 𝑠𝑑 ) + F ( 𝑖 hyp 𝑝 hyp ) ,and hence a homotopy F ( 𝜙 ) ∼ F ( 𝑠𝑑 𝜙 ) + F ( 𝑖 hyp 𝑝 hyp 𝜙 ) = F ( 𝑠𝑓 ) + F ( 𝑖 hyp 𝑘 hyp ) of maps F ( C , Ϙ ) → F (Q ( D , Φ)) . Post composing with the map F ( 𝑑 ) ∶ F (Q ( D , Φ)) → F ( D , Φ) we obtaina homotopy F ( 𝑔 ) = F ( 𝑑 𝜙 ) ∼ F ( 𝑑 𝑠𝑓 ) + F ( 𝑑 𝑖 hyp 𝑘 hyp ) = F ( 𝑓 ) + F (N 𝑘 ) of maps F ( C , Ϙ ) → F ( D , Φ) , as desired. (cid:3) Corollary.
For a group-like additive functor F ∶ Cat p∞ → E , the inversion map on F ( C , Ϙ ) is inducedby the sum of the endofunctors (id C , −id Ϙ ) and NΩ of ( C , Ϙ ) . Proof.
We resurrect the bent cylinder bcyl∶ ( C , Ϙ ) ⟶ Q ( C , Ϙ ) with underlying functor 𝑋 ⟼ [ 𝑋 ⊕ 𝑋 Δ ←←←←←←←←← 𝑋 → from Construction 2.2.8. By construction it is a nullcobordism of id ( C , Ϙ ) + (id C , −id Ϙ ) . We obtain theconclusion from Proposition 3.1.7 by observing that the fibre of the diagonal 𝑋 → 𝑋 ⊕ 𝑋 is naturallyequivalent to Ω 𝑋 . (cid:3) Next, we use Corollary 3.1.4 to determine the fundamental group of | Cob F ( C , Ϙ ) | . We base the calcula-tion on the well-known analogue for the categories Span G ( C ) for a small stable ∞ -category C and an additivefunctor G ∶ Cat ex∞ → S (i.e. one that sends split Verdier squares to cartesian squares), that we recalled inProposition 2.6.6.Analogous to the construction in the non-hermitian case we consider the diagram 𝜋 F (Hyp( C )) 𝜋 F (Met( C , Ϙ )) 𝜋 F ( C , Ϙ ) 𝜋 | Cob F (Hyp( C )) | 𝜋 | Cob F (Met( C , Ϙ )) | 𝜋 | Cob F ( C , Ϙ ) | , lag met with the vertical maps induced by various instances of Hom
Cob F ( C , Ϙ ) (0 ,
0) ≃ F ( C , Ϙ ) . The lower left horizontal map is an isomorphism by Corollary 3.1.4. Inverting it gives the commutativesquare in the following:3.1.9.
Theorem.
For a Poincaré ∞ -category ( C , Ϙ ) and an additive functor F ∶ Cat p∞ → S the naturalsquare 𝜋 F Met( C , Ϙ ) 𝜋 F ( C , Ϙ ) 𝜋 F Hyp( C ) 𝜋 | Cob F ( C , Ϙ ) | metlag of commutative monoids is cocartesian. Since the map lag is (split) surjective, this in particular describes 𝜋 | Cob F ( C , Ϙ ) | as the quotient monoidof 𝜋 F ( C , Ϙ ) identifying all metabolic objects with the hyperbolic objects on their lagrangians. We thus, inparticular, obtain an isomorphism 𝜋 | Cob( C , Ϙ ) | ≅ GW ( C , Ϙ ) with the Grothendieck-Witt group constructed in §[I].2.4. We will discuss this further in §4 below.For the proof we will need:3.1.10. Proposition.
The boundary map of the algebraic Genauer sequence | Cob F ( C , Ϙ [−1] ) | ⟶ | Cob F (Met( C , Ϙ )) | ⟶ | Cob F ( C , Ϙ ) | participates in a commutative diagram F ( C , Ϙ )Ω | Cob F ( C , Ϙ ) | | Cob F ( C , Ϙ [−1] ) | 𝜕 with the right hand map arising from the inclusion into the core, and the left hand map from the inclusionas the endomorphism of F ( C , Ϙ [1] ) . Since the map 𝜋 F ( C , Ϙ ) → 𝜋 | Cob F ( C , Ϙ ) | is surjective by Proposition 3.1.9, this in particular deter-mines the effect of the boundary map 𝜋 | Cob F ( C , Ϙ ) | → 𝜋 | Cob F ( C , Ϙ [−1] ) | . Before giving the proof, werecord, that from Lemma 2.3.7 and the discussion thereafter we have: ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 71 Lemma.
For a Poincaré ∞ -category ( C , Ϙ ) , an additive F ∶ Cat p∞ → S and 𝑋 ∈ F ( C , Ϙ [1] ) we have Cob F ( C , Ϙ ) 𝑋 ∕ ≃ f ib ( dec( F Q( C , Ϙ [1] )) ⟶ F ( C , Ϙ [1] ) ) where the arrow extracts the object positioned at (0 ≤ and thus in particular Cob F ( C , Ϙ ) ≃ F (Null( C , Ϙ [1] )) . Here
Null( C , Ϙ [1] ) = f ib ( dec Q( C , Ϙ [1] ) ⟶ ( C , Ϙ [1] ) ) denotes the higher metabolic categories from Definition 2.3.8. Proof of Lemma 3.1.10.
Recall that one way to describe the boundary map in a fibre sequence 𝐴 → 𝐵 → 𝐶 is as the induced map on pullbacks of [ Ω 𝐶 → P 𝐶 ← P 𝐵 ] ⟹ [ ∗ → 𝐶 ← 𝐵 ] , where P denotes the spaces of paths starting at the basepoints, and the transformation from left to right isgiven by evaluation at the endpoint. For the Bott-Genauer sequence the the left side is given by Hom | Cob F ( C , Ϙ ) | (0 , ⟶ | Cob F ( C , Ϙ ) | 𝜕 ⟵ | Cob F (Met( C , Ϙ ) | and the right is ⟶ | Cob F ( C , Ϙ ) | 𝜕 ⟵ | Cob F (Met( C , Ϙ ) | with the induced maps the canonical projections. The composition from the statement is then given bymapping F ( C , Ϙ ) ⟶ | Cob F ( C , Ϙ ) | 𝜕 ⟵ | Cob F (Met( C , Ϙ ) | to the former of these diagrams via F ( C , Ϙ ) ⟶ Hom
Cob F ( C , Ϙ ) (0 , ⟶ Hom | Cob F ( C , Ϙ ) | (0 , . But this composite transformation completes to a transformation of cartesian squares F ( C , Ϙ ) | Cob F (Met( C , Ϙ )) | | Cob F ( C , Ϙ ) | | Cob F (Met( C , Ϙ )) | F ( C , Ϙ ) | Cob F ( C , Ϙ ) | | Cob F ( C , Ϙ ) | id met met as follows: Using Corollary 3.1.11 the dashed map is given by the Poincaré functor ( C , Ϙ ) ⟶ Q (Met( C , Ϙ [1] )) which sends 𝑥 to the diagram 𝑥 𝑥 𝑥 , representing another bent cylinder, whose forms by definition are given by the limit of Ϙ [1] 𝑥 Ϙ [1] 𝑥 Ϙ [1] 𝑥 which is Ϙ 𝑥 , giving the hermitian structure. It is readily checked that this functor is Poincaré. Finally, rewriting the squares above as the realisations of const F ( C , Ϙ ) F Null(Met( C , Ϙ )) F Q( C , Ϙ ) F Q(Met( C , Ϙ ))const F ( C , Ϙ ) F Null( C , Ϙ ) 0 F Q( C , Ϙ ) id met met using Corollary 3.1.11 one finds a transformation from the left to the right via inclusion as the -simplicesin the top left corner and Null ⟹ dec(Q) 𝑑 ⟹ Q on the right hand side. (cid:3) Proof of Proposition 3.1.9.
Denote by 𝐺 ( C , Ϙ ) the pushout of the diagram 𝜋 F Hyp( C ) ⟵ 𝜋 F Met( C , Ϙ ) ⟶ 𝜋 F ( C , Ϙ ) , and similarly 𝑊 ( C , Ϙ ) the pushout of ⟵ 𝜋 F Met( C , Ϙ ) ⟶ 𝜋 F ( C , Ϙ ) , giving a canonical map 𝐺 ( C , Ϙ ) → 𝑊 ( C , Ϙ ) . By construction there is a natural map 𝐺 ( C , Ϙ ) → 𝜋 | Cob( C , Ϙ ) | .Now the discussion of the non-Poincaré case in Proposition 2.6.6 implies that this map is an equivalencefor hyperbolic categories: The square in Proposition 3.1.9 for F ∶ Cat p∞ → S and input Hyp( C ) becomesthat for F ◦ Hyp ∶ Cat ex∞ → S and input category C , under the equivalences Met(Hyp( C )) ≃ Hyp(Ar( C )) and Hyp( C ) ≃ Hyp(Hyp( C )) from Corollary [I].2.3.23 and Remark [I].7.4.15.Let us now construct a diagram 𝜋 | Cob F ( C , Ϙ [−1] ) | 𝜋 | Cob F (Met( C , Ϙ )) | 𝜋 | Cob F ( C , Ϙ ) | 𝜋 | Cob F ( C , Ϙ [−1] ) | 𝐺 ( C , Ϙ [−1] ) 𝐺 (Hyp( C )) 𝐺 ( C , Ϙ ) 𝑊 ( C , Ϙ ) ≅ ≅ , whose upper sequence is induced by the metabolic fibre sequence via additivity and thus exact. Furthermore,the rightmost map of the top sequence is surjective as indicated, since the next term in the sequence is 𝜋 | Cob F (Met( C , Ϙ )) | , which vanishes by Corollary 2.2.10. The vertical maps are the evident ones (seeCorollary 2.2.9 for the right most one), except the second one, which is the composition 𝐺 (Hyp( C )) can ←←←←←←←←←←←←←→ 𝐺 (Met( C , Ϙ )) ⟶ 𝜋 | Cob F (Met( C , Ϙ )) | . The left two horizontal maps in the lower sequence are 𝐺 ( C , Ϙ [−1] ) ⟶ 𝐺 (Met( C , Ϙ )) lag ←←←←←←←←←←←←→ 𝐺 (Hyp( C )) and hyp∶ 𝐺 (Hyp( C )) can ←←←←←←←←←←←←←→ 𝐺 (Met( C , Ϙ )) met ←←←←←←←←←←←←←←→ 𝐺 ( C , Ϙ ) , respectively. The right one is that constructed above. The right vertical map is an isomorphism by Corol-lary 2.2.9 and the second one by Corollary 3.1.4 and the claim for hyperbolic categories established above.Now the middle square commutes by construction, the left one by Corollary 3.1.4 and the right byLemma 3.1.10. Furthermore, the lower sequence is exact at 𝐺 ( C , Ϙ ) in the sense that two elements 𝑥, 𝑦 ∈ 𝐺 ( C , Ϙ ) have the same image in 𝑊 ( C , Ϙ ) if and only if there are elements 𝑤, 𝑧 in the image of 𝐺 (Hyp( C )) such that 𝑥 + 𝑤 = 𝑧 + 𝑦 : By the surjectivity of 𝜋 F (Hyp C ) → 𝐺 (Hyp C ) this follows straight from thecocartesian diagram 𝜋 F Met( C , Ϙ ) 𝜋 F ( C , Ϙ ) 𝜋 F Hyp( C ) 𝐺 ( C , Ϙ ) by taking horizontal cokernels. It then follows formally that 𝐺 ( C , Ϙ ) is in fact a group: Since 𝑊 ( C , Ϙ ) isone, there is for every 𝑎 ∈ 𝐺 ( C , Ϙ ) an element 𝑎 ′ ∈ 𝐺 ( C , Ϙ ) such that 𝑎 + 𝑎 ′ maps to in 𝑊 ( C , Ϙ ) . But ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 73 then by exactness there are 𝑏, 𝑏 ′ ∈ 𝐺 (Hyp( C )) with 𝑎 + 𝑎 ′ + 𝑏 ′ = 𝑏 , from which we can subtract 𝑏 to get aninverse to 𝑎 , since 𝐺 (Hyp( C )) is group.Furthermore, the composition 𝐺 ( C , Ϙ [−1] ) ⟶ 𝐺 (Hyp( C )) ⟶ 𝐺 ( C , Ϙ ) vanishes: By construction the map met ∶ 𝐺 (Met( C , Ϙ )) → 𝐺 ( C , Ϙ ) factors as 𝐺 (Met( C , Ϙ )) lag ←←←←←←←←←←←←→ 𝐺 (Hyp( C )) hyp ←←←←←←←←←←←←←←→ 𝐺 ( C , Ϙ ) which identifies the composition above with 𝐺 ( C , Ϙ [−1] ) ⟶ 𝐺 (Met( C , Ϙ )) met ←←←←←←←←←←←←←←→ 𝐺 ( C , Ϙ ) which vanishes already at the level of categories. It is a bit tedious to check that the lower sequence is infact exact at 𝐺 (Hyp( C )) . Luckily, we get away without doing so directly:We deduce Proposition 3.1.9 by two applications of the -lemma. Applying one half of it to the rightthree columns (extended by to the right) gives surjectivity of the map 𝐺 ( C , Ϙ ) → 𝜋 | Cob 𝐹 ( C , Ϙ ) | for every ( C , Ϙ ) , in particular also for the left most column. This formally implies exactness at 𝐺 (Hyp( C )) by a shortdiagram chase, whence the other half of the 4-lemma gives injectivity and thus the claim. (cid:3) Isotropic decompositions of Poincaré ∞ -categories. We now describe a rather general situationwhich gives rise to cobordisms of Poincaré functors. We will use it to analyse the categories Q 𝑛 ( C , Ϙ ) , seeProposition Proposition 3.2.15, below.Let now ( C , Ϙ ) be a Poincaré ∞ -category. Given a full subcategory L ⊆ C we will denote by L ⟂ ⊆ C thefull subcategory spanned by the objects 𝑦 ∈ C such that B Ϙ ( 𝑥, 𝑦 ) ≃ 0 for every 𝑥 ∈ L . Using B Ϙ ( 𝑥, 𝑦 ) ≃Hom C ( 𝑥, D Ϙ ( 𝑦 )) we immediately see that D Ϙ ( L ⟂ ) ⊆ C is the full subcategory of C consisting of the objects 𝑧 ∈ C that are right orthogonal to L , i.e. for which Map C ( 𝑥, 𝑧 ) ≃ 0 for every 𝑥 ∈ L .3.2.1. Definition.
By an isotropic subcategory of ( C , Ϙ ) we shall mean a full stable subcategory L ⊆ C withthe following properties:i) Ϙ vanishes on L .ii) The composite functor L op ⟶ C op D Ϙ ←←←←←←←←←←←←→ C ∕ L ⟂ is an equivalence.The first condition in particular implies L ⊆ L ⟂ , and the second expresses a unimodularity condition on L . In the ordinary theory of quadratic forms the analogue of this condition is equivalent to the requirementthat an isotropic subspace be a direct summand. It admits a convenient reformulation:3.2.2. Lemma.
For a stable subcategory L ⊆ C the composite functor L op ⟶ C op D Ϙ ←←←←←←←←←←←←→ C ∕ L ⟂ is anequivalence if and only if the inclusion of L into C admits a right adjoint. Furthermore, in this case L =( L ⟂ ) ⟂ . In particular, Lagrangians as considered in Definition [I].7.3.10, are examples of isotropic subcategories;we will recall their definition in Definition 3.2.7 below.
Proof.
The composite being an equivalence is clearly equivalent to L → C → C ∕D Ϙ ( L ⟂ ) being one. Since D Ϙ ( L ⟂ ) consists exactly of the right orthogonal of L it is closed under retracts in C and thus gives a Verdierinclusion into C by Proposition A.1.9. Both the equivalence of the conditions in the statement and the laststatement are then instances of Corollary A.2.8. (cid:3) Remark.
Applying the remainder of Corollary A.2.8 in the situation at hand, we find that the kernelof the right adjoint 𝑝 ∶ C → L is given by D Ϙ ( L ⟂ ) and thus L ⟂ is the kernel of 𝑝 ◦ D Ϙ .3.2.4. Remark.
The condition that L = ( L ⟂ ) ⟂ or even L = L ⟂ , does not imply condition ii) of the definitionof an isotropic category. For a concrete counterexample, take C = D p ( 𝐾 [ 𝑇 ]) , for 𝐾 a field of characteristicdifferent from . The involution sending 𝑇 to − 𝑇 provides 𝐾 [ 𝑇 ] with the structure of a ring with involution,and we can consider the symmetric Poincaré structure this involution provides. Fix then an ≠ 𝑎 ∈ 𝐾 and consider the subcategory L = D p ( 𝐾 [ 𝑇 ]) 𝑇 − 𝑎 spanned by those complexesthat become contractible after inverting 𝑇 − 𝑎 , i.e. whose homology is 𝑇 − 𝑎 -power torsion. We first claimthat Ϙ s vanishes on L : For example from the universal coefficient sequence, one finds that the homology of D Ϙ s ( 𝑋 ) is 𝑇 + 𝑎 -power torsion for 𝑋 ∈ L . But then, since 𝑇 + 𝑎 and 𝑇 − 𝑎 generate the unit ideal, 𝑇 − 𝑎 acts invertibly on D Ϙ s 𝑋 , so Ϙ s ( 𝑋 ) ≃ B Ϙ s ( 𝑋, 𝑋 ) hC ≃ Hom 𝐾 [ 𝑇 ] ( 𝑋, D Ϙ s ( 𝑋 )) hC ≃ 0 . Similarly, 𝑋 ∈ L ⟂ if and only if 𝑋 is left orthogonal to all perfect 𝑇 + 𝑎 -torsion complexes. Since every 𝑇 + 𝑎 -torsion complex is a colimit of perfect ones by Example 1.4.2 𝑋 is thus left orthogonal to the entiretyof ( Mod 𝐾 [ 𝑇 ] ) 𝑇 + 𝑎 . But the (non-small) Verdier sequence ( Mod 𝐾 [ 𝑇 ] ) 𝑇 + 𝑎 ⟶ Mod 𝐾 [ 𝑇 ] ⟶ Mod 𝐾 [ 𝑇 , ( 𝑇 + 𝑎 ) −1 ] is split, with left adjoint to the localisation given by the inclusion Mod 𝐾 [ 𝑇 , ( 𝑇 + 𝑎 ) −1 ] → Mod 𝐾 [ 𝑇 ] . The imageof this left adjoint is the left orthogonal to the Verdier kernel by Lemma A.2.3. In total then L ⟂ consistsexactly of those perfect complexes over 𝐾 [ 𝑇 ] on which 𝑇 + 𝑎 acts invertibly. Since this can be checkedon homology it follows easily from the classification of finitely generated modules over the principal idealdomain 𝐾 [ 𝑇 ] , that these are exactly the perfect 𝐾 [ 𝑇 ] -complexes that become contractible when localisedaway from the prime ideal ( 𝑇 + 𝑎 ) . Repeating the argument above by localising at the complement of ( 𝑇 − 𝑎 ) instead of inverting 𝑇 + 𝑎 then shows L = ( L ⟂ ) ⟂ .But the inclusion of perfect 𝑇 − 𝑎 -power torsion complexes into all perfect 𝐾 [ 𝑇 ] -complexes cannothave a right adjoint: If a map 𝑅 ( 𝑀 ) → 𝑀 from a 𝑇 − 𝑎 -power torsion module induces an equiva-lence Hom 𝐾 [ 𝑇 ] ( 𝑋, 𝑅 ( 𝑀 )) ≃ Hom 𝐾 [ 𝑇 ] ( 𝑋, 𝑀 ) for all perfect 𝑇 − 𝑎 -power torsion modules 𝑋 , then thisin fact holds for all 𝑇 − 𝑎 -power torsion modules. But then 𝑅 ( 𝑀 ) necessarily agrees with the imageof 𝑀 under the right adjoint to the inclusion ( Mod 𝐾 [ 𝑇 ] ) 𝑇 − 𝑎 → Mod 𝐾 [ 𝑇 ] , which is given by 𝑋 ↦ f ib ( 𝑋 → 𝑋 [( 𝑇 − 𝑎 ) −1 ] ) . But even for 𝑋 = 𝐾 [ 𝑇 ] , this is not a perfect 𝐾 [ 𝑇 ] -module.To upgrade this example to one where L = L ⟂ simply replace 𝐾 [ 𝑇 ] by its localisation at the complementof ( 𝑇 − 𝑎 ) ∪ ( 𝑇 + 𝑎 ) .A similar construction generally works for a Dedekind domain with an involution that swaps two maximalideals.3.2.5. Definition.
For an isotropic subcategory L of a Poincaré ∞ -category ( C , Ϙ ) , we define the homologycategory Hlgy( L ) to be the cofibre of the inclusion ( L , Ϙ ) → ( L ⟂ , Ϙ ) in Cat h∞ .Thus the underlying category is L ⟂ ∕ L and the hermitian structure is the left Kan extension of Ϙ | ( L ⟂ ) op along the projection ( L ⟂ ) op → ( L ⟂ ∕ L ) op . The next proposition, in particular, shows that Ϙ | ( L ⟂ ) op in factdescends along the projection ( L ⟂ ) op → ( L ⟂ ∕ L ) op and gives a Poincaré structure on Hlgy( L ) .3.2.6. Proposition.
Let L be an isotropic subcategory of a Poincaré ∞ -category ( C , Ϙ ) . Then both (B Ϙ ) | ( L ⟂ × L ⟂ ) op and (L Ϙ ) | ( L ⟂ ) op descend along the projection ( L ⟂ ) op → ( L ⟂ ∕ L ) op and give the bilinear and linear part ofthe hermitian structure on Hlgy( L ) , which is Poincaré. The duality on Hlgy( L ) is induced by the functor L ⟂ → L ⟂ sending 𝑋 to f ib(D Ϙ 𝑋 → D Ϙ 𝑝𝑋 ) , where 𝑝 denotes the right adjoint to L ⊆ C and the arrow isinduced by the counit.In particular, the composite (34) L ⟂ ∩ D( L ⟂ ) ⟶ L ⟂ ⟶ L ⟂ ∕ L = Hlgy( L ) canonically refines to an equivalence of Poincaré ∞ -categories using the restriction of Ϙ on the source. In particular,
Hlgy( L ) is equivalent to a full Poincaré subcategory of ( C , Ϙ ) , which one may think of asthe subcategory of harmonic objects for L . We denote by(35) 𝜄 ∶ Hlgy( L ) ⟶ ( C , Ϙ ) . the arising fully-faithful Poincaré functor. Proof.
The first two statements follow from the general analysis of Kan-extended hermitian structures: ByLemma [I].1.4.3 the linear and bilinear parts are given by the left Kan-extensions along ( L ⟂ ) op → ( L ⟂ ∕ L ) op of the restriction to L ⟂ . But they in fact descend along the projection: This is immediate from Condition i) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 75 of Definition 3.2.1 for the linear part and from the definition of L ⟂ in the case of the bilinear part. Notethat this implies via the decomposition into linear and bilinear parts that Ϙ | ( L ⟂ ) op also descends along theprojection ( L ⟂ ) op → ( L ⟂ ∕ L ) op , as claimed above. It furthermore implies that the hermitian structure on Hlgy( L ) is also right Kan extended along this map, which we will use below.For the equivalence of hermitian ∞ -categories claimed in the statement, note first that by Lemma A.2.5and the comments thereafter the cofibre of the counit 𝑝𝑋 → 𝑋 constitutes a right adjoint 𝑞 to the localisation L ⟂ → Hlgy( L ) . In fact, Lemma A.2.5 implies that 𝑞 is an equivalence onto the kernel of 𝑝 ∶ L ⟂ → L ,which is D Ϙ ( L ⟂ ) ∩ L ⟂ by REmark 3.2.3. In particular, 𝑞 is also a right adjoint to the composite 𝑐 ∶ L ⟂ ∩ D( L ⟂ ) → Hlgy( L ) from the statement, which is thus also an equivalence. Now right Kan extensions are computed by pullbackalong left adjoints, so the hermitian structure on Hlgy( L ) is given by Ϙ ◦ 𝑞 op , which upgrades 𝑞 and thus 𝑐 to an equivalence of hermitian ∞ -categories.Finally, L ⟂ ∩ D( L ⟂ ) is evidently closed under D Ϙ so forms a Poincaré subcategory of C , whence also Hlgy( L ) is Poincaré. The statement about the duality in Hlgy( L ) then follows from the formula for theinverse 𝑞 of 𝑐 . (cid:3) Definition.
Let L ⊆ C be an isotropic subcategory of a Poincaré ∞ -category ( C , Ϙ ) . We will saythat L is a Lagrangian if Hlgy( L ) = 0 . We will say that ( C , Ϙ ) is metabolic if it contains a Lagrangiansubcategory.As mentioned, Remark 3.2.2 shows that this definition of Lagrangian agrees with that discussed in Def-inition [I].7.3.10.3.2.8. Remark.
By Lemma A.1.8, an isotropic subcategory L ⊆ C is a Lagrangian if and only if the inclusion L ⊆ L ⟂ is an equivalence. Condition ii) of Definition 3.2.1 therefore yields a Verdier sequence L ⟶ C ⟶ L op exhibiting C as an extension of L by L op , where the right functor takes 𝑋 to f ib(D Ϙ 𝑋 → D Ϙ 𝑝𝑋 ) (and 𝑝 denotes the right adjoint of the inclusion L ⊆ C ). Furthermore Lemma A.2.5 shows that the right functorin this Verdier sequence admits a right adjoint as well.3.2.9. Examples. i) We showed in Proposition [I].7.3.11 that a Poincaré ∞ -category ( C , Ϙ ) is metabolic if and only if itis of the form Pair( D , Φ) for some hermitian ∞ -category ( D , Φ) . In fact, the Lagrangians in ( C , Ϙ ) are in one-to-one correspondence with representations of ( C , Ϙ ) as a category of pairings. Particularexamples are the inclusion C → Met( C , Ϙ ) as the equivalences, and C × 0 ⊂ Hyp( C ) .ii) Extending the Lagrangian of the metabolic category, the full subcategory inclusion 𝑖 ∶ C ↪ Q ( C ) of (29) sending 𝑥 to ← 𝑥 → 𝑥 gives an isotropic subcategory L , the adjoint 𝑝 witnessing Condition ii)of Definition 3.2.1 given by [ 𝑋 ← 𝑌 → 𝑍 ] ⟼ [0 ← f ib( 𝑌 → 𝑋 ) → f ib( 𝑌 → 𝑋 )] . Thus D Ϙ L ⟂ = ker( 𝑝 ) is spanned by all diagrams with left pointing arrow an equivalence, whereas L ⟂ itself consists of all diagram with right hand arrow an equivalence. Thus Hlgy( L ) ≃ ( C , Ϙ ) embeddedas the constant diagrams.iii) More generally one can consider the inclusion 𝑗 𝑛 ∶ C → Q 𝑛 ( C ) as those diagrams which vanish awayfrom {( 𝑖 ≤ 𝑛 ) ∣ 𝑖 ∈ {0 , ...𝑛 }} , and are constant on that subposet, i.e. ⋯ ⋯ ⋯ 𝑋 𝑋 ⋯ 𝑋 Formally this can be given by taking the embedding C → Q ( C ) considered in the previous exampleand composing with the degeneracy [ 𝑛 ] → [1] sending 𝑛 to and everything else to . Using the Segalproperty of Lemma 2.1.5 it is not difficult to see that the requisite adjoint 𝑝 is given by taking a diagram 𝜑 ∈ Q 𝑛 ( C ) to image of the fibre of the last left pointing arrow, namely 𝜑 ( 𝑛 −1 ≤ 𝑛 ) → 𝜑 ( 𝑛 −1 ≤ 𝑛 −1) .It follows that D Ϙ 𝑛 ( L ⟂ ) = ker( 𝑝 ) consists of all those diagrams 𝜑 with the arrow 𝜑 ( 𝑛 − 1 ≤ 𝑛 ) → 𝜑 ( 𝑛 − 1 ≤ 𝑛 − 1) an equivalence (and thus all arrows 𝜑 ( 𝑖 ≤ 𝑛 ) → 𝜑 ( 𝑖 ≤ 𝑛 − 1) equivalences as well).From the explicit formula for the duality of Q ( C , Ϙ ) from Example 2.1.3 i) it then follows that L ⟂ isspanned by the diagrams with the last right pointing arrow 𝜑 ( 𝑛 − 1 ≤ 𝑛 ) → 𝜑 ( 𝑛 ≤ 𝑛 ) an equivalence,and so in total Hlgy( L ) ≃ Q 𝑛 −1 ( C , Ϙ ) embedded in Q 𝑛 ( C , Ϙ ) via the degeneracy 𝑠 𝑛 −1 .iv) There are several other interesting isotropic subcategories of Q 𝑛 ( C , Ϙ ) : For example, let L + 𝑛 ⊆ Q 𝑛 ( C ) be the full subcategory spanned by those diagrams 𝜑 ∶ TwAr[ 𝑛 ] op → C for which 𝜑 (0 ≤
0) = 0 and 𝜑 (0 ≤ 𝑗 ) → 𝜑 ( 𝑖 ≤ 𝑗 ) is an equivalence for 𝑖 ≤ 𝑗 ∈ [ 𝑛 ] , i.e. 𝑋 … 𝑋 𝑛 𝑋 𝑋 … 𝑋 𝑛 𝑋 𝑋 … 𝑋 𝑛 Then L + 𝑛 ≃ Fun(Δ 𝑛 −1 , C ) is an isotropic subcategory: To give the right adjoint 𝑝 𝑛 of the inclu-sion L + 𝑛 ↪ Q 𝑛 ( C ) , let 𝜌 + 𝑛 ∶ Δ 𝑛 → TwAr(Δ 𝑛 ) denote the functor 𝑘 ↦ (0 ≤ 𝑘 ) . Then 𝑝 𝑛 sends 𝜑 ∶ TwAr([ 𝑛 ]) op → C to the left Kan extension along 𝜌 + 𝑛 ∶ [ 𝑛 ] → TwAr([ 𝑛 ]) of the functor [ 𝑛 ] ⟶ C 𝑗 ↦ f ib( 𝜑 (0 ≤ 𝑗 ) → 𝜑 (0 ≤ . In particular, the category D( L + 𝑛 ) ⟂ consists exactly of those diagrams that are right Kan extended fromthe image of 𝜌 + 𝑛 and ( L + 𝑛 ) ⟂ is dually spanned by those diagrams that are left Kan extended from thesubposet spanned by the various ( 𝑖 ≤ 𝑛 ) . The homology Hlgy( L + 𝑛 ) ⊆ Q 𝑛 ( C , Ϙ ) is consequently givenby the full Poincaré subcategory of constant diagrams.v) The isotropic subcategory L + 𝑛 +1 ≃ Fun(Δ 𝑛 , C ) from the previous example agrees with that from theproof of Proposition 2.3.9 upon restriction to Null 𝑛 ( C , Ϙ ) ⊆ Q 𝑛 +1 ( C , Ϙ ) . We showed there, that it is aLagrangian in Null 𝑛 ( C , Ϙ ) , and this follows again from the considerations above.In generalisation of Corollary 3.1.3 we now set out to prove:3.2.10. Theorem (Isotropic decomposition theorem) . Let ( C , Ϙ ) be a Poincaré ∞ -category and 𝑖 ∶ L → C be the inclusion of an isotropic subcategory. Let F ∶ Cat p∞ → E be a group-like additive functor. Then thePoincaré functors 𝑖 hyp ∶ Hyp( L ) ⟶ ( C , Ϙ ) and 𝜄 ∶ Hlgy( L ) ⟶ ( C , Ϙ ) from (35) induce an equivalence (36) F (Hyp( L )) × F (Hlgy( L )) ⟶ F ( C , Ϙ ) . We will explicitly construct an inverse to the map appearing in the theorem.3.2.11.
Construction.
Fix a Poincaré ∞ -category ( C , Ϙ ) and the inclusion 𝑖 ∶ L → C of an isotropic sub-category with right adjoint 𝑝 . We note that the counit 𝑖𝑝 → id C defines a surgery datum on the Poincaréobject id ( C , Ϙ ) of Fun ex (( C , Ϙ ) , ( C , Ϙ )) . Performing surgery as in Proposition 2.3.3, we obtain a Poincaré ob-ject in Q (Fun ex (( C , Ϙ ) , ( C , Ϙ ))) , in other words, a Poincaré functor 𝜙 ∶ ( C , Ϙ ) → Q ( C , Ϙ ) . By construction, 𝑑 ◦ 𝜙 = id , and we denote by ℎ the composite 𝑑 ◦ 𝜙 ∶ ( C , Ϙ ) → ( C , Ϙ ) , that is, the result of surgery, giving in total a cobordism(37) 𝑔 id ℎ. 𝛽𝛼 By construction 𝜙 ( 𝑐 ) ∶ ( 𝑐 ← 𝑔 ( 𝑐 ) → ℎ ( 𝑐 ) ) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 77 is obtained first by forming the fiber 𝑔 ( 𝑐 ) → 𝑐 of the composite map 𝑐 ≃ D Ϙ D Ϙ ( 𝑐 ) → D Ϙ ( 𝑖𝑝 D Ϙ ( 𝑐 )) wherethe second map is the dual of the counit, and then by forming the cofiber 𝑔 ( 𝑐 ) → ℎ ( 𝑐 ) of the canonicallyinduced map 𝑖𝑝 ( 𝑐 ) → 𝑔 ( 𝑐 ) .3.2.12. Lemma.
The functor ℎ ∶ C → C factors through the inclusion L ⟂ ∩ D Ϙ ( L ⟂ ) ⊆ C , and the Poincaréenhancement furnished by Construction 3.2.11 canonically factors as ( C , Ϙ ) ̃ℎ ←←←←←←→ Hlgy( L ) 𝜄 ←←←←→ ( C , Ϙ ) and ̃ℎ ◦ 𝜄 ≃ id Hlgy( L ) . In particular, if L is Lagrangian then ℎ = 0 .Proof. For the first part we observe that both the cofibre of 𝑖𝑝𝑐 → 𝑐 and D Ϙ 𝑖𝑝 D Ϙ 𝑐 belong to D Ϙ ( L ⟂ ) : Theformer because D Ϙ ( L ⟂ ) = ker( 𝑝 ) by Remark 3.2.3 and for the latter we simply note 𝑝 D Ϙ 𝑋 ∈ L ⊆ L ⟂ .Since ℎ𝑐 participates in a cofibre sequence ℎ𝑋 ⟶ cof( 𝑖𝑝𝑐 → 𝑐 ) ⟶ D Ϙ ( 𝑖𝑝 D Ϙ ( 𝑐 )) , also ℎ𝑐 ∈ D Ϙ ( L ⟂ ) . Since ℎ commutes with the duality its image is then also contained in L ⟂ .For the second claim, note that 𝑖𝑝 ( 𝑐 ) ≃ 0 for 𝑐 ∈ L ⟂ ∩ D Ϙ ( L ⟂ ) , since ker( 𝑝 ) = D Ϙ ( L ⟂ ) by Remark 3.2.3.Thus the cobordism (37) consists of equivalences in this case. The third claim is immediate from Proposi-tion 3.2.6. (cid:3) Proposition.
The functors 𝑝 hyp ∶ ( C , Ϙ ) ⟶ Hyp( L ) and ( C , Ϙ ) ̃ℎ ←←←←←←→ Hlgy( L ) combine into a left inverse of the Poincaré functor ( 𝑖 hyp , 𝜄 ) ∶ Hyp( L ) ⊕ Hlgy( L ) ⟶ ( C , Ϙ ) from Theorem 3.2.10.Proof. Consider the composite
Hyp( L ) ⊕ Hlgy( L ) ( 𝑖 hyp ,𝜄 ) ←←←←←←←←←←←←←←←←←←←←←←←←→ ( C , Ϙ ) ( 𝑝 hyp ,̃ℎ ) ←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ Hyp( L ) ⊕ Hlgy( L ) . We will analyse all four components in turn. That ̃ℎ𝜄 ≃ id ∶ Hlgy( L ) → Hlgy( L ) is part of Lemma 3.2.12.For the self-map of the Hyp( L ) -component we have that 𝑝 hyp 𝑖 hyp ( 𝑥, 𝑦 ) = 𝑝 hyp ( 𝑖 ( 𝑥 ) ⊕ D Ϙ 𝑖 ( 𝑦 )) = ( 𝑝𝑖 ( 𝑥 ) ⊕ 𝑝 D Ϙ 𝑖 ( 𝑦 ) , 𝑝 D Ϙ 𝑖 ( 𝑥 ) ⊕ 𝑝𝑖 ( 𝑦 )) . Since 𝑖 ∗ Ϙ vanishes it follows that Hom C ( 𝑖 ( 𝑥 ) , D Ϙ 𝑖 ( 𝑥 )) = Hom C ( 𝑖 ( 𝑦 ) , D Ϙ 𝑖 ( 𝑦 )) = 0 and hence 𝑝 D Ϙ 𝑖 ( 𝑥 ) = 𝑝 D Ϙ 𝑖 ( 𝑦 ) = 0 . We may then conclude that 𝑝 hyp 𝑖 hyp ( 𝑥, 𝑦 ) = ( 𝑝𝑖 ( 𝑥 ) , 𝑝𝑖 ( 𝑦 )) = Hyp( 𝑝𝑖 ) ∶ Hyp( L ) ⟶ Hyp( L ) and hence the unit equivalence id L → 𝑝𝑖 induces an equivalence id Hyp( L ) → 𝑝 hyp 𝑖 hyp , as desired.The map 𝑝 hyp ◦ 𝜄 ∶ Hlgy( L ) → Hyp( L ) vanishes since 𝑝 hyp 𝜄 = ( 𝑝𝜄 ) hyp and L ⟂ ∩ D Ϙ L ⟂ ⊆ ker( 𝑝 ) by Remark 3.2.3.Finally, we similarly have ̃ℎ ◦ 𝑖 hyp ≃ ( ̃ℎ ◦ 𝑖 ) hyp , and ̃ℎ ◦ 𝑖 ≃ 0 , since 𝑝 D Ϙ 𝑖 ( 𝑥 ) ≃ 0 as observed above so that 𝑐 → 𝑔 ( 𝑐 ) is an equivalence. (cid:3) Proof of Theorem 3.2.10.
It only remains to show that the composite map F ( C , Ϙ ) F (Hyp( L )) × F (Hlgy( C )) F ( C , Ϙ ) ( 𝑝 hyp∗ ,̃ℎ ∗ ) ( 𝑖 hyp∗ ,𝜄 ∗ ) is homotopic to the identity. But this now readily follows by applying Proposition 3.1.7 to the cobordismof Construction 3.2.11 and observing that 𝑖𝑝 is identified by construction with the fibre of 𝛽 ∶ 𝑔 → ℎ . (cid:3) In generalisation of Corollary 3.1.4 we thus find:3.2.14.
Corollary.
Let F ∶ Cat p∞ → E be a group-like additive functor and let ( C , Ϙ ) be a Poincaré ∞ -category. If ( C , Ϙ ) is metabolic with Lagrangian L ⊆ C then F ( C , Ϙ ) ≃ F (Hyp( L )) . Next, we use Theorem 3.2.10 to analyse the values of Q 𝑛 ( C , Ϙ ) under group-like additive functors. Tostate the result consider the functor 𝑓 𝑛 ∶ Q 𝑛 ( C , Ϙ ) → C 𝑛 taking fibres of the left pointing maps along thebottom of a diagram 𝑋 , i.e. 𝑋 ⟼ [ f ib ( 𝑋 (0 ≤ → 𝑋 (0 ≤ ) , … , f ib ( 𝑋 ( 𝑛 − 1 ≤ 𝑛 ) → 𝑋 ( 𝑛 − 1 ≤ 𝑛 − 1) )] . We then have:3.2.15.
Proposition.
The functors 𝑣 𝑛 ∶ Q 𝑛 ( C , Ϙ ) → ( C , Ϙ ) and 𝑓 hyp 𝑛 ∶ Q 𝑛 ( C , Ϙ ) → Hyp( C ) 𝑛 the former induced by the inclusion [0] → [ 𝑛 ] , combine into an equivalence F (Q 𝑛 ( C , Ϙ )) ≃ F (Hyp( C )) 𝑛 ⊕ F ( C , Ϙ ) for every group-like additive F ∶ Cat p∞ → E . In fact, these equivalences give an identification of the simpli-cial E ∞ -group F Q( C , Ϙ ) in E with the bar construction of F (Hyp( C )) acting on F ( C , Ϙ ) via the hyperboli-sation map hyp ∶ F (Hyp( C )) ⟶ F ( C , Ϙ ) . In particular, it follows that
Cob F ( C , Ϙ ) ≃ | F Q( C , Ϙ [1] ) | is a groupoid, provided F is group-like (andadditive). Proof.
We proceed by induction. For 𝑛 = 0 there is nothing to show. Using the isotropic subcategory 𝑗 𝑛 +1 ∶ C → Q 𝑛 +1 ( C ) described in Example 3.2.9 iii) we find an equivalence (( 𝑗 𝑛 +1 ) hyp , 𝑠 𝑛 ) ∶ F Hyp( C ) × F Q 𝑛 ( C , Ϙ ) ⟶ F Q 𝑛 +1 ( C , Ϙ ) as a consequence of Theorem 3.2.10. It is readily checked that this equivalence translates the map ( 𝑓 hyp 𝑛 +1 , 𝑣 𝑛 +1 ) to the matrix ⎛⎜⎜⎝ 𝑓 hyp 𝑛 id Hyp( C ) 𝑣 𝑛 ⎞⎟⎟⎠ ∶ F Hyp( C ) × F Q 𝑛 ( C , Ϙ ) ⟶ F (Hyp( C )) 𝑛 × F Hyp( C ) × F ( C , Ϙ ) . This matrix represents an equivalence by inductive assumption, which implies the first claim.To obtain an identification with the bar construction, we first note that the bar construction B( 𝑀, 𝑅, 𝑁 ) of an action of 𝑅 on 𝑁 from the left and on 𝑀 from the right in a semi-additive category is the left Kanextension along the inclusion of the coequaliser diagram (Δ ≤ ) op into Δ op of the diagram 𝑀 ⊕ 𝑅 ⊕ 𝑁 𝑀 ⊕ 𝑁 containing the two action maps; this follows directly by evaluation of the pointwise formulae for left Kanextensions. By the calculations above 𝑑 ∶ F Q ( C , Ϙ )) → F ( C , Ϙ ) is identified with the projection F (Hyp) × F ( C , Ϙ ) → F ( C , Ϙ ) and it is readily checked that 𝑑 ∶ F Q ( C , Ϙ ) → F ( C , Ϙ ) gets identified with the sum of theidentity of F ( C , Ϙ ) and the hyperbolisation map under the equivalence of Proposition 3.2.15. We thereforeobtain a map of simplicial objects B(0 , F (Hyp( C )) , F ( C , Ϙ )) ⟶ F Q( C , Ϙ )) and one readily unwinds the construction to find it given by the maps we just checked to be equivalences. (cid:3) Recall the higher metabolic categories
Null 𝑛 ( C , Ϙ ) = f ib(Q 𝑛 ( C , Ϙ ) 𝑑 ←←←←←←←→ ( C , Ϙ )) , where 𝑑 is induced bythe inclusion [0] → [1 + 𝑛 ] , from Definition 2.3.8:3.2.16. Corollary.
The functors 𝑓 hyp 𝑛 +1 ∶ Null 𝑛 ( C , Ϙ ) → Hyp( C ) 𝑛 induce an equivalence F Q(Null 𝑛 ( C , Ϙ )) ≃ B F (Hyp( C )) for every group-like additive F ∶ Cat p∞ → E .Proof. Note only that the sequence defining
Null 𝑛 ( C , Ϙ ) is in fact a split Poincaré-Verdier sequence byLemma 2.1.7, so gives rise to a fibre sequence after applying F . The result then follows immediately fromProposition Proposition 3.2.15. (cid:3) Remarks.
ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 79 i) The identification from Proposition 3.2.15 also shows that for group-like F the Segal space F Q( C , Ϙ ) iscomplete if and only if F (Hyp C ) vanishes (see §3.5 below for a detailed discussion of such functors):For a bar construction as above, the entirety of B( 𝑀, 𝑅, 𝑁 ) = 𝑀 ⊕ 𝑅 ⊕ 𝑁 consists of equivalences,so it is complete if and only if 𝑅 = 0 .ii) We based the proof of Proposition 3.2.15 on the isotropic decomposition theorem 3.2.10, but Propo-sition 3.2.15 can also be obtained directly using the Segal property of the simplicial space F Q( C , Ϙ ) and the bar construction, together with the computation of F Q ( C , Ϙ ) from Corollary 3.1.3; we leavethe details to the reader.iii) In Example 3.2.9 v) we constructed a Lagrangian Fun(Δ 𝑛 , C ) → Null 𝑛 ( C , Ϙ ) and Theorem 3.2.10therefore directly yields F Null 𝑛 ( C , Ϙ ) ≃ F Hyp Fun(Δ 𝑛 , C ) . This formula also implies Corollary 3.2.16 by an iterative application of the splitting lemma; a similardiscussion applies to F (Q 𝑛 ( C , Ϙ )) , we again leave the details to the reader. Interestingly, our proof ofCorollary Corollary 3.2.16 does not yield the assertion that Null 𝑛 ( C , Ϙ ) is metabolic; indeed for 𝑛 ≥ we are not aware of an isotropically embedded C 𝑛 → Q 𝑛 ( C , Ϙ ) .3.3. The group-completion of an additive functor.
Our goal in this section is to study the behavior ofspace-valued additive functors under the hermitian Q -construction, or equivalently of the assignment F ↦ | Cob F (−) | . In the present section we show that this procedure is the internal suspension in the (non locallysmall) category Fun add (Cat p∞ , S ) , see Theorem 3.3.4, which plays a key role in the study of the Grothendieck-Witt spectrum in §4.1. This is based on the following observation: For any additive functor F ∶ Cat p∞ → S ,there is a natural cartesian square F ( C , Ϙ ) Cob F ( C , Ϙ ) {0} Cob F ( C , Ϙ ) in Cat ∞ , since Hom
Cob F ( C , Ϙ ) (0 ,
0) ≃ F ( C , Ϙ ) , which is immediate from our discussion of Segal spaces in§2.2. We will show that the realisation of this square is always cocartesian and even bicartesian if F isgroup-like. Since the upper right corner becomes contractible upon realisation this gives the claim.As before, to carry out the requisite analysis we consider the corresponding statement at the level of theSegal spaces F Q( C , Ϙ ) . We start by constructing the corresponding model for the cartesian square above.Recall the decalage dec( 𝑆 ) of a simplicial object 𝑆 , i.e dec( 𝑆 ) 𝑛 ≃ 𝑆 𝑛 , and that we have Cob F ( C , Ϙ ) ≃ asscat( F Null( C , Ϙ )) from Corollary 3.1.11, where the higher metabolic categories Null 𝑛 ( C , Ϙ ) are given as the fibre of(38) dec(Q( C , Ϙ )) ⟶ Q ( C , Ϙ ) = ( C , Ϙ ) . Considering the face map 𝑑 ∶ [ 𝑛 ] → [1 + 𝑛 ] as a natural transformation Δ 𝑛 ⇒ Δ 𝑛 yields a map ofsimplicial objects(39) 𝜋 ∶ Null( C , Ϙ ) ⟶ Q( C , Ϙ ) . Lemma.
The simplicial objects
Null( C , Ϙ ) and dec(Q( C , Ϙ )) extend to a split simplicial objects overthe zero Poincaré ∞ -category and ( C , Ϙ ) , respectively. In particular, | F dec(Q( C , Ϙ )) | ≃ F ( C , Ϙ ) by [Lur09a, Lemma 6.1.3.16] (which also defines split simplicialobjects). Proof.
By construction the augmented simplicial object (38) is split, which gives both results. (cid:3)
Now, to describe the final map from the square, let 𝜄 𝑛 ∶ C → Null 𝑛 ( C , Ϙ ) be the simplicial map which atlevel 𝑛 is given by the exact functor which sends 𝑥 ∈ C to the diagram 𝜑 𝑥 ∶ TwAr[ 𝑛 ] → C given by 𝜑 𝑥 ( 𝑖 ≤ 𝑗 ) = { 𝑥 𝑖 < 𝑗 otherwise in which all the maps between the various 𝑥 ’s are identities. We note that the image of 𝜄 𝑛 is contained in thekernel of(40) 𝜋 𝑛 ∶ Null 𝑛 ( C , Ϙ ) ⟶ Q 𝑛 ( C , Ϙ ) Lemma.
The functor 𝜄 𝑛 ∶ C → Null 𝑛 ( C , Ϙ ) determines an equivalence of stable ∞ -categories between C and the kernel of (40) . In addition, the restriction of the quadratic functor of Null 𝑛 ( C , Ϙ ) to C along 𝜄 𝑛 isnaturally equivalent to Ϙ [−1] .Proof. By definition, the kernel of (40) consists of those 𝜑 ∶ TwAr[ 𝑛 + 1] op → C in Q 𝑛 +1 ( C , Ϙ ) such that 𝜑 ( 𝑖 ≤ 𝑗 ) = 0 if either ( 𝑖 ≤ 𝑗 ) = (0 ≤ or 𝑖 ≥ . The only non-zero entries of such a functor arehence 𝜑 (0 ≤ 𝑗 ) for 𝑗 ≥ , and for ≤ 𝑖 ≤ 𝑗 the maps 𝜑 (0 ≤ 𝑗 ) → 𝜑 (0 ≤ 𝑖 ) are equivalences by theexactness conditions of Definition 2.1.1, since 𝜑 (1 ≤ 𝑖 ) = 𝜑 (1 ≤ 𝑗 ) = 0 . Conversely, every functor 𝜑 ∶ TwAr[ 𝑛 + 1] op → C which satisfies these vanishing conditions and for which 𝜑 (0 ≤ 𝑗 ) → 𝜑 (0 ≤ 𝑖 ) are equivalences satisfies all the exactness conditions of Definition 2.1.1. We may hence conclude that 𝜄 𝑛 +1 yields an equivalence between C and kernel of (40), since the elements (0 ≤ 𝑗 ) span a contractible category.To finish the proof we note that for 𝑥 ∈ C we have lim ( 𝑖 ≤ 𝑗 )∈TwAr[ 𝑛 +1] op Ϙ ( 𝜑 𝑥 ( 𝑖 ≤ 𝑗 )) = lim ( 𝑖 ≤ 𝑗 )∈ I op 𝑛 +1 Ϙ ( 𝜑 𝑥 ( 𝑖 ≤ 𝑗 )) ≃ 0 × Ϙ ( 𝑥 ) Ϙ ( 𝑥 ) where I 𝑛 +1 ⊆ TwAr[ 𝑛 + 1] is the cofinal full subposet of the twisted arrow category spanned by the arrowsof the form ( 𝑖 ≤ 𝑗 ) for 𝑗 ≤ 𝑖 + 1 , see Examples 2.1.3. (cid:3) In light of Lemma 3.3.2 we now obtain a fibre sequence of simplicial Poincaré ∞ -categories(41) const( C , Ϙ [−1] ) 𝜄 ⟶ Null( C , Ϙ ) 𝜋 ⟶ Q( C , Ϙ ) . Remark.
The sequence (41) is a split Poincaré-Verdier sequence in each degree: By Lemma 2.1.7,the maps 𝑑 ∶ Q 𝑛 ( C , Ϙ ) → Q 𝑛 ( C , Ϙ ) are split Poincaré-Verdier projections, and the left adjoint of 𝑑 isgiven via extension by and thus factors through the underlying categories of Null 𝑛 ( C , Ϙ ) → Q 𝑛 ( C , Ϙ ) ,whence Corollary 1.2.3 gives the claim.As desired applying an additive functor F ∶ Cat p∞ → S levelwise to the sequence (41) yields a sequenceof Segal spaces which corresponds to the fibre sequence of ∞ -categories F ( C , Ϙ ) ⟶ Cob F ( C , Ϙ ) ⟶ Cob F ( C , Ϙ ) . Here the second functor is the canonical projection as in Lemma 2.3.7 and the first functor is informallygiven by sending a Poincaré object 𝑥 to the cobordism [0 ← 𝑥 → . In total we have thus modelled thesquare from the start of this section.We can now formulate the main result of the present section:3.3.4. Theorem.
Let F ∶ Cat p∞ → S be an additive functor and consider the commutative square of spacevalued functors (42) F | F Null(− [1] ) | ∗ | F Q(− [1] ) | obtained from the sequence (41) . Then we have:i) The square is cocartesian in Fun add (Cat p∞ , S ) , and so exhibits | F Q(− [1] ) | ≃ | Cob F (−) | as the suspen-sion of F in Fun add (Cat p∞ , S ) , since the upper right corner is contractible.ii) If F is group-like then the square is also cartesian, yielding an equivalence 𝜏 F ∶ F ⟶ Ω | Cob F (−) | . in Fun add (Cat p∞ , S ) . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 81 Before giving the proof of Theorem 3.3.4 let us give some of its direct consequences. Given an additivefunctor F ∶ Cat p∞ → S , the square (42) determines a natural map(43) F ⟶ Ω | Cob F (−) | in Fun add (Cat p∞ , S ) . The codomain of (43), being the loop of another additive functor, is always group-like. Our goal is to show that (43) exhibits Ω | Cob F (−) | as universal among group-like additive functorsreceiving a map from F . In other words, we claim that the association F ↦ Ω | Cob F (−) | realises thegroup-completion of F in the semi-additive category Fun add (Cat p∞ , S ) . We need a general lemma:3.3.5. Proposition.
Let E be a semi-additive ∞ -category which admits suspensions and loops, and let E grp ⊆ E be the full subcategory spanned by the group-like objects. Then the following holds:i) The full subcategory E grp ⊆ E is closed under any limits and colimits that exist in E , and both thesuspension and loop functors Σ , Ω ∶ E → E have their image contained in E grp . In particular, we mayconsider the monad ΩΣ ∶ E → E as a functor from E to E grp .ii) If the suspension functor Σ ∶ E grp → E grp is fully-faithful then the unit map 𝑢 ∶ id ⇒ ΩΣ exhibits ΩΣ as left adjoint to the inclusion E grp → E .iii) For every object 𝐴 ∈ E , the suspension of the unit Σ 𝑢 ∶ Σ 𝐴 → ΣΩΣ 𝐴 is an equivalence.Proof. The first claim follows from the fact that 𝑥 ∈ E being group-like can be detected on the level ofboth the represented functor Map(− , 𝑥 ) and the corepresented functor Map( 𝑥, −) (which automatically takevalues in monoid objects since E is semi-additive), and that loop spaces are always group-like.To prove the second claim, it suffices to check that under the given assumptions the natural transforma-tions 𝑢 ΩΣ 𝑥 , ΩΣ 𝑢 𝑥 ∶ ΩΣ 𝑥 → ΩΣΩΣ 𝑥 are both equivalences [Lur09a, Proposition 5.2.7.4]. But since ΩΣ isa monad these two natural transformations admit a common section (the multiplication of the monad) and Σ ∶ E grp → E grp being fully-faithful implies that 𝑢 is a natural equivalence on all group-like objects of E .The final claim now follows by adjunction: For any 𝐵 ∈ E , the induced map Hom E (ΣΩΣ 𝐴, 𝐵 ) (Σ 𝑢 ) ∗ ←←←←←←←←←←←←←←←←←←←←→ Hom E (Σ 𝐴, 𝐵 ) identifies with Hom E (ΩΣ 𝐴, Ω 𝐵 ) 𝑢 ∗ ←←←←←←←←←←→ Hom E ( 𝐴, Ω 𝐵 ) , which is an equivalence by the first parts. (cid:3) Since
Fun add (Cat p∞ , S ) is semi-additive by Lemma 1.5.7, we obtain the universal property of the hermit-ian Q -construction:3.3.6. Corollary.
The natural map F → Ω | Cob F (−) | exhibits Ω | Cob F (−) | as universal among group-likeadditive functors receiving a map from F ; that is, the operation F ↦ Ω | Cob F (−) | is left adjoint to theinclusion Fun add (Cat p∞ , Grp E ∞ ( S )) ⊆ Fun add (Cat p∞ , Mon E ∞ ( S )) ≃ Fun add (Cat p∞ , S ) , of group-like additive functors inside all additive functors. We will therefore also denote Ω | Cob F (−) | as F grp and refer to it as the group-completion of F . Proof.
Note only that Part ii) of Theorem 3.3.4 implies that the unit id ⇒ Ω | Cob − | is an equivalence on allgroup-like additive functors. Thus | Cob − | restricts to a fully faithful functor on Fun add (Cat p∞ , Grp E ∞ ( S )) and the previous proposition gives the claim. (cid:3) Part iii) of Proposition 3.3.5 together with Theorem 3.3.4 also immediately implies:3.3.7.
Corollary.
For every additive functor F ∶ Cat p∞ → S and ( C , Ϙ ) ∈ Cat p∞ , the natural map | Cob F ( C , Ϙ ) | ⟶ Cob F grp ( C , Ϙ ) is an equivalence. Remark.
While we described the unit map F → Ω | F Q(− [1] ) | of the adjunction arising from Theo-rem 3.3.4 already at the beginning of this section, the counit | Ω F Q(− [1] ) | → F is more elusive. One thingwe can say about it is that the composite(44) | Ω F Q(− [1] ) | → F → Ω | F Q(− [1] ) | of the counit and unit can be identified with the negative of the canonical limit-colimit interchange map(as we will show below). In case F is group-like, the unit map is an equivalence by Theorem 3.3.4, so thisdetermines the counit for such F .Note also that in the cases F = Pn , Cr or Cr hC , or more generally any additive F for which the componentof in F ( C , Ϙ ) is always contractible, the source of the counit simply vanishes. These cases cover all additivefunctors of interest to us.To see the claim about the composite map, observe that the limit-colimit interchange map 𝜎 ∶ | Ω F Q(− [1] ) | → Ω | F Q(− [1] ) | can also be described as the Beck-Chevalley map associated to the square Fun add (Cat p∞ , S ) Fun add (Cat p∞ , S )Fun add (Cat p∞ , S ) Fun add (Cat p∞ , S ) F ↦ | F Q(− [1] ) | Σ Σ F ↦ | F Q(− [1] ) | Ω Ω where Σ F here denotes the suspension of F in Fun add (Cat p∞ , S ) , not the valuewise suspension. By Theo-rem 3.3.4 this is equivalent to | F Q(− [1] ) | , but it will be notationally advantageous to keep the notations forthe horizontal and vertical arrows separate for a moment. By definition, the Beck-Chevalley map dependson the commutativity data of the square involving the down facing vertical arrows, which itself is given bythe canonical map 𝜏 ∶ Σ | F Q(− [1] ) | → | (Σ F ) Q(− [1] ) | exchanging the order of the colimits. This map is anequivalence, since the geometric realisations occuring on both sides (which by construction are valuewise!)actually compute the colimits in Fun add (Cat p∞ , S ) : from Proposition 1.4.14 we find that each F Q 𝑛 (−) is addi-tive, and Theorem 2.4.1 implies that the valuewise realisation, which computes the colimit in Fun(Cat p∞ , S ) ,already lies in Fun add (Cat p∞ , S ) . Now unwinding the definitions using | F Q(− [1] ) | ≃ Σ F , 𝜏 becomes the selfequivalence of Σ F which switches the two suspension coordinates. In particular, the square involving thedown facing vertical arrows can be endowed with two different commutativity structures, correspondingto the swap 𝜏 and the identity of Σ F . These determine two corresponding Beck-Chevalley maps given,respectively, by ΣΩ F 𝑢 ΣΩ F ←←←←←←←←←←←←←←←←←←←←→ ΩΣΣΩ F Ω 𝜏 Ω F ←←←←←←←←←←←←←←←←←←←←←←→ ΩΣΣΩ F ΩΣ 𝑐 F ←←←←←←←←←←←←←←←←←←←←←←→ ΩΣ F and ΣΩ F 𝑢 ΣΩ F ←←←←←←←←←←←←←←←←←←←←→ ΩΣΣΩ F id ←←←←←←←←→ ΩΣΣΩ F ΩΣ 𝑐 F ←←←←←←←←←←←←←←←←←←←←←←→ ΩΣ F , where 𝑢 and 𝑐 denote the unit and counit of the adjunction Σ ⊣ Ω . The second of these is equivalent to (44),since ΩΣ 𝑐 F ◦ 𝑢 ΣΩ F ≃ 𝑢 F ◦ 𝑐 F by naturality. The first, which we showed to be the colimit-limit interchange map above, is its negative sincethe map 𝜏 Ω F ∶ Σ Ω F → Σ Ω F is homotopic to the negative of the identity map.We now turn to the proof of Theorem 3.3.4. It requires us to consider the dual Q -construction, denoted d Q( C , Ϙ ) , which we discuss next.3.3.9. Lemma.
The functors Q 𝑛 ∶ Cat p∞ → Cat p∞ and Q 𝑛 ∶ Cat ex∞ → Cat ex∞ admit left adjoints d Q 𝑛 , givenby tensoring with the poset I 𝑛 . These adjoints make both diagrams
Cat p∞ Cat p∞ Cat ex∞
Cat ex∞d Q 𝑛 fgt fgtd Q 𝑛 Cat p∞ Cat p∞ Cat ex∞
Cat ex∞d Q 𝑛 d Q 𝑛 Hyp Hyp
ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 83 commute, since the analogous diagrams involving the respective Q -constructions commute and the dia-grams in questions are then obtained by passing to left adjoints everywhere. Proof.
Recall that for [ 𝑛 ] ∈ Δ we have denoted by I 𝑛 the full subposet of TwAr(Δ 𝑛 ) spanned by thearrows of the form ( 𝑖 ≤ 𝑗 ) for 𝑗 ≤ 𝑖 + 1 . From Examples 2.1.3 we find Q 𝑛 ≃ (−) I 𝑛 , which by Proposi-tion [I].6.4.4 has (−) I 𝑛 as a left adjoint when regarded as a functor Cat h∞ → Cat h∞ . As an application ofProposition [I].6.6.1 we find, however, that ( C , Ϙ ) I 𝑛 is Poincaré whenever ( C , Ϙ ) is and from Remark [I].6.4.6and Proposition [I].6.2.2 we then find an equivalence of Poincaré ∞ -categories Fun ex (( C , Ϙ ) I 𝑛 , ( D , Φ)) ≃ Fun ex (( C , Ϙ ) , ( D , Φ) I 𝑛 ) which according to Corollary [I].6.2.12 gives the claim by passing to Poincaré objects. (cid:3) Now recall that there is a canonical equivalence
Fun L (Cat p∞ , Cat p∞ ) ≃ Fun R (Cat p∞ , Cat p∞ ) op for exampleas an immediate consequence of Lurie’s straightening equivalences, which makes both ∞ -categories equiv-alent to that of bicartesian fibrations over Δ with both fibres identified with Cat p∞ ; the superscripts L and R indicate left and right adjoint functors, respectively. In particular, as the Q -construction is a simplicialobject the left adjoints above assemble into a cosimplicial object.3.3.10. Definition.
Let ( C , Ϙ ) be a hermitian ∞ -category. We will denote by d Q( C , Ϙ ) the cosimplicialhermitian ∞ -category obtained by applying the left adjoint of Q 𝑛 in each degree.3.3.11. Remark.
The proof of Lemma 3.3.9 does not make the functoriality of d Q( C , Ϙ ) very apparent sincethe categories I 𝑛 do not form a cosimplicial object.To remedy this defect, we offer the following description of d Q 𝑛 ( C , Ϙ ) : By the discussion in Exam-ples 2.1.3 a diagram 𝜙 ∶ TwAr(Δ 𝑛 ) → C lies in Q 𝑛 ( C ) ⊆ C TwAr(Δ 𝑛 ) if and only if it lies in the image ofthe right Kan extension along the inclusion 𝜄 𝑛 ∶ I 𝑛 → TwAr(Δ 𝑛 ) . The Poincaré ∞ -category d Q 𝑛 ( C , Ϙ ) isdually given by instead considering the quotient in Cat h∞ of ( C , Ϙ ) TwAr(Δ 𝑛 ) by the kernel of the left adjoint 𝜄 ∗ 𝑛 ∶ C TwAr[ 𝑛 ] → C I 𝑛 of the canonical map ( 𝜄 𝑛 ) ∗ ∶ C I 𝑛 → C TwAr[ 𝑛 ] on the tensoring construction. Under theidentifications C I 𝑛 ≃ Fun( I op 𝑛 , C ) of Proposition [I].6.5.8 and its analogue for TwAr(Δ 𝑛 ) the kernel of 𝜄 ∗ 𝑛 consists of those 𝜑 ∶ TwAr[ 𝑛 ] op → C for which 𝜑 ( 𝑖 < 𝑗 ) = 0 whenever | 𝑗 − 𝑖 | ≤ .One can check that this description directly assembles d Q( C , Ϙ ) into a cosimplicial object of Cat p∞ , whichis left adjoint to Q( C , Ϙ ) , but we shall not need this description, so leave details to the reader.3.3.12. Definition.
Let ( C , Ϙ ) be a hermitian ∞ -category. We define dNull 𝑛 ( C , Ϙ ) to be the Poincaré-Verdierquotient of d Q 𝑛 +1 ( C , Ϙ ) by the image of the functor C = d Q ( C ) → d Q 𝑛 ( C ) induced by the inclusion [0] → [1 + 𝑛 ] .Note that Proposition 1.4.11 shows, that there is then indeed a Poincaré-Verdier sequence ( C , Ϙ ) ⟶ d Q 𝑛 ( C , Ϙ ) ⟶ dNull 𝑛 ( C , Ϙ ) . Remark.
The functor dNull 𝑛 is by definition the cofibre of the natural transformation d Q {0} ⇒ d Q 𝑛 +1 , while the functor Null 𝑛 (−) is the fibre of the natural transformation Q 𝑛 +1 → Q . We conclude thatthe association ( C , Ϙ ) ↦ dNull 𝑛 ( C , Ϙ ) is left adjoint to ( D , Φ) ↦ Null 𝑛 ( D , Φ) .For the proof of Theorem 3.3.4 we will employ the Rezk’s equifibration criterion for colimits to fit intopullback squares from [Rez14, Proposition 2.4]:3.3.14. Lemma.
Let
𝑋 𝑌𝑍 𝑊 𝜏 be a cartesian square of functors from some small category 𝐼 to S , such that the transformation 𝜏 ∶ 𝑌 ⇒ 𝑊 is equifibred, i.e. such that 𝑌 ( 𝑖 ) 𝑊 ( 𝑖 ) 𝑌 ( 𝑗 ) 𝑊 ( 𝑗 ) 𝜏 𝑖 𝜏 𝑗 is cartesian for every 𝑖 → 𝑗 in 𝐼 . Then the square colim 𝑋 colim 𝑌 colim 𝑍 colim 𝑊 𝜏 is cartesian as well.Proof. This follows from S being an ∞ -topos: By [Lur09a, Lemma 6.1.3.14] we may apply [Lur09a, The-orem 6.1.3.9 (4)] to the category S (Lurie calls an equifibred transformation cartesian). This gives us thatany extension of 𝜏 to the cone of 𝐼 , such that the extension of 𝑊 is a colimit cone, is again equifibred ifand only if the the extension of 𝑌 is also a colimit cone. Applying the backwards direction we find 𝑌 ( 𝑖 ) colim 𝑌𝑊 ( 𝑖 ) colim 𝑊 and therefore also 𝑋 ( 𝑖 ) colim 𝑌𝑍 ( 𝑖 ) colim 𝑊 is cartesian for every 𝑖 ∈ 𝐼 . Cancelling one pullback, it follows that also 𝑋 ( 𝑖 ) colim 𝑍 × colim 𝑊 colim 𝑌𝑍 ( 𝑖 ) colim 𝑍 is cartesian. But then it follows from [Lur09a, Lemma 6.1.3.2], that the extension of the transformation 𝑋 ⇒ 𝑍 to the cone of 𝐼 via the right hand column of the last diagram is also equifibred. A forwardsapplication of [Lur09a, Theorem 6.1.3.9 (4)] now gives the claim. (cid:3) We are finally ready for the proof the main result of this subsection.
Proof of Theorem 3.3.4.
We begin with the first claim. We first note that the square F | F Null(− [1] ) | ∗ | F Q(− [1] ) | is the colimit in Fun(Cat p∞ , S ) of the simplicial diagram of squares(45) F F
Null 𝑛 (− [1] )∗ F Q 𝑛 (− [1] ) in Fun add (Cat p∞ , S ) , since F Q 𝑛 and F Null 𝑛 are additive by Proposition 1.4.14, the description of Q 𝑛 in Ex-ample 2.1.3 iv) (and the analogous statement for Null 𝑛 ), and colimits in functor categories being computedpointwise. As the individual colimits are then contained in Fun add (Cat p∞ , S ) by the Additivity Theorem,specifically Theorem 2.4.1, it is also a colimit of squares in this smaller category. It will hence suffice toshow that for each 𝑛 the square (45) is cocartesian in Fun add (Cat p∞ , S ) . Since F ↦ Q 𝑛 F is obtained byprecomposition with Q 𝑛 ∶ Cat p∞ → Cat p∞ , and Q 𝑛 has a left adjoint d Q 𝑛 ∶ Cat p∞ → Cat p∞ , it follows that F ↦ F Q 𝑛 is left adjoint to F ↦ F d Q 𝑛 . Similarly, F ↦ F Null 𝑛 is left adjoint to F ↦ F dNull 𝑛 . Wemay thus conclude that all the entries of the square (45) depend on F ∈ Fun add (Cat p∞ , S ) in a colimit pre-serving manner (note also that the terminal functor ∗ is also initial in Fun add (Cat p∞ , S ) by Lemma 1.5.7). ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 85 Since
Fun add (Cat p∞ , S ) is generated under (large) colimits by corepresentables (see Remark 3.3.16 below)it will suffice to show that (45) is cocartesian when F is of the form Hom
Cat p∞ (( C , Ϙ ) , −) for some Poincaré ∞ -category ( C , Ϙ ) . Writing 𝑗 ∶ (Cat p∞ ) op → Fun add (Cat p∞ , S ) for the Yoneda embedding and using againthe adjunctions d Q 𝑛 ⊣ Q 𝑛 and dNull 𝑛 ⊣ Null 𝑛 it will now suffice to show that the square(46) 𝑗 ( C , Ϙ ) 𝑗 (dNull 𝑛 ( C , Ϙ [−1] )) 𝑗 (0) 𝑗 (d Q 𝑛 ( C , Ϙ [−1] )) is coCartesian in Fun add (Cat p∞ , S ) for every Poincaré ∞ -category ( C , Ϙ ) . Mapping the square (46) to a testfunctor G ∈ Fun add (Cat p∞ , S ) this is equivalent to saying that any additive G sends(47) d Q 𝑛 ( C , Ϙ [−1] ) ⟶ dNull 𝑛 ( C , Ϙ [−1] ) ⟶ ( C , Ϙ ) to a fibre sequence in spaces. Indeed, this is true because this sequence is a split Poincaré-Verdier sequence:Using Remark 1.2.4 this statement can be obtained entirely formally by ( Cat h∞ -enriched) adjunction fromthe sequence(48) ( C , Ϙ ) ⟶ Null 𝑛 ( C , Ϙ [1] ) ⟶ Q 𝑛 ( C , Ϙ [1] ) being split Poincaré-Verdier by Observation 3.3.3, but we shall give a more direct argument. It is immediatefrom adjointness that (47) is a cofibre sequence in Cat p∞ , so it remains to check that the composite d Q 𝑛 ( C , Ϙ [−1] ) 𝑑 ←←←←←←←←←←→ d Q 𝑛 ( C , Ϙ [−1] ) ⟶ dNull 𝑛 ( C , Ϙ [−1] ) is a Poincaré-Verdier inclusion. But from the equivalence d Q 𝑛 ( C , Ϙ ) ≃ ( C , Ϙ ) I 𝑛 we find the first map suchan inclusion by Proposition 1.4.11. Thus the Poincaré structure on d Q 𝑛 ( C , Ϙ [−1] ) is obtained from that on d Q 𝑛 ( C , Ϙ [−1] ) by pullback along 𝑑 or equivalently by left Kan extension along (the opposite of) the rightadjoint d Q 𝑛 ( C , Ϙ [−1] ) → d Q 𝑛 ( C , Ϙ [−1] ) to 𝑑 . We will therefore be done, if we show that this right adjointfactors through dNull 𝑛 ( C , Ϙ [−1] ) . But this follows from the corresponding statement for the left adjoint of 𝑑 ∶ Q 𝑛 ( C , Ϙ [1] ) → Q 𝑛 ( C , Ϙ [1] ) factoring through Null 𝑛 ( C , Ϙ [1] ) in Observation 3.3.3, since the adjunction d Q 𝑛 ⊢ Q 𝑛 is compatible with the passage to underlying categories by the discussion after Lemma 3.3.9(and the same argument gives the claim for dNull 𝑛 ⊢ Null 𝑛 ).We now prove the second claim of Theorem 3.3.4. We hence add the assumption that F if group-like.We need to show that for every Poincaré ∞ -category ( C , Ϙ ) the square of simplicial spaces(49) F ( C , Ϙ ) F (Null( C , Ϙ [1] ))∗ F (Q( C , Ϙ [1] )) realizes to a cartesian square of spaces. We first note that since F is additive the square (49) is levelwisecartesian. By Lemma 3.3.14 it will hence suffice to show that the map F (Null( C , Ϙ [1] )) → F (Q( C , Ϙ [1] )) isequifibered in the sense that for any map [ 𝑚 ] → [ 𝑛 ] in Δ the corresponding square(50) F (Null 𝑛 ( C , Ϙ ) [1] ) F (Q 𝑛 ( C , Ϙ [1] )) F (Null 𝑚 ( C , Ϙ [1] )) F (Q 𝑚 ( C , Ϙ [1] )) 𝜌 ∗ 𝜌 ∗ is cartesian. In fact, it is enough to consider only the maps 𝑑 𝑖 ∶ [ 𝑛 − 1] → [ 𝑛 ] in Δ : this follows fromthe pasting lemma for pullback squares and the fact that the 2-out-of-3 closure of these maps includes allarrows in Δ . But for 𝑑 𝑖 the result is immediate from Proposition 3.2.15 and Corollary 3.2.16. (cid:3) Remark.
Let us remark, that the equifibering condition in the previous proof can also be verifiedmore directly by appealing to the Segal property of Q( C , Ϙ ) , which reduces the claim to the three bound-ary maps 𝑑 𝑖 ∶ Q ( C , Ϙ ) → ( C , Ϙ ) . For 𝑖 = 1 , the square (50) is in fact a Poincaré-Verdier square byLemma 2.1.7 and Corollary 1.2.6 and for 𝑖 = 0 one can argue as follows: The boundary maps are not only split Poincaré-Verdier projections, but also admit splits in Cat p∞ (by the degeneracies), so induce surjections 𝜋 F Q ( C , Ϙ ) → 𝜋 F ( C , Ϙ ) . It therefore suffices to check that the vertical fibres of (50) agree for 𝜌 = 𝑑 .But the left vertical map has fibre F Hyp( C ) whereas the right has fibre F Met( C , Ϙ ) and the induced map is can , which we showed an equivalence in Corollary 3.1.4. This proof of Theorem 3.3.4 does not rely on thetheory of isotropic decompositions and the computation of F Q 𝑛 ( C , Ϙ ) for group-like F .3.3.16. Remark.
In the proof of Theorem 3.3.4 above we make use of the fact that every additive functor F ∶ Cat p∞ → S is a colimit in Fun(Cat p∞ , S ) (and hence also in the full subcategory Fun add (Cat p∞ , S ) ) ofrepresentables. Since Cat p∞ is large this colimit is a-priori indexed by a large category. This might seemproblematic since Fun(Cat p∞ , S ) does not admit all large colimits. To see what is going on let us unwindfor a minute what one means by working with the large ∞ -category Cat p∞ . In effect, one is choosing alarge inaccessible cardinal 𝜅 , and defines small to mean “of size < 𝜅 ”. The ∞ -category Fun(Cat p∞ , S ) is then actually the ∞ -category Fun(Cat p∞ ,𝜅 , S 𝜅 ) of functors from 𝜅 -small Poincaré ∞ -categories to 𝜅 -small spaces. Choosing a larger inaccessible cardinal 𝜏 > 𝜅 one may embed this ∞ -category in the ∞ -category Fun(Cat p∞ ,𝜅 , S 𝜏 ) of functors to 𝜏 -small spaces, which itself admits 𝜏 -small colimits. Then anyfunctor F ∶ Cat p∞ ,𝜅 → S 𝜏 is the colimit of the associated canonical diagram of representables indexed by (Cat p∞ ,𝜅 ) op∕ F , and the latter is 𝜏 -small provided 𝜏 is chosen to be sufficiently large with respect to 𝜅 . If F happens to take values in 𝜅 -small spaces then this colimit (which is equivalent to F ) is contained in Fun(Cat p∞ ,𝜅 , S 𝜅 ) and is hence also the colimit there. Put differently, any object of Fun(Cat p∞ ,𝜅 , S 𝜏 ) is a 𝜏 -small colimit of representables, and this property is inherited by any full subcategory Fun(Cat p∞ ,𝜅 , S 𝜏 ) which contains the representables, whether this subcategory admits all 𝜏 -small colimits or not.3.4. The spectrification of an additive functor.
In §3.3 we showed that for any additive functor F ∶ Cat p∞ → S the commutative square(51) F | Cob F | | Cob F | exhibits | Cob F | as the suspension of F in Fun add (Cat p∞ , S ) . Iterating this procedure we obtain for eachadditive functor F a model for the suspension pre-spectrum of F ∈ Fun add (Cat p∞ , S ) . To set the stage wefirst note that, for each 𝑛 ≥ , we have an 𝑛 -fold simplicial object in Cat p∞ given by Q ( 𝑛 ) ( C , Ϙ ) ∶ (Δ op ) 𝑛 ⟶ Cat p∞ ([ 𝑚 ] , ..., [ 𝑚 𝑛 ]) ↦ Q 𝑚 Q 𝑚 ... Q 𝑚 𝑛 ( C , Ϙ ) . By Lemmas 2.1.6 and 2.1.5 Q ( 𝑛 ) ( C , Ϙ ) is an 𝑛 -fold Segal object of Cat p∞ , the 𝑛 -fold iterated hermitian Q -construction of ( C , Ϙ ) . As a multiple Segal object it presents an (∞ , 𝑛 ) -category, though we shall not attemptto make this precise. We simply set:3.4.1. Definition.
For F ∶ Cat p∞ → S additive, we shall call the 𝑛 -fold Segal space F Q ( 𝑛 ) ( C , Ϙ [ 𝑛 ] ) the F -based 𝑛 -extended cobordism category Cob F 𝑛 ( C , Ϙ ) of ( C , Ϙ ) .In particular, Cob F ( C , Ϙ ) = F Q( C , Ϙ [1] ) really is the Segal space giving rise to the cobordism category Cob F ( C , Ϙ ) , and Cob F ( C , Ϙ ) = F ( C , Ϙ ) . Furthermore, there are canonical equivalences | Cob | Cob F 𝑗 | 𝑖 ( C , Ϙ ) | ≃ | Cob F 𝑗 + 𝑖 ( C , Ϙ ) | . The multiple Segal space
Pn Q ( 𝑛 ) ( C , Ϙ [ 𝑛 ] ) models the (∞ , 𝑛 ) -category informally described as having Poincaréobjects of ( C , Ϙ [ 𝑛 ] ) as objects, their cobordisms as morphisms, cobordisms between cobordisms as -morphismsand so on up to degree 𝑛 .3.4.2. Remark.
The analogous 𝑛 -fold topological category Cob 𝑛𝑑 (note the unfortunate index switch) forcobordism categories of 𝑑 -manifolds first appeared in [BM14], ironically inspired by the ordinary iterated Q -construction of Quillen, and served to produce cobordism theoretic deloopings of | Cob 𝑑 | . In particular,Bökstedt and Madsen showed that | Cob 𝑛𝑑 | ≃ Ω ∞− 𝑛 MTSO( 𝑑 ) , extending the theorem of Galatius, Madsen, ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 87 Tillmann and Weiss from the case 𝑛 = 1 . They used this description to give an entirely cobordism theoreticmodel for the spectrum MTSO( 𝑑 ) , which endows it with an interesting map to A (BSO( 𝑑 )) , studied exten-sively by Raptis and the 9’th author in [RS14, RS17, RS20], where it was used to give a short proof of theDwyer-Weiss-Williams index theorem [DWW03]. We will take up the study of the evident refinements ofthis map in a sequel to the present paper.The higher categorical incarnations of these extended cobordism categories are of course also the mainobjects of study in Lurie’s (sketch of a) solution to the cobordism hypothesis [Lur09c], and the results ofBökstedt-Madsen have been reproven in the language of higher categories by Schommer-Pries in [SP17].Now, denote by PS 𝑝 the category of pre-spectra, that is the lax limit of the diagram … Ω ←←←←←←←←→ S ∗ Ω ←←←←←←←←→ S ∗ Ω ←←←←←←←←→ S ∗ , consisting of sequences ( 𝑋 𝑛 ) 𝑛 ∈ ℕ of pointed spaces together with structure maps 𝑋 𝑛 → Ω 𝑋 𝑛 +1 . There isa fully faithful inclusion S 𝑝 ⊆ PS 𝑝 , which admits a left adjoint we will refer to as spectrification. It doesnot affect the homotopy groups. Furthermore, the evaluation functors 𝑒𝑣 𝑛 ∶ PS 𝑝 → S ∗ commute with bothlimits and colimits, and restrict to the functors Ω ∞− 𝑛 ∶ S 𝑝 → S ∗ (which still preserve limits, but only filteredcolimits).3.4.3. Definition.
Let F ∶ Cat p∞ → S be a functor. We will denote by ℂ ob F ( C , Ϙ ) = [Cob F ( C , Ϙ ) , | Cob F ( C , Ϙ ) | , | Cob F ( C , Ϙ ) | , …] the corresponding functor from Cat p∞ to pre-spectra with the structure maps determined by the square (51)applied to the functors | Cob F 𝑖 | .Since the ’th object in the pre-spectrum ℂ ob( C , Ϙ ) is F ( C , Ϙ ) itself, we obtain a natural map(52) F ( C , Ϙ ) ⟶ Ω ∞ ℂ ob F ( C , Ϙ ) , where the right hand side refers to the -th space of the spectrification of ℂ ob F ( C , Ϙ ) .3.4.4. Remark. i) There is another possible definition of the bonding maps of ℂ ob F ( C , Ϙ ) : One could take the map F → Ω | Cob F | provided by (51) and form | Cob F 𝑖 | ⟶ | Cob | ΩCob F | 𝑖 | ⟶ Ω | Cob | Cob F | 𝑖 | ≃ Ω | Cob F 𝑖 | These differ from the bonding maps we chose by a coordinate flip in the (1 + 𝑖 ) -fold simplicial object Cob F 𝑖 . Since iterated application of the Q -construction models the suspension in Fun add (Cat p∞ , S ) byTheorem 3.3.4, such a coordinate flip induces the negative of the identity on realisations. In particular,this choice of bonding maps gives a pre-spectrum naturally equivalent to ℂ ob( C , Ϙ ) .ii) In fact, the coordinate flips endow ℂ ob( C , Ϙ ) with the structure of an ( ∞ -categorical version of a)symmetric pre-spectrum, just as the more classical construction of K -theory spectra. We will nothave to make use of this observation, which is classically used to produce multiplicative structureson K -spectra, since we argue instead by universal properties to construct multiplicative structures inPaper [IV].iii) By Theorem 3.3.4 | Cob F 𝑛 | is a model for the 𝑛 -fold suspension of F in Fun add (Cat p∞ , S ) . Considering ℂ ob F as a pre-spectrum object in Fun add (Cat p∞ , S ) it is hence the suspension pre-spectrum of F .3.4.5. Proposition.
Let F be an additive functor Cat p∞ → S and ( C , Ϙ ) ∈ Cat p∞ . Then:i) The functor ℂ ob F ∶ Cat p∞ → PS 𝑝 is again additive and takes values in positive Ω -spectra, i.e. thestructure map | Cob F 𝑛 ( C , Ϙ ) | → Ω | Cob F 𝑛 +1 ( C , Ϙ ) | is an equivalence for every 𝑛 ≥ .ii) If F is group-like, then ℂ ob F ( C , Ϙ ) is in fact an ( Ω -)spectrum, and ℂ ob F is then additive when con-sidered as a functor ℂ ob F ∶ Cat p∞ → S 𝑝 .iii) The natural map ℂ ob F ( C , Ϙ ) → ℂ ob F grp ( C , Ϙ ) exhibits the right hand side as the spectrification of theleft. In particular, we obtain equivalences F grp ( C , Ϙ ) ≃ Ω ∞ ℂ ob F ( C , Ϙ ) and | Cob F 𝑛 ( C , Ϙ ) | ≃ Ω ∞− 𝑛 ℂ ob F ( C , Ϙ ) for 𝑛 ≥ . From Theorem 3.3.4 and Part ii) of this proposition we thus obtain the following universal property forthe iterated hermitian Q -construction:3.4.6. Corollary.
For a group-like additive functor F ∶ Cat p∞ → S the functor ℂ ob F ∶ Cat p∞ → S 𝑝 is theinitial additive functor under 𝕊 [ F ] , the pointwise suspension spectrum of F . In other words, ℂ ob ∶ Fun add (Cat p∞ , Grp E ∞ ) ⟶ Fun add (Cat p∞ , S 𝑝 ) is left adjoint to the forgetful functor, i.e. composition with Ω ∞ . Also, ℂ ob ◦ (−) grp ∶ Fun add (Cat p∞ , S ) ⟶ Fun add (Cat p∞ , S 𝑝 ) is left adjoint to the forgetful functor. An explicit description of the counit of the former adjunction is easily derived from Remark 3.3.8.
Proof.
For the proof note, that transformations 𝕊 [ F ] ⇒ G of functors Cat p∞ → S 𝑝 correspond naturally totransformations F ⇒ Ω ∞ G of functors to both E ∞ -monoids and plain spaces by Lemma 1.5.7. On the otherhand, the space of transformations ℂ ob F ⇒ G is given by lim 𝑛 ∈ ℕ Nat(Ω ∞− 𝑛 ℂ ob F , Ω ∞− 𝑛 G ) ≃ lim 𝑛 ∈ ℕ Nat( | Cob F 𝑛 | , Ω ∞− 𝑛 G ) . But since | Cob F 𝑛 | is the 𝑛 -fold suspension of F by Theorem 3.3.4, this colimit system is constant with value Nat( F , Ω ∞ G ) , which gives the claim. (cid:3) Proof of Proposition Proposition 3.4.5.
By Proposition 2.2.6, we have that | Cob F 𝑛 | ≃ | Cob | Cob F | 𝑛 −1 | is group-like as soon as 𝑛 ≥ , and hence in this case the structure map | Cob F 𝑛 | → Ω | Cob F 𝑛 +1 | is an equivalence byTheorem 3.3.4 ii). Of course, if F is group-like then this holds also at the ’th level. Furthermore, since byTheorem 2.4.1 all functors | Cob F 𝑛 | are additive so is ℂ ob , as fibre sequences in (pre-)spectra are detecteddegreewise. This gives the first two statements.To obtain the third statement just observe that by Part ii) the spectrification of ℂ ob( C , Ϙ ) is given by [ Ω | Cob F ( C , Ϙ ) | , | Cob F ( C , Ϙ ) | , | Cob F ( C , Ϙ ) | , … ] which by Corollaries 3.3.6 and 3.3.7 agrees with [ F grp ( C , Ϙ ) , | Cob F grp ( C , Ϙ ) | , | Cob F grp ( C , Ϙ ) | , … ] , since | Cob F 𝑛 +1 | = | Cob | Cob F | 𝑛 | . (cid:3) Part iii) of Proposition 3.4.5 identifies the non-negative homotopy groups of ℂ ob F ( C , Ϙ ) with those of F grp ( C , Ϙ ) . While these are generally very difficult to understand, we can determine the negative homotopygroups of the spectrum ℂ ob F ( C , Ϙ ) , much more easily:3.4.7. Proposition.
For every additive F ∶ Cat p∞ → S , Poincaré ∞ -category ( C , Ϙ ) , 𝑛 ≥ and ≤ 𝑘 < 𝑛 the iterated bonding maps of the pre-spectrum ℂ ob F ( C , Ϙ ) induce isomorphisms 𝜋 𝑘 | Cob F 𝑛 ( C , Ϙ ) | ≅ 𝜋 | Cob F ( C , Ϙ [ 𝑛 − 𝑘 −1] ) | and 𝜋 − 𝑛 ℂ ob F ( C , Ϙ ) ≅ 𝜋 | Cob F ( C , Ϙ [ 𝑛 −1] ) | . In other words, 𝜋 𝑘 | Cob F 𝑛 ( C , Ϙ ) | for 𝑘 < 𝑛 is just the F -based cobordism group of ( C , Ϙ [ 𝑛 − 𝑘 ] ) and similarlyfor the negative homotopy groups of ℂ ob F . Proof.
Part i) of Proposition 3.4.5 reduces the claim about the left hand side to the case 𝑘 = 0 . By realisingthe 𝑛 -fold simplicial object Cob F 𝑛 iteratively, this case follows from Corollaries 2.2.9 and 2.2.10 by inductionon 𝑛 . The statement for the right hand side is now immediate from Proposition 3.4.5 iii) and Corollary 3.3.7. (cid:3) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 89 Corollary.
For any grouplike additive F ∶ Cat p∞ → S the spectrum ℂ ob F ( C , Ϙ ) is connective when-ever ( C , Ϙ ) admits a lagrangian subcategory, in particular ℂ ob F Met( D , Φ) and ℂ ob F Hyp( E ) are alwaysconnective.In fact, the functor ℂ ob ∶ Fun add (Cat p∞ , Grp E ∞ ) ⟶ Fun add (Cat p∞ , S 𝑝 ) is fully faithful and its essential image consists precisely of the functors whose values on all metabolicPoincaré ∞ -categories ( C , Ϙ ) is connective. The essential image of ℂ ob can equivalently be described by the condition that F Met( D , Φ) be con-nective for all Poincaré ∞ -categories ( D , Φ) or that F Hyp( E ) be connective for all small stable E : For thelatter condition this is immediate from Corollary 3.2.14, and for the former it then follows from F Hyp( E ) being a retract of F Hyp(Hyp( E )) ≃ F (Met(Hyp( E )) ). Proof.
The first part is a consequence of Proposition 3.4.7, Corollary 3.2.14 and Corollary 2.2.10. That ℂ ob is fully faithful follows from Corollary 3.4.6, since the unit F ⇒ Ω ∞ ℂ ob F is an equivalence byProposition 3.4.5 if F is grouplike. To see the statement about the essential image note that the counit ℂ ob Ω ∞ F → F of the adjunction is an equivalence after applying Ω ∞ by the triangle identities, and thereforean equivalence on non-negative homotopy groups. Applying this counit transformation to the metabolicfibre sequence ( C , Ϙ ) → Met( C , Ϙ [1] ) → ( C , Ϙ [1] ) , we conclude inductively on 𝑖 that the transformation is an equivalence on 𝜋 − 𝑖 for all 𝑖 ≥ . (cid:3) Remark.
Completely analogous definitions and arguments work in the non-hermitian set-up to givethe 𝑛 -fold Segal spaces Span F 𝑛 ( C ) and (pre-)spectra 𝕊 pan F ( C ) , with the K -theory functor Cat ex∞ → S 𝑝 beingthe (pointwise) spectrification of 𝕊 pan Cr or equivalently 𝕊 pan Cr grp . As a consequence of Proposition 2.6.3one here finds that 𝕊 pan F ( C ) is always a connective (pre-)spectrum. The analogue of the above corollaryis the statement that 𝕊 pan ∶ Fun add (Cat ex∞ , Grp E ∞ ) ⟶ Fun add (Cat ex∞ , S 𝑝 ) is fully faithful with essential image the functors taking values in connective spectra. In particular, thenon-connectivity of the iterated Q -construction is an entirely hermitian phenomenon.Let us also record the relationship between the Q -construction and suspension in Fun add (Cat p∞ , S 𝑝 ) . Tothis end, consider again the squares(53) const Met( C , Ϙ ) dec(Q( C , Ϙ )){0} Q( C , Ϙ ) const( C , Ϙ [−1] ) Null( C , Ϙ ){0} Q( C , Ϙ ) consisting of split Poincaré-Verdier sequences in each simplicial degree. Applying an additive F ∶ Cat p∞ → S 𝑝 one obtains a levelwise cartesian square of simplicial spectra. As this is also cocartesian by stability, itfollows that also F Met( C , Ϙ ) | F dec(Q( C , Ϙ )) | {0} | F Q( C , Ϙ ) | F ( C , Ϙ [−1] ) | F Null( C , Ϙ ) | {0} | F Q( C , Ϙ ) | are bicartesian squares of spectra. As the simplicial objects in the top right corner are split by Lemma 3.3.1over and ( C , Ϙ ) , respectively, we obtain a canonical equivalence(54) 𝕊 ⊗ F ( C , Ϙ [−1] ) ⟶ | F Q( C , Ϙ ) | and a natural bifibre sequence F Met( C , Ϙ ) met ←←←←←←←←←←←←←←→ F ( C , Ϙ ) ⟶ | F Q( C , Ϙ ) | . As the right square tautologically maps to the left one, one finds that under the equivalence (54) this fibresequence is a rotation of the metabolic fibre sequence F ( C , Ϙ [−1] ) ⟶ F Met( C , Ϙ ) met ←←←←←←←←←←←←←←→ F ( C , Ϙ ) . Furthermore, the functor ( C , Ϙ ) ↦ | F Q( C , Ϙ ) | is again additive, by the same argument, so we find:3.4.10. Corollary.
The endofunctor F → | F Q − | on Fun add (Cat p∞ , S 𝑝 ) is the internal suspension functor,or equivalently postcomposition with the suspension functor in S 𝑝 . Remark.
The geometric realisation | F Q | occuring in the previous statement may be taken bothobjectwise as a geometric realisation of simplicial spectra, or as a colimit in the category Fun add (Cat p∞ , S 𝑝 ) itself, since we noted above that the objectwise colimit (which is also the colimit in the category of allfunctors Cat p∞ → S 𝑝 ) is already additive.However, we warn the reader that in general | F Q( C , Ϙ ) | cannot be computed levelwise via the realisationof the simplicial spaces Ω ∞− 𝑛 F Q( C , Ϙ ) : Consider for example the functor 𝕂 ∶ Cat p∞ → S 𝑝 extracting thenon-connective K -theory of the underlying stable ∞ -category C . Then Ω ∞ | 𝕂 Q( C ) | ≃ Ω ∞−1 𝕂 ( C ) need notbe connected, whereas | Ω ∞ 𝕂 Q( C ) | ≃ | K (Q( C ) idem ) | ≃ | K (Q( C idem )) | ≃ Ω ∞−1 K( C idem ) is always connected.The notable exception to this discrepancy are the functors F = ℂ ob G for some group-like additive G ∶ Cat p∞ → S : For these functors the colimit of F Q( C , Ϙ ) ∶ Δ op → PS 𝑝 (which is formed levelwise) isautomatically an Ω -spectrum, and thus also a colimit in spectra. To see this, observe that by switching theorder of the realisations we find ( | ℂ ob G Q( C , Ϙ ) | ) 𝑛 = || G Q ( 𝑛 ) (Q( C , Ϙ [ 𝑛 ] )) || = ℂ ob | G Q − | ( C , Ϙ ) 𝑛 and the latter terms form an Ω -spectrum by Proposition 3.4.5, so ultimately by the additivity theorem.Finally, we use these observations to study the effect of shifting the Poincaré structure on the C -equivariant spectrum F (Hyp( C )) acted on by the duality of Ϙ . This will ultimately lead to our generalisationof Karoubi’s periodicity theorem in Corollary 4.5.5 below. To this end, recall that the composite functor Cat p∞ → Cat ex∞ Hyp ←←←←←←←←←←←←←←←←→
Cat p∞ refines to a functor Hyp ∶ Cat p∞ → Fun(BC , Cat p∞ ) via the action of the duality,see Remark [I].7.4.14.3.4.12. Definition.
Given a functor F ∶ Cat p∞ → E define the hyperbolisation F hyp ∶ Cat p∞ → Fun(BC , E ) of F as F ◦ Hyp .3.4.13.
Proposition (Naive Karoubi periodicity) . There is a canonical equivalence of C -spectra F hyp ( C , Ϙ [−1] ) ≃ 𝕊 𝜎 −1 ⊗ F Hyp ( C , Ϙ ) , natural in the Poincaré ∞ -category ( C , Ϙ ) and the additive functor F ∶ Cat p∞ → S 𝑝 . Furthermore, underthis equivalence the boundary map F hyp ( C , Ϙ ) → 𝕊 ⊗ F hyp ( C , Ϙ [−1] ) of the metabolic fibre sequence is induced by the inclusion S → S 𝜎 as the fixed points. Here 𝕊 𝜎 denotes the C -spectrum equivalently described as the suspension spectrum of S 𝜎 , the -spherewith complex conjugation action, or the functor BC = BO(1) → BO J ←←←←←→ Pic( 𝕊 ) ⊆ S 𝑝. Proof.
We recall that under the equivalence F (Met( C , Ϙ )) ≃ F (Hyp( C )) induced by can ∶ Hyp( C ) → Met( C , Ϙ ) the map met ∶ Met( C , Ϙ ) → C identifies with hyp ∶ Hyp( C ) → ( C , Ϙ ) . Using the metabolicPoincaré-Verdier sequence we may therefore identify 𝕊 ⊗ F hyp ( C , Ϙ [−1] ) with the cofibre of the map F hyp ∶ F hyp (Hyp( C )) ⟶ F hyp ( C , Ϙ ) . Now there is a the natural equivalence
Hyp(Hyp( C )) ≃ Hyp( C × C op ) ≃ Hyp( C ) ⊗ C ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 91 which translates the action of D Ϙ on the left into the flip action on the right, see Remark [I].7.4.15. We maythen identify the map F hyp with the map F (Hyp( C )) ⊗ C ⟶ F (Hyp( C )) obtained from the map C → ∗ of C -spaces, whose cofibre is S 𝜎 . We therefore obtain a natural equivalence 𝕊 ⊗ F hyp ( C , Ϙ [−1] ) ≃ 𝕊 𝜎 ⊗ F hyp ( C , Ϙ ) which is the claim. (cid:3) Bordism invariant functors.
In the next two subsections, we will introduce the notion of a bordisminvariant functor out of
Cat p∞ , the main examples being various flavours of L -theory. We will then show thateach additive functor F ∶ Cat p∞ → S admits an initial bordism invariant functor F bord equipped with a map F → F bord , the bordification of F , and show that any group-like F can then be described in terms of thisbordification and the hyperbolisation F hyp = F ◦ Hyp from the previous section. This yields a version of ourTheorem Main Theorem, first part, for arbitrary additive functors in Corollary 3.6.7; it will be specialisedto F = Pn in section 4.To get started recall the notion of a cobordism between Poincaré functors from Definition 3.1.1: It is aPoincaré functor ( C , Ϙ ) → Q ( C ′ , Ϙ ′ ) projecting correctly to the endpoints of Q .3.5.1. Definition.
A Poincaré functor ( 𝐹 , 𝜂 ) ∶ ( C , Ϙ ) → ( C ′ , Ϙ ′ ) is called a bordism equivalence if there existsa Poincaré functor ( 𝐺, 𝜗 ) ∶ ( C ′ , Ϙ ′ ) → ( C , Ϙ ) such that the composites ( 𝐹 , 𝜂 ) ◦ ( 𝐺, 𝜃 ) and ( 𝐺, 𝜃 ) ◦ ( 𝐹 , 𝜂 ) arecobordant to the respective identities.3.5.2. Example.
Let ( C , Ϙ ) be a Poincaré ∞ -category and L ⊆ C an isotropic subcategory (see Defini-tion 3.2.1). Then the inclusion Hlgy( L ) ⊆ ( C , Ϙ ) of the homology ∞ -category is a bordism equivalence.This follows directly from Construction 3.2.11.3.5.3. Definition.
Given a category with finite products E , we say that an additive functor F ∶ Cat p∞ → E is bordism invariant if it sends bordism equivalences of Poincaré ∞ -categories to equivalences in E . Weshall denote by Fun bord (Cat p∞ , E ) the full subcategory of Fun add (Cat p∞ , E ) spanned by the bordism invariantfunctorsIn particular, such a functor vanishes on all metabolic Poincaré ∞ -categories, i.e. those that admit aLagrangian. For (group-like) additive functors, and these are the only ones we will investigate in any detailhere, the converse holds as well:3.5.4. Lemma.
Let F ∶ Cat p∞ → E be a group-like additive functor. Then the following are equivalent:i) F is bordism invariant.ii) F takes the degeneracy map 𝑠 ∶ ( C , Ϙ ) → Q ( C , Ϙ ) to an equivalence for every Poincaré ∞ -category ( C , Ϙ ) .iii) F vanishes on all metabolic Poincaré ∞ -categories.iv) F (Met( C , Ϙ )) ≃∗ for any Poincaré ∞ -category ( C , Ϙ ) .v) F (Hyp( C )) ≃∗ for any stable ∞ -category C .Proof. The functors in ii) are bordism equivalences (essentially by definition) so i) ⇒ ii) and it follows im-mediately from Corollary 3.1.3 that ii) ⇒ iv). By Example 3.5.2 all metabolic categories are bordism equiv-alent to , so i) ⇒ iii) and since Met( C , Ϙ ) really is metabolic, we have iii) ⇒ iv). To obtain iv) ⇒ v) observethat F Hyp( C ) is a retract of F Hyp( C × C op ) , which by Corollary 3.1.4 is equivalent to F Met(Hyp( C )) ≃∗ .Finally, by Proposition 3.1.7, if F vanishes on hyperbolics, then cobordant Poincaré functors induce homo-topic maps after applying F , and so F is bordism invariant giving v) ⇒ i). (cid:3) Example.
The L -theory space provides a bordism invariant functor L ∶ Cat p∞ → S . The fact that itis invariant under bordism equivalences can be seen by direct analysis of its homotopy groups: In degree 𝑛 they are given by bordism classes of Poincaré objects in ( C , Ϙ [− 𝑛 ] ) , see [Lur11, Lecture 7, Theorem 9] for aproof in the present language. Thus by definition the two maps 𝑑 , 𝑑 ∶ L (Q ( C , Ϙ )) → L ( C , Ϙ ) induce thesame map on L -groups. Consequently, so do any two cobordant functors and thus bordism equivalencesinduce inverse isomorphisms on L -groups, compare §[I].2.3. The same statements apply to the L -theoryspectrum. We will discuss this example, and additivity of both functors L and L , in §4.4. To discuss the second important example, recall from Definition 3.4.12, the hyperbolisation F hyp ( C , Ϙ ) = F (Hyp( C )) taking values in the category S 𝑝 hC = Fun(BC , S 𝑝 ) of C -spectra via the action of the duality D Ϙ on Hyp( C ) .3.5.6. Example.
Given an additive functor F ∶ Cat p∞ → S 𝑝 the Tate construction (−) tC ∶ S 𝑝 hC → S 𝑝 produces a functor ( F hyp ) tC ∶ Cat p∞ → S 𝑝 which is bordism invariant; to see this, we invoke the naturalequivalence(55) Hyp(Hyp( C )) ⟶ Hyp( C ) ⊗ C from [I].7.4.15 again, which shows that F hyp (Hyp( C )) is an induced C -spectrum. It then follows that forevery stable ∞ -category C we have ( F hyp ) tC Hyp( C ) ≃ 0 , since the Tate construction generally vanisheson induced C -spectra.Let us also record for later use:3.5.7. Lemma. If F ∶ Cat p∞ → E is arbitrary and G ∶ Cat p∞ → E is bordism-invariant, then the spaces Nat( F hyp , G ) and Nat( F hyphC , G ) are contractible (assuming E admits sufficient colimits to form the homotopyorbits in the second case).Proof. Since
Hyp ∶ Cat ex∞ → Cat p∞ is both left and right adjoint to the forgetful functor by Corollary [I].7.2.20,it follows that the composite Cat p∞ fgt → Cat ex∞ Hyp → Cat p∞ is both left and right adjoint to itself and hence theassociation F ↦ F hyp is both left and right adjoint to itself. Since G hyp ≃∗ for any bordism invariant functorit follows that the mapping space from F hyp to any bordism invariant functor is trivial.The computation Nat( F hyphC , G ) ≃ Nat( F hyp , G ) hC ≃∗ gives the second claim. (cid:3) For the next statement recall the metabolic Poincaré-Verdier sequence ( C , Ϙ [−1] ) ⟶ Met( C , Ϙ ) ⟶ ( C , Ϙ ) from Example 1.2.5.3.5.8. Proposition.
Suppose that F ∶ Cat p∞ → E is a bordism invariant functor. Then the natural map Ω F ( C , Ϙ ) ⟶ F ( C , Ϙ [−1] ) arising from the metabolic Poincaré-Verdier sequence is an equivalence. In particular, F is automaticallygroup-like.If E is stable then the converse holds in the sense that an additive functor F ∶ Cat p∞ → E is bordisminvariant if and only if this map is an equivalence for all Poincaré ∞ -categories ( C , Ϙ ) . In particular, we find 𝜋 𝑖 F ( C , Ϙ ) = 𝜋 F ( C , Ϙ [− 𝑖 ] ) for every space or spectrum valued bordism invariantfunctor. Furthermore, by Corollary 3.1.8 the inversion map on F ( C , Ϙ ) is induced by the Poincaré functor (id C , −id Ϙ ) . Proof.
By Lemma 3.5.4, F is bordism invariant if and only if F Met( C , Ϙ ) ≃∗ for all Poincaré ∞ -categories ( C , Ϙ ) , from which we obtain a fibre sequence F ( C , Ϙ [−1] ) ⟶ ∗ ⟶ F ( C , Ϙ ) which gives the first claim. Conversely, if E is stable, then the map in question being an equivalence impliesthat F Met( C , Ϙ ) vanishes for every Poincaré ∞ -category. (cid:3) In particular, bordism invariant functors can be delooped simply by shifting the Poincaré structure, i.e.by considering [ F ( C , Ϙ ) , F ( C , Ϙ [1] ) , F ( C , Ϙ [2] ) , … ] with the structure maps provided by the Proposition 3.5.8. We next show that this delooping agrees withthat from the previous section. In fact, we have as the main result of this subsection: ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 93 Theorem.
The forgetful functor
Fun bord (Cat p∞ , S 𝑝 ) ⟶ Fun bord (Cat p∞ , S ) , i.e. postcomposition with Ω ∞ , is an equivalence with inverse F ⟼ ℂ ob F . In particular, any additive bordism invariant functor F ∶ Cat p∞ → S admits an essentially unique lift toanother such functor Cat p∞ → S 𝑝 . The same is not true for arbitrary group-like additive F ∶ Cat p∞ → S as the examples ( C , Ϙ ) ⟼ K( C idem ) and 𝕂 ( C ) , which have equivalent infinite loopspaces, show.For the proof we need:3.5.10. Remark. If F ∶ Cat p∞ → S is additive and bordism invariant, then so is | Cob F | . This follows straightfrom the definitions, as a cobordism of Poincaré functors ( C , Ϙ ) → Q ( C ′ , Ϙ ′ ) , induces one Q 𝑛 ( C , Ϙ ) → Q 𝑛 (Q ( C ′ , Ϙ ′ )) ≅ Q (Q 𝑛 ( C ′ , Ϙ ′ )) so a bordism equivalence ( C , Ϙ ) → ( C ′ , Ϙ ′ ) gives an equivalence of simplicial objects F Q( C , Ϙ ) → F Q( C ′ , Ϙ ′ ) ,and thus an equivalence on realisations. Proof of Theorem 3.5.9.
That the essential image of Ω ∞ is contained in the bordism invariant functors isclear. If now F ∶ Cat p∞ → S is bordism invariant, then so is ℂ ob F ∶ Cat p∞ → S 𝑝 : By Proposition 3.5.8 F is group-like, so by Proposition 3.4.5 ℂ ob F takes values in spectra and is additive. To check bordisminvariance it suffices, by induction and the equivalences Ω ∞− 𝑛 ℂ ob F ≃ | Cob F 𝑛 | ≃ | Cob | Cob F | 𝑛 −1 | to show that | Cob F | is again bordism invariant, which we did above. Thus the adjunction between Ω ∞ and ℂ ob restricts as claimed and Ω ∞ is essentially surjective.Finally, to obtain full faithfulness of Ω ∞ , we check that the counit 𝑐 ∶ ℂ ob Ω ∞ F ( C , Ϙ ) ⇒ F ( C , Ϙ ) is anequivalence for every bordism invariant F ∶ Cat p∞ → S 𝑝 and Poincaré ∞ -category ( C , Ϙ ) . By Proposi-tion 3.4.5 Ω ∞ 𝑐 is an equivalence for all ( C , Ϙ ) , but as both domain and target of 𝑐 are bordism invari-ant functors Proposition 3.5.8 then implies, that Ω ∞− 𝑛 𝑐 is an equivalence for all 𝑛 ≥ , which gives theclaim. (cid:3) The bordification of an additive functor.
In this subsection we will establish the following theoremand deduce a formal version of our Theorem Main Theorem in Corollary 3.6.7.3.6.1.
Theorem.
The inclusions
Fun bord (Cat p∞ , S 𝑝 ) ⊆ Fun add (Cat p∞ , S 𝑝 ) and Fun bord (Cat p∞ , S ) ⊆ Fun add (Cat p∞ , S ) of the bordism invariant into all additive functors admit left adjoints. Definition.
We will refer to these left adjoint functors as bordification and denote their values on anadditive functor F ∶ Cat p∞ → S 𝑝 or F ∶ Cat p∞ → S by F bord .This result, Theorem 3.5.9 and Corollary 3.4.6 may be summarized by the following commutative squareof forgetful functors and their left adjoints (displayed by curved arrows) as follows: Fun bord (Cat p∞ , S 𝑝 ) Fun add (Cat p∞ , S 𝑝 )Fun bord (Cat p∞ , S ) Fun add (Cat p∞ , S ) (−) bord ℂ ob ℂ ob ◦ (−) grp (−) bord whose left hand vertical arrows are inverse equivalences. Thus the existence of the upper horizontal adjointimplies the existence of the lower one. We will therefore mostly restrict attention to the case of functorstaking values in spectra in this section. Remark.
We will frequently conflate the bordifications of a space valued F and of the spectrumvalued Cob F grp , as these can be reconstructed from one another by the commutative diagram above.We warn the reader, however, that conversely for a spectrum-valued additive F the map ℂ ob (Ω ∞ F ) bord = ( ℂ ob Ω ∞ F ) bord ⟶ F bord induced by the counit of the adjunction ℂ ob ⟂ Ω ∞ is not generally an equivalence (unless F ≃ ℂ ob G forsome group-like additive G , in which case this map is the identity); a concrete counterexample is providedby the Karoubi-Grothendieck-Witt functor of §[IV].2.2.We will give three distinct formulae for the spectral bordification functor in Proposition 3.6.6, Corol-lary 3.6.12 and Corollary 3.6.18, and it really is the comparison between these that is most relevant for ourwork. While this comparison can be established by direct calculations, that route does not lead to shorterarguments and the present framework allows for a more conceptual intrepretation.Before getting started, we can already record the following special cases:3.6.4. Lemma.
Let F ∶ Cat p∞ → S 𝑝 be additive. Then we have ( F hyp ) bord ≃ 0 and ( F hyphC ) bord ≃ 0 , whereasthe natural map ( F hyp ) hC ⇒ ( F hyp ) tC descends to an equivalence (( F hyp ) hC ) bord ≃ ( F hyp ) tC . To interpret the statement one can either assume the existence of a bordification functor already (thepresent lemma will not enter the proof of existence below), or better one can simply interpret the definitionof bordifications as a pointwise statement about left adjoint objects. In this case the present lemma, inparticular, provides the existence of bordifications for the functors F hyp , F hyphC and ( F hyp ) hC . Proof.
The first two statements are immediate from Lemma 3.5.7. But then consider the cofibre sequence F hyphC ⟶ ( F hyp ) hC ⟶ ( F hyp ) tC . For some bordism invariant G , it induces a fibre sequence Nat(( F hyp ) tC , G ) ⟶ Nat(( F hyp ) hC , G ) ⟶ Nat( F hyphC , G ) But the right hand term vanishes by Lemma 3.5.7, whereas ( F hyp ) tC is already bordism invariant by Ex-ample 3.5.6. The claim follows. (cid:3) Example. If F ∶ Cat ex∞ → S 𝑝 is additive, then the bordification of the composite Cat p∞ → Cat ex∞ → S 𝑝 vanishes: In this case the C -spectrum F hyp ( C , Ϙ ) itself is induced from F ( C ) , whence the map F hyp ( C , Ϙ ) hC → F ( C ) is an equivalence. This implies for example that the bordifications of Cr , K , 𝕂 , THH , TC and similarfunctors all vanish.Expressed differently, bordification is a genuinely hermitian concept that has no classical counterpart.We now introduce the first construction of bordifications. We first recall from Corollary [I].7.4.18 thatthe hyperbolic and forgetful maps refine to C -equivariant maps Hyp( C ) ⟶ ( C , Ϙ ) ⟶ Hyp( C ) , where ( C , Ϙ ) is considered with the trivial C -action. It then follows that the induced natural maps(56) F hyp ⟶ F ⟶ F hyp refine to maps of the form(57) F hyphC ⟶ F ⟶ ( F hyp ) hC . Note also that the composition of the two maps in (57) coincides with the norm map F hyphC → ( F hyp ) hC associated to the C -action on F hyp . Now from Lemma 3.5.7 we find Nat( F hyphC , G ) ≃∗ if G is bordism invariant. In particular, assuming the existence of a bordification, there must be a sequence(58) F hyphC ⟶ F ⟶ F bord ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 95 whose composition admits an essentially unique null-homotopy. There is a universal way to produce sucha sequence:3.6.6. Proposition.
Consider the functor
Φ ∶ Fun add (Cat p∞ , S 𝑝 ) → Fun add (Cat p∞ , S 𝑝 ) given by the formula Φ F = cof ( F hyphC ⟶ F ) . Then the canonical transformations F ⇒ Φ F exhibit Φ as a bordification.Proof. Let F be an additive functor. We first verify that Φ F is bordism invariant. By lemma 3.5.4 we needto check that Φ F (Hyp( C )) ≃ 0 , or, equivalently, that the canonical transformation(59) F HyphC (Hyp( C )) ⟶ F (Hyp( C )) is an equivalence. Indeed, by the equivalence (55) we can identify (59) with a map of the form(60) [ F (Hyp( C )) ⊗ C )] hC ⟶ F (Hyp( C )) . It will therefore suffice to check that the pre-composition of (60) with the equivalence F (Hyp( C )) → [ F (Hyp( C )) ⊗ C )] hC given by the inclusion of a component is an equivalence; by direct inspection itis the identity.Now suppose that G is any bordism invariant functor. We need to show that the induced map Nat(Φ F , G ) ⟶ Nat( F , G ) is an equivalence. Indeed, by construction we have a fibre sequence Nat(Φ F , G ) ⟶ Nat( F , G ) ⟶ Nat(( F Hyp ) hC , G ) and Nat(( F Hyp ) hC , G ) ≃∗ by Lemma 3.5.7. (cid:3) Applying bordification to the natural map F → ( F hyp ) hC and using Lemma 3.6.4 we find an abstractversion of our main result, the fundamental fibre square :3.6.7. Corollary.
For every additive functor F ∶ Cat p∞ → S 𝑝 and Poincaré ∞ -category ( C , Ϙ ) there is afibre sequence F hyphC ( C , Ϙ ) hyp ←←←←←←←←←←←←←←→ F ( C , Ϙ ) bord ←←←←←←←←←←←←←←←←←→ F bord ( C , Ϙ ) , which canonically extends to a bicartesian square F ( C , Ϙ ) F bord ( C , Ϙ )( F hyp ) hC ( C , Ϙ ) ( F hyp ) tC ( C , Ϙ ) . Proof.
It suffices to check that the induced map on horizontal fibres is an equivalence. But both of theseare given by F hyphC and the induced map is necessarily the identity by Lemma 3.5.7 (cid:3) The construction of bordification via the hyperbolisation map F hyphC → F in Proposition 3.6.6 is, however,not very suitable for computations of F bord . Therefore we present two more formulae, both of which we putto use in the next section. To verify that these really give bordifications we employ the following criterion:3.6.8. Lemma.
Suppose that
B ∶ Fun add (Cat p∞ , S 𝑝 ) → Fun add (Cat p∞ , S 𝑝 ) is a functor equipped with a nat-ural transformation 𝛽 ∶ id ⇒ B . Suppose the following conditions hold:i) B commutes with colimits;ii) if F is bordism invariant then 𝛽 F ∶ F ⇒ B F is an equivalence;iii) B( F hyp ) ≃ 0 for every additive F ∶ Cat p∞ → S 𝑝 .Then 𝛽 exhibits B as a bordification functor. Let us explicitly point out that we do not assume a priori that B takes values in bordism invariant functors.The price is that we have to invest that we already know that there exists a bordification functor into theproof. Direct arguments are also certainly possible, but slightly more cumbersome. Proof.
Let F ∶ Cat p∞ → S 𝑝 be an additive functor. Applying B to the fibre sequence F hyphC → F → F bord from Corollary 3.6.7 yields a commutative rectangle(61) F hyphC F F bord B( F hyphC ) B F B( F bord ) in which both rows are bifibre sequences and the vertical maps are all the respective components of 𝛽 . Bydefinition, F bord is bordism invariant and hence by property ii) we get that the right most vertical map in (61)is an equivalence. This implies that the left square is bicartesian. On the other hand, by properties i) and iii)the lower left corner of (61) is equivalent to , hence the lower right map is an equivalence as well. Theright hand square thus exhibits B as equivalent to bord under the identity of Fun add (Cat p∞ , S 𝑝 ) . (cid:3) Our second formula for bordification is modelled on the classical definition of L -theory spectra via ad-spaces. Its starting point is the 𝜌 -construction : For [ 𝑛 ] ∈ Δ we denote by T 𝑛 = P ([ 𝑛 ]) op the opposite ofthe poset of nonempty subsets of [ 𝑛 ] . We observe that T 𝑛 depends functorially on [ 𝑛 ] ∈ Δ , giving rise to acosimplical category 𝜌 ( C , Ϙ ) : Given a Poincaré ∞ -category ( C , Ϙ ) denote 𝜌 𝑛 ( C , Ϙ ) = (Fun( T 𝑛 , C ) , Ϙ T 𝑛 ) the cotensor of ( C , Ϙ ) by T 𝑛 . Since T 𝑛 is the reverse face poset of Δ 𝑛 we find from Proposition [I].6.6.1that the hermitian ∞ -categories 𝜌 𝑛 C are Poincaré for every [ 𝑛 ] ∈ Δ and from Proposition [I].6.6.2 that thehermitian functor 𝜎 ∗ ∶ 𝜌 𝑛 C → 𝜌 𝑚 C is Poincaré for every 𝜎 ∶ [ 𝑚 ] → [ 𝑛 ] in Δ . We may hence consider 𝜌 ( C , Ϙ ) as a simplicial object in Cat p∞ .3.6.9. Definition.
Let E be an ∞ -category with sifted colimits. Given a functor F ∶ Cat p∞ → E we denoteby ad F ∶ Cat p∞ → E the functor given by ad F ( C , Ϙ ) = | F 𝜌 ( C , Ϙ ) | . Using the functoriality of the cotensor construction we may promote the association F ↦ ad F to afunctor(62) ad ∶ Fun(Cat p∞ , E ) ⟶ Fun(Cat p∞ , E ) . The inclusion of vertices then equips ad with a natural transformation 𝑏 F ∶ F → ad F .In this section we consider the ad -construction only in the case when E = S 𝑝 , as this entails greatsimplifications (though the case E = S is fundamental for the discussion of L -theory in §4.4). The keyis that for stable E the collection of additive functors from Cat p∞ to E is closed under colimits inside thecategory of all functors. Since in addition, the functor 𝜌 𝑛 ∶ Cat p∞ → Cat p∞ preserves split Poincaré-Verdiersequences by Proposition 1.4.14 it follows that ad F is additive whenever F ∶ Cat p∞ → S 𝑝 is.In particular, we may consider ad as a functor(63) ad ∶ Fun add (Cat p∞ , S 𝑝 ) ⟶ Fun add (Cat p∞ , S 𝑝 ) . Remark.
The analogous statement with target category S requires an additivity theorem for the 𝜌 -construction. Lurie showed in [Lur11, Lecture 8, Corollary 9] (see Theorem 4.4.2) that L = ad(Pn) is evenVerdier-localising, generalising results of Ranicki in more classical language, see e.g. [Ran92, Proposition13.11].We now set out to show:3.6.11. Proposition.
Let F ∶ Cat p∞ → S 𝑝 be an additive functor. Theni) if F is bordism invariant then the map 𝑏 F ∶ F ⇒ ad F is an equivalence.ii) ad( F hyp ) ≃∗ . Combining this with Lemma 3.6.8 and the fact that ad evidently commutes with colimits we obtain:3.6.12. Corollary.
The natural transformation 𝑏 exhibits ad as a bordification functor. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 97 For the proof of Proposition 3.6.11, we denote by P ([ 𝑛 ]) the full power set of [ 𝑛 ] and endow Fun( P ([ 𝑛 ]) op , C ) with the hermitian structure Ϙ tf that sends a cubical diagram 𝜑 ∶ P ([ 𝑛 ]) op → C to the total fibre of Ϙ [1] ◦ 𝜑 op ;through the isomorphism P ([ 𝑛 ]) op ≅ ∏ 𝑛𝑖 =0 [1] the hermitian ∞ -category (Fun( P ([ 𝑛 ]) op , C ) , Ϙ tf ) is equivalent to Met ( 𝑛 +1) ( C , Ϙ [1] ) , but in the form given itis clear that it assembles into a functor Cat h∞ → sCat h∞ . Through the identification as an iterated metabolicobject, it is, however, easy to check that it restricts to Cat p∞ → sCat p∞ .3.6.13. Lemma.
The sequence (64) 𝜌 𝑛 ( C , Ϙ ) ⟶ (Fun( P ([ 𝑛 ]) op , C ) , Ϙ tf ) ev ∅ ←←←←←←←←←←←←←→ ( C , Ϙ [1] ) is a Poincaré-Verdier sequence for all Poincaré ∞ -categories ( C , Ϙ ) and 𝑛 ∈ ℕ . Furthermore, there areequivalences Fun( P ([−]) op , Ar( C )) , Ϙ tfmet ) ≃ dec(Fun( P ([−]) op , C ) , Ϙ tf ) ≃ Met(Fun( P ([−]) op , C ) , Ϙ tf ) of simplicial Poincaré ∞ -categories. Note that the left term in the second display is just
Fun( P ([−]) op , D ) , Φ tf ) for ( D , Φ) = Met( C , Ϙ ) . Proof.
The map ev ∅ is given by evaluation of the cubical diagram 𝜑 at ∅ , together with the canonicalprojection of hermitian functors. Under the equivalence of the middle term with Met ( 𝑛 +1) ( C , Ϙ [1] ) , thesecond map is the ( 𝑛 + 1) -fold iteration of the map met ∶ Met( C , Ϙ [1] ) → ( C , Ϙ [1] ) ; thus it is a split Poincaré-Verdier projection. Its kernel is equivalent to the first term by restriction along T 𝑛 = P ([ 𝑛 ]) op ⊂ P ([ 𝑛 ]) op and the equivalence lim ∅ ≠ 𝐴⊆ [ 𝑛 ] Ϙ ◦ 𝜑 op ( 𝐴 ) ≃ f ib ( ⟶ lim ∅ ≠ 𝐴⊆ [ 𝑛 ] Ϙ [1] ◦ 𝜑 op ( 𝐴 ) ) , since for 𝜑 ∈ ker(ev ∅ ) the second term is equivalent to the total fibre of Ϙ [1] ◦ 𝜑 op .For the second claim note that commuting limits and functor categories gives equivalences Fun( P ([−]) op , Ar( C )) , Ϙ tfmet ) ≃ Fun( P ([−]) op × Δ , C ) , Ϙ tf ) ≃ Met(Fun( P ([−]) op , C ) , Ϙ tf ) and the middle term is the requisite décalage by inspection. (cid:3) Proof of Proposition 3.6.11. If F is bordism invariant, then it vanishes on the middle term of (64) (which isan iterated metabolic construction); we conclude that the left term becomes constant in 𝑛 , after applicationof F . This shows i).To show ii) we note that for a Poincaré ∞ -category ( C , Ϙ ) F hyp (Fun( P [−] , C )) = F (Hyp(Fun( P [−] , C ))) ≃ F ( Met(Fun( P [−] , C ) , Ϙ tf ) ) by 3.1.4. But this is the décalage of F (Fun( P [−] , C ) , Ϙ tf ) by 3.6.13 with augmentation induced by ev ∅ .Interpreting this map as a map of split simplicial objects, we conclude that its fibre F hyp ( 𝜌 𝑛 ( C , Ϙ )) is splitover , and therefore has contractible realisation. (cid:3) Remark.
To see that F 𝜌 ( C , Ϙ ) is a constant simplicial space if F is bordism invariant, one can al-ternatively observe that the degeneracy maps ( C , Ϙ ) = 𝜌 ( C , Ϙ ) → 𝜌 𝑛 ( C , Ϙ ) is the inclusion of the homologycategory Hlgy( L + 𝑛 ) for the following isotropic subcategory L + 𝑛 ⊆ 𝜌 𝑛 ( C , Ϙ ) : Let T 𝑛 ⊆ T 𝑛 be the subposetspanned by those 𝑆 ⊆ [ 𝑛 ] which contain and M + 𝑛 ⊆ Fun( T 𝑛 , C ) be the full subcategory spanned by those di-agrams 𝜑 ∶ T 𝑛 → C which are left Kan extensions of their restriction to T 𝑛 , i.e. such that 𝜑 ( 𝑆 ∪{0}) → 𝜑 ( 𝑆 ) is an equivalence for every 𝑆 ⊆ {1 , ..., 𝑛 } . Then L + 𝑛 ⊆ M + 𝑛 may be taken to consist of those diagrams whichadditionally satisfy 𝜑 ({0}) ≃ 0 .One readily checks that for 𝜑 ∈ M + 𝑛 there is an equivalence Ϙ T 𝑛 ( 𝜑 ) ≃ Ϙ ( 𝜑 (0)) , so L + 𝑛 really is isotropic. Furthermore, ( L + 𝑛 ) ⟂ = M − 𝑛 and D Ϙ T 𝑛 ( L + 𝑛 ) ⟂ = M + 𝑛 , where M − 𝑛 ⊆ Fun( T 𝑛 , C ) isthe full subcategory spanned by those diagrams 𝜑 ∶ T 𝑛 → C whose restriction to T 𝑛 is constant.Thus Hlgy( L + 𝑛 ) ≃ M + 𝑛 ∩ M − 𝑛 consists precisely of the constant diagrams as desired. Weiss and Williams in [WW98, Lemma 9.3] give a direct verification that ad(K) ≃ 0 , and their proof im-mediately generalises to give a different argument for the vanishing of bordifications of all additive functorsof the form
Cat p∞ → Cat ex∞ → S 𝑝 . To use the bordification procedure ad directly in other circumstances,however, one would have to investigate the effect of the 𝜌 -construction on an arbitrary additive functor F ∶ Cat p∞ → S . In particular, one would have to provide an additivity theorem in this generality, to obtain ahandle on the geometric realisation occuring in the ad -construction (essentially for the reasons spelled outin Remark 3.4.11). As mentioned in Remark 3.6.10, such a statement was worked out in the case F = Pn byLurie (see [Lur11, Lecture 8, Corollary 9]) and we will refrain from exhibiting further details in the presentpaper.Instead, we present a third bordification procedure, that is more in line with the methods developed here.It is obtained by iterating the boundary map F ( C , Ϙ ) → 𝕊 ⊗ F ( C , Ϙ [−1] ) of the metabolic fibre sequence.3.6.15. Definition.
Let F ∶ Cat p∞ → S 𝑝 be an additive functor. We define its stabilization stab F by theformula (stab F )( C , Ϙ ) = colim( F ( C , Ϙ ) ⟶ 𝕊 ⊗ F ( C , Ϙ [−1] ) ⟶ 𝕊 ⊗ F ( C , Ϙ [−2] ) ⟶ … ) , with structure maps the shifts of the boundary map for F , and we denote by 𝜎 ∞ F ∶ F ⟶ stab F the arising natural transformation.Recall from the discussion preceding Corollary 3.4.10, that the boundary map F ( C , Ϙ ) ⟶ 𝕊 ⊗ F ( C , Ϙ [−1] ) of the metabolic fibre sequence F ( C , Ϙ [−1] ) ⟶ F (Met( C , Ϙ )) met ←←←←←←←←←←←←←←→ F ( C , Ϙ ) is also modelled by the inclusion of vertices 𝜎 F ∶ F ( C , Ϙ ) ⟶ | F Q( C , Ϙ ) | . So we equally well find, that stab F ( C , Ϙ ) ≃ colim( F ( C , Ϙ ) 𝜎 F ←←←←←←←←←←←←→ | F Q( C , Ϙ ) | | 𝜎 F Q | ←←←←←←←←←←←←←←←←←←←←←←←←←←→ | F Q (2) ( C , Ϙ ) | ⟶ … ) , arises from another iteration of the Q -construction.3.6.16. Remark.
Again, there is another equally sensible choice for the structure maps in the colimit systemin Definition 3.6.15, namely the boundary maps for the functors 𝕊 𝑖 ⊗ F (− [− 𝑖 ] ) . These translate to 𝜎 | F Q ( 𝑖 ) | under the equivalence described above, and thus differ from the ones we choose to employ by a sign (−1) 𝑖 ,compare Remark 3.4.4. Therefore, the choice has no effect on the colimit stab F .3.6.17. Proposition.
Let F ∶ Cat p∞ → S 𝑝 be an additive functor. Theni) if F is bordism invariant then the map 𝜎 ∞ F ∶ F → stab F is an equivalence.ii) stab( F hyp ) ≃ 0 .Proof. Property i) follows immediately from Corollary 3.5.8. To prove ii) it will suffice to show that forany additive F ∶ Cat p∞ → S 𝑝 and any stable ∞ -category C the boundary map F (Hyp( C )) ⟶ 𝕊 ⊗ F (Hyp( C ) [−1] ) is null-homotopic. But this follows immediately from the metabolic functor met ∶ Met(Hyp( C )) → Hyp( C ) being split by Corollary [I].2.3.23. (cid:3) Since stab evidently commutes with colimits and preserves additivity, we can apply Lemma 3.6.8 andobtain:3.6.18.
Corollary.
The transformation 𝜎 ∞ exhibits stab as a bordification. The filtration provided by the arising equivalence F bord ( C , Ϙ ) = colim 𝑑 𝕊 𝑑 ⊗ F ( C , Ϙ [− 𝑑 ] ) allows us to access the homotopy groups of the bordification of a space-valued F : ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 99 Corollary.
For every space valued additive F ∶ Cat p∞ → S the structure maps in the colimit ofDefinition 3.6.15 induce isomorphisms 𝜋 𝑖 F bord ( C , Ϙ ) ≅ 𝜋 | Cob F ( C , Ϙ [−( 𝑖 +1)] ) | , for all 𝑖 ∈ ℤ . In particular, the induced maps 𝜋 𝑖 ℂ ob F ( C , Ϙ ) ⟶ 𝜋 𝑖 F bord ( C , Ϙ ) are isomorphisms for 𝑖 < and for 𝑖 = 0 become the canonical projection under the identification Propo-sition 3.1.9 of the source. In other words, the group 𝜋 𝑖 F bord ( C , Ϙ ) is the F -based cobordism group of ( C , Ϙ [− 𝑖 ] ) . In fact, the proofwill show that the colimit description for F bord ( C , Ϙ ) stabilises on 𝜋 𝑖 after step 𝑖 . Here we follow our gen-eral convention not to distinguish notationally between a space-valued bordism invariant functor and itsspectrification. Proof.
By Proposition 3.5.8 we need only consider the case 𝑖 = −1 to obtain the first statement and wemay, furthermore, assume F group-like since both sides of the claimed isomorphism only depend on F grp (see Corollary 3.3.7 for the right hand side). But then by Corollary 3.4.8 the spectra ℂ ob F Met( C , Ϙ ) areconnective so all maps in the colimit sequence F bord ( C , Ϙ ) = colim( ℂ ob F ( C , Ϙ ) ⟶ 𝕊 ⊗ ℂ ob F ( C , Ϙ [−1] ) ⟶ 𝕊 ⊗ ℂ ob F ( C , Ϙ [−2] ) ⟶ … ) , induce isomorphisms on 𝜋 −1 , as their fibres are given by 𝕊 𝑘 ⊗ F (Met( C , Ϙ [− 𝑘 ] )) . We conclude using Proposi-tion 3.4.7. The claim about the induced map in 𝜋 follows from Lemma 3.1.10 by unwinding definitions. (cid:3) For a general additive F ∶ Cat p∞ → S 𝑝 we do not know how to compute the homotopy groups of F bord ( C , Ϙ ) in terms of those of F ( C , Ϙ ) . For 𝑖 ≥ the tautological map 𝜋 𝑖 F ( C , Ϙ ) ⟶ 𝜋 𝑖 F bord ( C , Ϙ ) factors canonically as 𝜋 𝑖 F ( C , Ϙ ) ⟶ 𝜋 𝑖 ((Ω ∞ F ) bord ( C , Ϙ )) ⟶ 𝜋 𝑖 F bord ( C , Ϙ ) , and we already noted in Remark 3.6.3 that the right hand map is not an equivalence in general.Finally, let us mention that one can also use the stab -construction and naive Karoubi periodicty to provideanother proof of Corollary 3.6.7 (without even investing that stab is a bordification). We can in fact showdirectly, that there is a bicartesian square F stab F ( F hyp ) hC ( F hyp ) tC 𝜎 ∞ F as follows: Consider the natural transformation F ⇒ ( F hyp ) hC for any additive F ∶ Cat p∞ → S 𝑝 and apply stab . Using naive Karoubi periodicity Proposition 3.4.13 we find stab(( F hyp ) hC )( C , Ϙ ) ≃ colim 𝑑 𝕊 𝑑 ⊗ F hyp ( C , Ϙ [− 𝑑 ] ) hC ≃ colim 𝑑 ( 𝕊 𝑑𝜎 ⊗ F hyp ( C , Ϙ ) ) hC with the structure maps in the final colimit induced by the inclusions 𝕊 → 𝕊 𝜎 as fixed points. But for any C -spectrum there is a canonical equivalence colim 𝑑 ( 𝕊 𝑑𝜎 ⊗ 𝑋 ) hC ≃ 𝑋 tC , in fact, this is essentially the classical definition of Tate spectra, say in [GM95]; to obtain it from thedefinition as the cofibre of the norm, note that the analogous colimit for the homotopy orbit spectra vanishes,since then the colimit can be permuted into the orbits and colim 𝑑 𝕊 𝑑𝜎 ⊗ 𝑋 ≃ 0 : The colimit is formed alongmaps 𝕊 → 𝕊 𝜎 , which are (non-equivariantly!) null-homotopic. This produces the Tate square above. Tosee that it is bicartesian, note that by construction stab preserves cofibre sequences. Now the cofibre of F ⇒ ( F hyp ) hC is easily checked to vanish on hyperbolic categories, so it is bordism invariant by Lemma 3.5.4.
00 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
Thus by Proposition 3.6.17 𝜎 ∞ F induces an equivalence on vertical cofibres of the Tate square. It is thereforecocartesian.From the fact that the Tate square is bicartesian, one can also obtain the fibre sequence F hyphC ⟶ F ⟶ stab F and conclude that stab really is a bordification functor, reversing the logic used in the original proof ofCorollary 3.6.7.3.7. The genuine hyperbolisation of an additive functor.
In this final subsection we recast the funda-mental fibre square Corollary 3.6.7 of an additive functor F ∶ Cat p∞ → S 𝑝 as the isotropy separation squareof a genuine C -spectrum, that is a spectral Mackey functor for the group C , refining the hyperbolisation F hyp ( C , Ϙ ) ∈ S 𝑝 hC . This allows for a convenient way of combining Karoubi periodicity with the shiftingbehaviour of bordism invariant functors, see Theorem 3.7.7 below. Note, however, that in the end this re-formulation does not yield additional information: The category of genuine C -spectra S 𝑝 gC participatesin a cartesian diagram S 𝑝 gC Ar( S 𝑝 ) S 𝑝 hC S 𝑝, (−) 𝜑 C2 ⇒ (−) tC2 u 𝑡 (−) tC2 where − 𝜑 C ∶ S 𝑝 gC → S 𝑝 extracts the geometric fixed points and u the underlying C -spectrum; we willgive a quick proof of this folklore result in Remark 3.7.5 below. In particular, the data F hyp ( C , Ϙ ) ∈ S 𝑝 hC and F bord ( C , Ϙ ) → F hyp ( C , Ϙ ) tC can be used to define the desired genuine refinement F ghyp ( C , Ϙ ) of F hyp ( C , Ϙ ) . The Mackey functor pointof view does, however, have the advantage that the requisite data can be constructed, once and for all, at thelevel of Poincaré ∞ -categories: In Corollary [I].7.4.18 we constructed (pre-)Mackey objects gHyp( C , Ϙ ) in Cat p∞ , see Theorem 3.7.1 below for the statement. The genuine C -spectrum F ghyp ( C , Ϙ ) just describedarises then by simply applying F to gHyp( C , Ϙ ) .Let us briefly recall the notion of a spectral Mackey functor. For a discrete group 𝐺 , we denote by Span( 𝐺 ) the span ∞ -category of finite 𝐺 -sets, introduced for the purposes of equivariant homotopy theoryin [Bar17, Df. 3.6] (under the name effective Burnside category).Then a Mackey object in an additive ∞ -category A is by definition a product preserving functor Span( 𝐺 ) → A . If A is taken to be S 𝑝 , the results of [Nar16, Appendix A] or [GM20, Appendix C] show, that the aris-ing ∞ -category underlies the model category of orthogonal 𝐺 -spectra classically used for the definition ofgenuine 𝐺 -spectra, see e.g. [Sch20]. We will treat spectral Mackey functors as the definition of the latterobjects and therefore put S 𝑝 gC = Fun × (Span(C ) , S 𝑝 ) . Evaluation at the finite C -set C then defines the functor u ∶ S 𝑝 gC → S 𝑝 hC , 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 . A genuine C -spectrum thus gives rise to a pair of spectra ( 𝐸 𝑔 C , 𝐸 ) , together with a C -action on 𝐸 and restriction andtransfer maps res ∶ 𝐸 gC → 𝐸 hC tr ∶ 𝐸 hC → 𝐸 gC coming from the spans(65) ∗ ← C ←←←←←←←←→ C and C ←←←←←←←←← C → ∗ together with a host of coherence data, which in particular identifies the composite tr ◦ res ∶ 𝐸 hC → 𝐸 hC with the norm map of 𝐸 , and similarly for other target categories.In Corollary [I].7.4.18, we showed: ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 101 Theorem.
The construction of hyperbolic categories canonically refines to a functor gHyp ∶ Cat p∞ ⟶ Fun × (Span(C ) , Cat p∞ ) together with natural equivalences of Poincaré ∞ -categories gHyp( C , Ϙ ) gC ≃ ( C , Ϙ ) , and C -Poincaré ∞ -categories u(gHyp( C , Ϙ )) ≃ Hyp C , such that transfer and restriction gHyp( C , Ϙ ) hC → gHyp( C , Ϙ ) gC and gHyp( C , Ϙ ) gC → gHyp( C , Ϙ ) hC , are naturally identified with hyp ∶ Hyp( C ) hC → ( C , Ϙ ) and f gt ∶ ( C , Ϙ ) → Hyp( C ) hC . Definition.
Let F ∶ Cat p∞ → S 𝑝 be an additive functor. Then we call the composite Cat p∞ gHyp ←←←←←←←←←←←←←←←←←←←←→
Fun × (Span(C ) , Cat p∞ ) F ←←←←←←←←→ Fun × (Span(C ) , S 𝑝 ) = S 𝑝 gC . the genuine hyperbolisation F ghyp of F .Now any genuine C -spectrum 𝑋 has an associated isotropy separation square 𝑋 gC 𝑋 𝜑 C 𝑋 hC 𝑋 tC , where the simplest description of the geometric fixed points − 𝜑 C for our purposes is as the cofibre of thetransfer 𝑋 hC → 𝑋 gC .3.7.3. Remark.
There are many other, more conceptual descriptions of the geometric fixed points. Forexample [Bar17, B.7] describes − 𝜑 C ∶ S 𝑝 gC → S 𝑝 as the left Kan extension along the fixed point functor (−) C ∶ Span(C ) → Span(Fin) , under the equivalence Fun × (Span(Fin) , S 𝑝 ) ≃ S 𝑝 and classically they are often defined as the cofibre of ( 𝑋 ⊗ 𝕊 [EC ]) gC → 𝑋 gC , where EC ∈ S C is the unique C -space with empty fixed points, whoseunderlying space is contractible, see e.g. [Sch20, Proposition 7.6]; here S C is the category of functors fromthe opposite of the orbit category O(C ) of C to S and the genuine suspension functor 𝕊 [−] ∶ S C → S 𝑝 gC is given as the composite S C 𝕊 [−] ←←←←←←←←←←←←←←←←←←→ S 𝑝 C Lan ←←←←←←←←←←←←←←→ S 𝑝 gC , where the second functor is left Kan extension along the evident inclusion O(C ) op → Span(C ) (it is alsothe left derived functor of the suspension spectrum functor in the classical model category picture). Thegenuine fixed points of the result are described by tom Dieck’s splitting [Sch20, Theorem 6.12] 𝕊 [ 𝑋 ] gC ≃ 𝕊 [ 𝑋 gC ] ⊕ 𝕊 [ 𝑋 hC ] , which can be recovered from the pointwise formula for the Kan extension.Geometric fixed points are in fact characterised in terms of this construction as the unique colimit pre-serving, symmetric monoidal functor S 𝑝 gC → S 𝑝 participating in a commutative square S C SS 𝑝 gC S 𝑝, gC 𝕊 [−] 𝕊 [−] 𝜑 C see [Sch20, Remark 7.15].
02 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
Now from the identification of the transfer in Theorem 3.7.1 and Proposition 3.6.6, there results anidentification F ghyp ( C , Ϙ ) 𝜑 C ≃ F bord ( C , Ϙ ) and by the universal property of bordifications this determinesthe entire isotropy separation square. We conclude:3.7.4. Corollary.
The isotropy separation square of the genuine C -spectrum F ghyp ( C , Ϙ ) is naturally iden-tified with the fundamental fibre square, in symbols F ghyp ( C , Ϙ ) gC F ghyp ( C , Ϙ ) 𝜑 C F ghyp ( C , Ϙ ) hC F ghyp ( C , Ϙ ) tC ≃ F ( C , Ϙ ) F bord ( C , Ϙ ) F Hyp( C ) hC F Hyp( C ) tC , for any additive functor F ∶ Cat p∞ → S and Poincaré ∞ -category ( C , Ϙ ) . In particular, combining this with the following remark, we find the functor F ghyp ∶ Cat p∞ → S 𝑝 gC isadditive again (although this is also readily checked straight from the definition).3.7.5. Remark.
That the extraction of isotropy separation squares leads to a cartesian square S 𝑝 gC Ar( S 𝑝 ) S 𝑝 hC S 𝑝, (−) 𝜑 C2 ⇒ (−) tC2 u 𝑡 (−) tC2 is a direct application of Proposition A.2.11: The forgetful functor u ∶ Sp gC → Sp hC admits both a leftand a right adjoint through Kan extension, whose images are often said to consist of the Borel (co)complete C -spectra. One readily checks the compositions starting and ending in Sp hC to be the identity. Thus u is a split Verdier projection (of non-small categories). The results of §A.2 together with some elementarymanipulations of the functors involved complete this to a stable recollement S 𝑝 S 𝑝 gC S 𝑝 hC , 𝑅 (−) 𝜑 C2 (−) s (−) q where 𝑅 is given by restriction along the fixed point functor Span(C ) → Span(Fin) , under the identification S 𝑝 ≃ Fun × (Span(Fin) , S 𝑝 ) , and the lower left functor takes 𝑋 to the fibre of 𝑋 gC → 𝑋 hC . The classifyingfunctor of this recollement is given by − tC ∶ S 𝑝 hC → S 𝑝 , so Proposition A.2.11 shows that the squareabove is cartesian. Furthermore, the resulting bicartesian square 𝑋 𝑅 ( 𝑋 𝜑 C )(u 𝑋 ) s 𝑅 (((u 𝑋 ) 𝑠 ) 𝜑 C ) recovers the isotropy separation square of 𝑋 upon applying genuine fixed points.Finally, we use the genuine spectrum F ghyp ( C , Ϙ ) to combine naive periodicity with the behaviour ofbordism invariant functors under shifting.3.7.6. Lemma.
Let F ∶ Cat p∞ → S 𝑝 be an additive functor and C a stable ∞ -category. Then the map ofgenuine C -spectra C ⊗ F (Hyp C ) → F ghyp (Hyp C ) adjoint to the diagonal F (Hyp C ) → F (Hyp C ) ⊕ F (Hyp C ) ≃ F (Hyp C × Hyp C ) ≃ F (Hyp(Hyp C )) is an equivalence. In particular, F ghyp (Met( C , Ϙ )) ≃ C ⊗ F (Hyp C ) . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 103 Proof.
The map is an equivalence both on underlying spectra and on geometric fixed points: On underlyingspectra this follows immediately from the corresponding statement
Hyp(Hyp( C )) ≃ C ⊗ Hyp( C ) on underlying C -Poincaré ∞ -categories from [I].7.4.15. Furthermore, both spectra have vanishing geo-metric fixed points: The left hand side by the symmetric monoidality of geometric fixed points togetherwith C = ∅ , the right hand side by bordism invariance. (cid:3) As a direct generalisation of Proposition 3.4.13 we then have:3.7.7.
Theorem (Genuine Karoubi periodicity) . Let ( C , Ϙ ) be a Poincaré ∞ -category and F ∶ Cat p∞ → S 𝑝 an additive functor. Then there is a natural equivalence of genuine C -spectra F ghyp ( C , Ϙ [−1] )) ≃ 𝕊 𝜎 −1 ⊗ F ghyp ( C , Ϙ ) , which translates the boundary map F ghyp ( C , Ϙ )) ⟶ 𝕊 ⊗ F ghyp ( C , Ϙ [−1] )) of the metabolic fibre sequence into the map induced by the inclusion 𝕊 → 𝕊 𝜎 as the fixed points. In particular, passing to geometric fixed points we recover the equivalence F bord ( C , Ϙ [ 𝑖 ] ) ≃ 𝕊 𝑖 ⊗ F bord ( C , Ϙ ) from Proposition 3.5.8. Proof.
Given the previous lemma, the proof of Proposition 3.4.13 applies essentially verbatim, when in-terpreted in the category of genuine C -spectra: Lemma 3.7.6 identifies the once-rotated metabolic fibresequence F ghyp (Met( C , Ϙ )) met ←←←←←←←←←←←←←←→ F ghyp ( C , Ϙ ) 𝜕 ←←←←←←→ 𝕊 ⊗ F ghyp ( C , Ϙ [−1] ) with C ⊗ F (Hyp( C )) ⟶ F ghyp ( C , Ϙ ) ⟶ 𝕊 𝜎 ⊗ F ghyp ( C , Ϙ ) obtained by tensoring F ghyp ( C , Ϙ ) with 𝕊 [C ] → 𝕊 → 𝕊 𝜎 . (cid:3) Alternatively, the statement of Theorem 3.7.7 can also be deduced from Proposition 3.4.13 together withProposition 3.5.8, via the interpretation of genuine C -spectra as isotropy separation squares: For everygenuine C -spectrum the canonical map 𝑋 ≃ 𝕊 ⊗ 𝑋 → 𝕊 𝜎 ⊗ 𝑋 induces an equivalence on geometric fixedpoints, for example by monoidality and ( 𝕊 𝜎 ) 𝜑 C ≃ 𝕊 .Therefore the effect of tensoring with 𝕊 𝜎 −1 on both geometric fixed points and the Tate constructionis a shift, which by Proposition 3.5.8 is also the effect of shifting the quadratic functors on these terms.Combined with the statement on underlying spectra Proposition 3.4.13 we obtain the claim.4. G ROTHENDIECK -W ITT THEORY
In this section we will define the central object of this paper, the Grothendieck-Witt spectrum
GW( C , Ϙ ) associated to a Poincaré ∞ -category ( C , Ϙ ) , and discuss its main properties. Most of the results are corol-laries of the results of §3.We will start out by defining the Grothendieck-Witt space GW ( C , Ϙ ) and record its properties, as speciali-sations of the general results of the previous section to the case F = Pn ∈ Fun add (Cat p∞ , S ) . We then proceedto analyse the Grothendieck-Witt spectrum in the same manner, in particular identifying its hyperbolisationas K -theory and its bordification as L -theory.This will lead to the identification of the homotopy type of the algebraic cobordism categories in Corol-lary 4.2.3, the fundamental fibre square in Corollary 4.4.14, localisation sequences for Grothendieck-Wittspectra of discrete rings in Corollary 4.4.18 and our generalisation of Karoubi periodicity in Corollar-ies 4.3.4 and Corollary 4.5.5, constituting the main results of the present paper.In the final subsection we spell out the relation of our constructions to the LA -theory of Weiss andWilliams from [WW14]. In particular, our results provide a cycle model for the infinite loop spaces of theirspectra.
04 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
The Grothendieck-Witt space.
In this section we will define the Grothendieck-Witt space of a Poincaré ∞ -category ( C , Ϙ ) , whose homotopy groups are by definition the higher Grothendieck-Witt groups of ( C , Ϙ ) .Recall that a functor Cat p∞ → E into an ∞ -category with finite limits is called additive if it carries splitPoincaré-Verdier squares to cartesian squares, see §1.5. Additive functors automatically take values in E ∞ -monoids (with respect to the cartesian product in E ) but may well not be grouplike; the functor Pn ∶ Cat p∞ → S taking Poincaré objects being the first example. Denoting by Fun add (Cat p∞ , E ) ⊆ Fun(Cat p∞ , E ) the fullsubcategory of additive functors, Corollary 3.3.6 asserts that the inclusion Fun add (Cat p∞ , Grp E ∞ ( S )) ⟶ Fun add (Cat p∞ , S ) admits a left adjoint (−) grp , the group completion functor.4.1.1. Definition.
We define the
Grothendieck-Witt space functor GW ∶ Cat p∞ ⟶ Grp E ∞ to be the group-completion GW ( C , Ϙ ) = Pn grp ( C , Ϙ ) , of the functor Pn ∈ Fun add (Cat p∞ , S ) . Furthermore, for a Poincaré ∞ -category ( C , Ϙ ) , we set GW 𝑖 ( C , Ϙ ) = 𝜋 𝑖 GW ( C , Ϙ ) , the Grothendieck-Witt-groups of ( C , Ϙ ) .We already introduced a group GW ( C , Ϙ ) explicitly in §[I].2.4 as the quotient of 𝜋 Pn( C , Ϙ ) given byidentifying every metabolic object with the hyperbolisation of its Lagrangian. We will see in Corollary 4.1.7below that this matches with the definition above.As a direct reformulation of the definition of Grothendieck-Witt functor we record:4.1.2. Observation.
The functor GW ∶ Cat p∞ → S is additive and grouplike, and it is the initial such functorunder Pn ∶ Cat p∞ → S . We will show in Corollary 4.4.15 that GW is in fact Verdier localising (and not just additive); that is, ittakes all Poincaré-Verdier squares to cartesian squares (and not just the split Poincaré-Verdier squares).4.1.3. Remark.
Let us explicitly warn the reader, that GW is the group-completion of Pn inside Fun add (Cat p∞ , S ) ,but not given as a levelwise group-completion, that is, GW ( C , Ϙ ) is generally not the group completion ofthe E ∞ -monoid Pn( C , Ϙ ) . Indeed, the levelwise group completion of Pn will not yield an additive functor.The functor GW can then be considered as the universal way of fixing this.From the results of the previous section we obtain several formulae for GW ( C , Ϙ ) : Recall from Def-inition 2.2.2 the cobordism category Cob( C , Ϙ ) associated to the Segal space Pn Q( C , Ϙ [1] ) given by thehermitian Q -construction. From Corollary 3.3.6 we find:4.1.4. Corollary.
There are canoncial equivalences GW ( C , Ϙ ) ≃ Ω | Cob( C , Ϙ ) | ≃ Ω | Pn Q( C , Ϙ [1] ) | natural in the Poincaré ∞ -category ( C , Ϙ ) . These formulae are in accordance with the usual definition of the K -theory space Ω | Span( C ) | ≃ Ω | Cr Q( C ) | of C .Classically, the Grothendieck-Witt space is often defined as the fibre of the forgetful functor from thehermitian to the usual Q -construction. We obtain such a description from the metabolic fibre sequence:Applying the hermitian Q -construction to the split Poincaré-Verdier sequence ( C , Ϙ ) ⟶ Met( C , Ϙ [1] ) ⟶ ( C , Ϙ [1] ) results in the fibre sequence | Pn Q( C , Ϙ ) | ⟶ | Pn Q Met( C , Ϙ [1] ) | met ←←←←←←←←←←←←←←→ | Pn Q( C , Ϙ [1] ) | , modelling the algebraic Genauer sequence | Cob( C , Ϙ [−1] ) | ⟶ | Cob 𝜕 ( C , Ϙ [−1] ) | 𝜕 ←←←←←←→ | Cob( C , Ϙ ) | , see Theorem 2.4.1 and Corollary 2.4.2. Now from Proposition 2.2.12 and Example 2.2.3 we obtain: ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 105 Corollary.
There are canonical equivalences | Pn Q Met( C , Ϙ [1] ) | ≃ | Pn Q Hyp( C ) | ≃ | Cr Q( C ) | under which the metabolic fibre sequence corresponds to | Pn Q( C , Ϙ ) | fgt ←←←←←←←←←←←←→ | Cr Q( C ) | hyp ←←←←←←←←←←←←←←→ | Pn Q( C , Ϙ [1] ) | . In particular, there are natural equivalences GW (Met( C , Ϙ )) ≃ K ( C ) and GW (Hyp( D )) ≃ K ( D ) for all Poincaré ∞ -categories ( C , Ϙ ) and stable ∞ -categories D . We also immediately obtain:4.1.6.
Corollary.
There are canonical equivalences GW ( C , Ϙ ) ≃ f ib( | Pn Q( C , Ϙ ) | fgt ←←←←←←←←←←←←→ | Cr Q( C ) | ) natural in the Poincaré ∞ -category ( C , Ϙ ) , and natural fibre sequences GW ( C , Ϙ [−1] ) fgt ←←←←←←←←←←←←→ K ( C ) hyp ←←←←←←←←←←←←←←→ GW ( C , Ϙ ) . This formula for the Grothendieck-Witt space is the transcription of the classical definition for examplefrom [Sch10a] into our framework.We can also use these formulae for an explicit description of GW ( C , Ϙ ) , giving another direct link. FromProposition 3.1.9 we find:4.1.7. Corollary.
The natural map 𝜋 Pn( C , Ϙ ) ⟶ GW ( C , Ϙ ) exhibits the target as the quotient of the source by the congruence relation generated by (66) [ 𝑥, 𝑞 ] ∼ [hyp( 𝑤 )] , where ( 𝑥, 𝑞 ) runs through the Poincaré objects of ( C , Ϙ ) with Lagrangian 𝑤 ⟶ 𝑥 . In particular, GW ( C , Ϙ ) is the quotient of 𝜋 Pn( C , Ϙ ) grp by the subgroup spanned by the differences [ 𝑥, 𝑞 ] − [hyp( 𝑤 )] , but part of the statement is that one does not need to complete 𝜋 Pn( C , Ϙ ) to a group inorder to obtain GW ( C , Ϙ ) as a quotient. From Corollary 3.1.8 or indeed from Lemma [I].2.4.3, we find thatfor [ 𝑥, 𝑞 ] ∈ GW ( C , Ϙ ) we have −[ 𝑥, 𝑞 ] = [ 𝑥, − 𝑞 ] + hyp(Ω 𝑋 ) . Finally, we showed in Corollary [I].5.2.8 that the functor
Pn ∶ Cat p∞ → Mon E ∞ ( S ) admits a canonical laxsymmetric monoidal structure with respect to the tensor product of Poincaré ∞ -categories on the left andthe tensor product of E ∞ -spaces on the right; informally it is simply given by tensoring Poincaré objects.Since 𝜋 ∶ Mon E ∞ ( S ) → CMon is also lax symmetric monoidal for the tensor products on both sides, thefunctor 𝜋 Pn ∶ Cat p∞ → CMon acquires a canonical lax symmetric monoidal structure. We showed inProposition [I].7.5.3:4.1.8.
Proposition.
The functor GW ∶ Cat p∞ → A 𝑏 admits a unique lax symmetric monoidal structure,making the transformation 𝜋 Pn → GW symmetric monoidal. In Paper [IV] we will enhance this to a lax symmetric monoidal structure on the functor GW itself, butfor the purposes of the present paper the above suffices.
06 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
The Grothendieck-Witt spectrum.
Our next goal is to deloop the Grothendieck-Witt-space into aspectrum valued additive functor. To this end recall from Corollary 3.3.6 that the forgetful functor Ω ∞ ∶ Fun add (Cat p∞ , S 𝑝 ) ⟶ Fun add (Cat p∞ , Grp E ∞ ) admits a left adjoint ℂ ob .4.2.1. Definition.
We define the
Grothendieck-Witt spectrum functor
GW ∶ Cat p∞ → S 𝑝 by GW( C , Ϙ ) = ℂ ob GW ( C , Ϙ ) , and denote by GW 𝑖 ( C , Ϙ ) its homotopy groups.We will see in Corollary 4.2.3 below, that for 𝑖 ≥ this conforms with the definition from Defini-tion 4.1.1.Again, we list the properties that are immediate from the results of the previous section. As a reformu-lation of the definition we find:4.2.2. Corollary.
The functor
GW ∶ Cat p∞ → S 𝑝 is additive, and it is the initial such functor equipped atransformation Pn ⇒ Ω ∞ GW of functors Cat p∞ → S . In fact, we show in Corollary 4.4.15 below, that GW is Verdier-localising and not just additive. Proof.
By Corollary 3.4.6, the functor GW is the initial additive functor to spectra equipped with a transfor-mation GW ⇒ Ω ∞ GW of E ∞ -groups. By the universal property of GW established in Observation 4.1.2,the functor GW is therefore also the initial additive functor to spectra with a transformation Pn ⇒ Ω ∞ GW of E ∞ -monoids. (cid:3) As additive functors to spaces carry unique refinements to
Mon E ∞ ( S ) , these statements remain true upondropping the E ∞ -structures, and by adjunction GW is also the initial additive functor GW ∶ Cat p∞ → S 𝑝 under 𝕊 [Pn] .Next, we identify the spaces Ω ∞− 𝑖 GW( C , Ϙ ) . To this end recall the 𝑖 -fold simplicial space Cob 𝑖 ( C , Ϙ ) = Pn Q ( 𝑖 ) ( C , Ϙ [ 𝑖 ] ) , given by the iterated hermitian Q -construction from Definition 2.1.1. These model the extended cobordismcategories of ( C , Ϙ ) and by Proposition 3.4.5 form a positive Ω -spectrum ℂ ob Pn ( C , Ϙ ) . The natural map ℂ ob Pn ( C , Ϙ ) ⟶ GW( C , Ϙ ) exhibits the right hand side as the spectrification of the left. From Proposition 3.4.5 we also find:4.2.3. Corollary.
For any Poincaré ∞ -category ( C , Ϙ ) there are canonical equivalences GW ( C , Ϙ ) ≃ Ω ∞ GW( C , Ϙ ) and | Cob 𝑖 ( C , Ϙ ) | ≃ Ω ∞− 𝑖 GW( C , Ϙ ) for any 𝑖 ≥ , that are natural in ( C , Ϙ ) . In particular, we obtain isomorphisms 𝜋 𝑖 GW( C , Ϙ ) ≅ GW 𝑖 ( C , Ϙ ) for all 𝑖 ≥ . Remark.
We propose to view the equivalences | Cob 𝑖 ( C , Ϙ ) | ≃ Ω ∞− 𝑖 GW( C , Ϙ ) for 𝑖 ≥ as a close analogue of the equivalence | Cob 𝑖𝑑 | ≃ Ω ∞− 𝑖 MTSO( 𝑑 ) , established by Galatius, Madsen, Tillmann and Weiss for 𝑖 = 1 , and Bökstedt and Madsen in general[GTMW09, BM14]. In particular, the sequence of spectra GW( C , Ϙ [− 𝑑 ] ) can be considered as an algebraicanalogue of the Madsen-Tillmann-spectra MTSO( 𝑑 ) .Of course, our arguments so far correspond only to the statement that the higher cobordism categories Cob 𝑛𝑑 deloop one another, i.e. that | Cob 𝑛𝑑 | ≃ Ω | Cob 𝑛 +1 𝑑 | for 𝑛 ≥ . The identification of the resulting ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 107 spectrum via a Pontryagin-Thom construction has no direct analogue in our work. We will describe thehomotopy type of GW( C , Ϙ ) by different means in Corollary 4.4.14 below.We shall make this connection more than an analogy in future work by promoting the association of itscochains or stable normal bundle to a manifold into functors 𝜎 ∶ Cob 𝑑 ⟶ Cob ( (Sp∕BSO( 𝑑 )) 𝜔 , Ϙ v− 𝛾 𝑑 ) ⟶ Cob ( D p ( ℤ ) , ( Ϙ s ) [− 𝑑 ] ) from the geometric to our algebraic cobordism categories. The Grothendieck-Witt spectrum of the mid-dle term has already appeared in manifold topology, see §4.6, and we expect the comma category of thecomposite functor over to be closely related to the category Cob L 𝑛 +1 from [HP19] for 𝑑 = 2 𝑛 + 1 .Just as the negative homotopy groups of the Madsen-Tillmann spectra are given by the cobordism groups,so are the negative homotopy groups of the Grothendieck-Witt spectrum. From Definition [I].2.3.11 werecall:4.2.5. Definition.
We define the L -group L ( C , Ϙ ) of a Poincaré ∞ -category as the quotient monoid of 𝜋 Pn( C , Ϙ ) by the submonoid of forms ( 𝑥, 𝑞 ) admitting a Lagrangian 𝑤 → 𝑥 .For the definition of a Lagrangian, see Definition [I].2.3.1. Also, L ( C , Ϙ ) really is a group: We showedin Corollary 2.2.9 that there is a canonical isomorphism 𝜋 | Cob( C , Ϙ [−1] ) | ≅ L ( C , Ϙ ) , and consequently, we find [ 𝑥, − 𝑞 ] + [ 𝑥, 𝑞 ] = 0 in L ( C , Ϙ ) from Proposition 2.2.6. In other words, L ( C , Ϙ ) is the cobordism group of Poincaré forms in ( C , Ϙ ) and inverses are given be reversing the orientation. From Proposition 3.4.7, we obtain:4.2.6. Corollary.
For 𝑖 > there are canonical isomorphisms 𝜋 − 𝑖 GW( C , Ϙ ) ≅ L ( C , Ϙ [ 𝑖 ] ) natural in the Poincaré ∞ -category ( C , Ϙ ) . By Proposition [I].7.5.3, we have:4.2.7.
Proposition.
The functor L ∶ Cat p∞ → A 𝑏 admits a unique lax symmetric monoidal structure, mak-ing the transformation 𝜋 Pn → L symmetric monoidal. In fact, this transformation then factors lax sym-metric monoidally over GW . We will use this fact in Proposition 4.6.4 below.4.3.
The Bott-Genauer sequence and Karoubi’s fundamental theorem.
In the present section we anal-yse the behaviour of the metabolic Poincaré-Verdier sequence ( C , Ϙ [−1] ) ⟶ Met( C , Ϙ ) met ←←←←←←←←←←←←←←→ ( C , Ϙ ) under the Grothendieck-Witt functor. From Example 2.2.3 and Corollary 3.1.4 we obtain:4.3.1. Corollary.
The functors lag ∶ Met( C , Ϙ ) ↔ Hyp( C ) ∶ can induce inverse equivalences GW ( Met( C , Ϙ ) ) ≃ GW ( Hyp( C ) ) for every Poincaré ∞ -category ( C , Ϙ ) and switching the order of the hyperbolic and Q -constructions givesan equivalence GW ( Hyp( D ) ) ≃ K( D ) for every stable ∞ -category D . In particular, for the hyperbolisation of the Grothendieck-Witt functor wefind GW hyp ≃ K . Now, applying GW to the metabolic sequence gives a fibre sequence GW( C , Ϙ [−1] ) fgt ←←←←←←←←←←←←→ K( C ) hyp ←←←←←←←←←←←←←←→ GW( C , Ϙ ) , of spectra, which we call the Bott-Genauer-sequence . It is a general version of the Bott-sequence appearingfor example in [Sch17, Section 6].
08 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
Remark.
We chose the present terminology to highlight the analogy with the fibre sequence
MTSO( 𝑑 + 1) ⟶ 𝕊 [BSO( 𝑑 + 1)] ⟶ MTSO( 𝑑 ) originally appearing in [GTMW09, Section 3], which complemented Genauer’s theorem in [Gen12, Section6] that | Cob 𝜕𝑑 | ≃ Ω ∞−1 𝕊 [BSO( 𝑑 )] . In particular, in the Bott-Genauer-sequence the algebraic K -theory spectrum really arises via the metaboliccategory, encoding objects with boundary, rather than the hyperbolic category. From this perspective theconnectivity of the algebraic K -theory spectrum corresponds to the fact, that the bordism groups of mani-folds with boundary vanish.Finally, we observe that the Bott-Genauer sequence gives a vast extension of Karoubi’s fundamentaltheorem: Following Karoubi and Schlichting [Kar80a, Sch17] we define functors U( C , Ϙ ) = f ib(K( C ) hyp ⟶ GW( C , Ϙ )) and V( C , Ϙ ) = f ib(GW( C , Ϙ ) fgt ⟶ K( C )) . Karoubi’s fundamental theorem [Kar80a, p. 260] compares these functors in the setting of discrete ringswith involution. In the setting of Poincaré ∞ -categories, this statement is a direct consequence of the Bott-Genauer sequence (we will specialise this abstract version to discrete rings in Corollary 4.3.4 below).4.3.3. Corollary (Karoubi’s fundamental theorem) . There is a canonical equivalence (67) U( C , Ϙ [2] ) ≃ 𝕊 ⊗ V( C , Ϙ ) natural in the Poincaré ∞ -category ( C , Ϙ ) .Proof. Simply note that the Bott-Genauer sequence allows us to identify both sides with
GW( C , Ϙ [1] ) . (cid:3) We next spell out the consequence of these abstract results for Grothendieck-Witt theory of rings. Recallthat these are integrated into our setup via their derived categories of modules. More generally, consideran E -ring 𝑅 and an invertible module with genuine involution ( 𝑀, 𝛼 ∶ 𝑁 → 𝑀 tC ) over 𝑅 . By Proposi-tion [I].3.4.2 there is a canonical equivalence of Poincaré ∞ -categories (Mod 𝜔𝑅 , ( Ϙ 𝛼𝑀 ) [ 𝑘 ] ) ≃ (Mod 𝜔𝑅 , Ϙ 𝛼 𝕊 − 𝑘𝜎 ⊗𝑀 ) refining the 𝑘 -fold loop functor. In particular, this yields equivalences (Mod 𝜔𝑅 , ( Ϙ 𝛼𝑀 ) [2 𝑘 ] ) ≃ (Mod 𝜔𝑅 , Ϙ 𝕊 𝑘 ⊗𝛼 𝕊 𝑘 − 𝑘𝜎 ⊗𝑀 ) and likewise for Mod f 𝑅 , if 𝑀 even belongs to Mod f 𝑅 . Now if 𝑅 is complex oriented, for example if 𝑅 is even periodic or (the Eilenberg-MacLane spectrum of) a discrete ring, then the module with involution 𝕊 𝑘 − 𝑘𝜎 ⊗ 𝑀 only depends on the parity of 𝑘 modulo up to a canonical equivalence induced by the complexorientation, see Example [I].3.4.6. Let us denote the common value for odd 𝑘 by − 𝑀 . If 𝑅 and 𝑀 arediscrete, then − 𝑀 is really given by changing the involution on 𝑀 to its negative.Using the arising equivalence (− 𝑀 ) tC ≃ 𝕊 ⊗ 𝑀 tC , we obtain equivalences (Mod 𝜔𝑅 , ( Ϙ s 𝑀 ) [2] ) ≃ (Mod 𝜔𝑅 , Ϙ s− 𝑀 ) , (Mod 𝜔𝑅 , ( Ϙ q 𝑀 ) [2] ) ≃ (Mod 𝜔𝑅 , Ϙ q− 𝑀 ) and, whenever 𝑅 is furthermore connective, (Mod 𝜔𝑅 , ( Ϙ ≥ 𝑚𝑀 ) [2] ) ≃ (Mod 𝜔𝑅 , Ϙ ≥ 𝑚 +1− 𝑀 ) . Note also, that if 𝑅 is even real oriented, for example a discrete ring of characteristic , then we even find 𝑀 ≃ − 𝑀 .Recall furthermore, that for c ∈ K ( 𝑅 ) = K (Mod 𝜔𝑅 ) a subgroup we denote by D c ( 𝑅 ) ⊆ D p ( 𝑅 ) thefull subcategory spanned by those 𝑅 -module complexes 𝑋 with [ 𝑋 ] ∈ c , the most interesting special casesbeing D K ( 𝑅 ) ( 𝑅 ) = D p ( 𝑅 ) and D ℤ ( 𝑅 ) = D f ( 𝑅 ) , where the integers on the right denote the image of ℤ → K ( 𝑅 ) , ↦ 𝑅 . We shall usually need to assumethat c is closed under the involution induced by 𝑀 . This is clearly always true in the former case, and inthe latter amounts to 𝑀 ∈ Mod f 𝑅 . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 109 Corollary.
For 𝑅 a complex oriented E -ring, for example a discrete ring, 𝑀 an invertible modulewith involution over 𝑅 , and c ⊆ K ( 𝑅 ) a subgroup closed under the involution induced by 𝑀 , there arecanonical equivalences U(Mod c 𝑅 , Ϙ q− 𝑀 ) ≃ 𝕊 ⊗ V(Mod c 𝑅 , Ϙ q 𝑀 ) and U(Mod c 𝑅 , Ϙ s− 𝑀 ) ≃ 𝕊 ⊗ V(Mod c 𝑅 , Ϙ s 𝑀 ) and if 𝑅 is furthermore connective, then also U(Mod c 𝑅 , Ϙ ≥ 𝑚 +1− 𝑀 ) ≃ 𝕊 ⊗ V(Mod c 𝑅 , Ϙ ≥ 𝑚𝑀 ) for arbitrary 𝑚 ∈ ℤ . Specialising the last equivalence further to a discrete ring 𝐴 , 𝑐 = K ( 𝐴 ) and either 𝑚 = 1 and 𝑚 = 2 ,we obtain the following extension of Karoubi’s fundamental theorem:4.3.5. Corollary.
For a discrete ring 𝐴 and a discrete invertible module with involution 𝑀 over 𝐴 , thereare canonical equivalences U( D p ( 𝐴 ) , Ϙ gq− 𝑀 ) ≃ 𝕊 ⊗ V( D p ( 𝐴 ) , Ϙ ge 𝑀 ) , U( D p ( 𝐴 ) , Ϙ ge− 𝑀 ) ≃ 𝕊 ⊗ V( D p ( 𝐴 ) , Ϙ gs 𝑀 ) . Given the comparisons in Appendix B, all of these equivalences collapse into the classical formulation ofKaroubi’s fundamental theorem upon restricting to discrete rings in which is invertible; if is not assumedinvertible they are, however, distinct. We will explore their uses for discrete rings in the third paper of thisseries.4.4. L -theory and the fundamental fibre square. In the present section we will prove our main result onthe homotopy type of the Grothendieck-Witt spectrum. In §3.6, we studied the bordification of an additivefunctor F ∶ Cat p∞ → S 𝑝 and in Corollary 4.4.14 produced a fibre square reconstructing F from its hyperboli-sation F hyp and its bordification F bord . In the previous subsection we obtained an equivalence GW hyp ≃ K ,and in the present section we will show GW bord ≃ L . To set the stage, recall the 𝜌 -construction fromDefinition 3.6.9.4.4.1. Definition.
The L -theory space is the functor Cat p∞ → S given by L ( C , Ϙ ) = | Pn 𝜌 ( C , Ϙ ) | obtained by applying the 𝜌 -construction to Pn .Since 𝜌 ( C , Ϙ ) = ( C , Ϙ ) , there is a canonical map Pn( C , Ϙ ) → L ( C , Ϙ ) . and by construction the -skeleta of the 𝜌 and Q construction agree, so from Corollary 2.2.9 we find thatthe natural map 𝜋 Pn( C , Ϙ ) ⟶ 𝜋 L ( C , Ϙ ) descends to an isomorphism L ( C , Ϙ ) ⟶ 𝜋 L ( C , Ϙ ) for all Poincaré ∞ -categories ( C , Ϙ ) .But much more is true: Generalising a classical result of Ranicki, Lurie showed in [Lur11, Lecture 7,Theorem 9], that there are canonical isomorphisms 𝜋 𝑖 L ( C , Ϙ ) = L ( C , Ϙ [− 𝑖 ] ) for all 𝑖 ≥ . While analogous to our results on bordifications, this is more difficult and fundamentally restson the fact that Pn 𝜌 ( C , Ϙ ) is a Kan simplicial space. In fact:4.4.2. Theorem.
Given a Poincaré-Verdier sequence ( C , Ϙ ) → ( D , Φ) → ( E , Ψ) the functor Pn 𝜌 ( D , Φ) → Pn 𝜌 ( E , Ψ) is a Kan fibration of simplicial spaces with fibre Pn 𝜌 ( C , Ϙ ) .In particular, the functor L ∶ Cat p∞ → S is Verdier-localising and bordism invariant. The above identification of homotopy groups is then a consequence of Proposition 3.5.8. The result itselfis the main content of [Lur11, Lectures 8 & Lecture 9], we give the proof here for completeness’ sake. Itrests on the following lemma:
10 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
Lemma.
Given a Poincaré-Verdier projection ( D , Φ) ( 𝑝,𝜂 ) ←←←←←←←←←←←←←←←←←→ ( E , Ψ) , an object 𝑥 ∈ D , a map 𝑓 ∶ 𝑦 → 𝑝 ( 𝑥 ) in E and a diagram 𝐾 ∗Ω ∞ Φ( 𝑥 ) Ω ∞ Ψ( 𝑝 ( 𝑥 )) Ω ∞ Ψ( 𝑦 ) 𝑞𝜂 𝑓 ∗ with 𝐾 ∈ S 𝜔 there exists an arrow 𝑔 ∶ 𝑧 → 𝑥 in D lifting 𝑓 together with a lift 𝐾 ∗Ω ∞ Φ( 𝑥 ) Ω ∞ Φ( 𝑧 ) 𝑟𝑔 ∗ of the original rectangle up to homotopy.Proof. Since 𝑝 is essentially surjective by Corollary A.1.7, there exists a 𝑣 in D with 𝑝 ( 𝑣 ) ≃ 𝑦 , and applying[NS18, Theorem I.3.3 ii)] we can then modify 𝑣 to find ℎ ∶ 𝑤 → 𝑦 lifting 𝑓 . From Remark 1.1.8 wefurthermore find Ψ( 𝑦 ) ≃ colim 𝑐 ∈ C 𝑤 ∕ Φ(f ib( 𝑤 → 𝑐 )) so putting 𝑢 = f ib( 𝑤 → 𝑐 ) for an appropriate 𝑐 , we find a lift 𝑠 ∈ Ω ∞ Φ( 𝑢 ) lifting 𝑞 , and the composite 𝑢 → 𝑤 → 𝑦 still lifts 𝑓 . To find a lift of the homotopy of maps 𝐾 → Ω ∞ Ψ( 𝑦 ) , note that the colimit aboveis filtered so since 𝐾 is assumed compact we also have Hom S ( 𝐾, Ω ∞+1 Ψ( 𝑦 )) ≃ colim 𝑐 ′ ∈ C 𝑢 ∕ Hom S ( 𝐾, Ω ∞+1 Φ(f ib( 𝑢 → 𝑐 ′ )) , which for appropriate 𝑐 yields all the desired data on for 𝑧 = f ib( 𝑢 → 𝑐 ′ ) and 𝑔 the composite 𝑧 → 𝑢 → 𝑦 . (cid:3) Proof of Theorem 4.4.2.
We need to show that each diagram Λ 𝑛𝑖 𝜌 ( D , Φ)Δ 𝑛 𝜌 ( E , Ψ) admits a filler up to homotopy (where we regard Δ 𝑛 and Λ 𝑛𝑖 as simplicial spaces via the inclusion Set ⊂ S ).To unwind this, recall that 𝜌 𝑛 ( D , Φ) = ( D , Φ) T 𝑛 , where T 𝑛 = P ([ 𝑛 ]) op is the opposite of the barycentricsubdivision sd(Δ 𝑛 ) of Δ 𝑛 . Denote then by 𝐻 𝑖𝑛 ⊆ T 𝑛 the opposite of the subdivision of the 𝑖 -horn, i.e. thecollection of subsets missing an element besides 𝑖 . Then the lifting problem above translates to showingthat the canonical map Pn ( ( D , Φ) T 𝑛 ) ⟶ Pn ( ( E , Ψ) T 𝑛 ) × Pn ( ( E , Ψ) 𝐻𝑖𝑛 ) Pn ( ( D , Φ) 𝐻 𝑖𝑛 ) is surjective on 𝜋 . To this end we first show the corresponding statement on spaces of hermitian objects, andthen explain how to adapt a lift in Fm ( ( D , Φ) T 𝑛 ) to a Poincaré one, provided its images in Fm ( ( E , Ψ) T 𝑛 ) and Fm ( ( D , Φ) 𝐻 𝑖𝑛 ) are Poincaré. The first claim even holds for boundary inclusions instead of horn inclu-sions, so denote by 𝐵 𝑛 the opposite of the subdivision of 𝜕 Δ 𝑛 and consider hermitian objects ( 𝐹 ∶ T 𝑛 → E , 𝑞 ) and ( 𝐺 ∶ 𝐵 𝑛 → D , 𝑟 ) and an equivalence between their images in Fm ( ( E , Ψ) 𝐵 𝑛 ) . Put then 𝑥 ∈ D as thelimit of 𝐺 . By construction there is then a canonical map 𝑓 ∶ 𝑦 → 𝑝 ( 𝑥 ) , where 𝑦 = 𝐹 ( 𝑏 ) with 𝑏 the barycen-tric vertex [ 𝑛 ] in T 𝑛 . Furthermore, regarding 𝑟 ∈ Φ 𝐵 𝑛 ( 𝐺 ) as a map 𝑟 ∶ ∗ → lim 𝐵 op 𝑛 Ω ∞ Φ ◦ 𝐺 op it is adjointto a transformation const ∗ ⇒ Ω ∞ Φ ◦ 𝐺 op , which gives rise to a map | 𝐵 op 𝑛 | ≃ colim 𝐵 op 𝑛 ∗ ⟶ colim 𝐵 op 𝑛 Ω ∞ Φ ◦ 𝐺 op ⟶ Ω ∞ Φ(lim 𝐵 𝑛 𝐺 ) = Ω ∞ Φ( 𝑥 ) whose composition down to Ω ∞ Ψ( 𝑦 ) is canonically identified with the constant map with value 𝑞 ∈ Ω ∞ Ψ T 𝑛 ( 𝐹 ) ≃Ω ∞ Ψ( 𝐹 ( 𝑏 )) , since 𝑏 = [ 𝑛 ] is initial in T op 𝑛 , so | T op 𝑛 | ≃∗ . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 111 We can therefore apply the previous lemma to obtain a lift 𝑔 ∶ 𝑧 → 𝑥 of 𝑓 , together with a lift 𝑠 ∈Ω ∞ Φ( 𝑧 ) of 𝑞 and an identification of the composite | 𝐵 op 𝑛 | ⟶ Ω ∞ Φ( 𝑥 ) 𝑔 ∗ ←←←←←←←←←←→ Ω ∞ Φ( 𝑧 ) with the constant map on 𝑠 , that lifts the identification above. Since T 𝑛 is the cone on 𝐵 𝑛 the map 𝑔 preciselydefines an extension of 𝐺 to a map ̃𝐺 ∶ T op 𝑛 → D , on which 𝑠 defines a hermitian form, and the remainderof the data produced bears witness to ( ̃𝐺, 𝑠 ) being a lift as desired.For the second step we need to modify a hermitian lift ( ̃𝐺, 𝑟 ) ∈ Fm ( ( D , Φ) T 𝑛 ) of ( 𝐹 , 𝑞 ) ∈ Pn ( ( E , Ψ) T 𝑛 ) ∈ and ( 𝐺, 𝑠 ) ∈ Pn ( ( D , Φ) 𝐻 𝑛𝑖 ) into a Poincaré lift. This is achieved by performing surgery as follows: Thealgebraic Thom construction from Corollary [I].2.3.20 gives an equivalence Fm ( ( D , Φ) T 𝑛 ) ≃ Pn(Met ( ( D , Φ [1] ) T 𝑛 ) refining the map taking ( ̃𝐺, 𝑠 ) to(68) D Φ T 𝑛 ( ̃𝐺 ) ⟶ cof ( ̃𝐺 𝑠 ←←←←←←←←←→ D Φ T 𝑛 ( ̃𝐺 )) . In particular, the Poincaré objects in ( D , Φ) T 𝑛 correspond precisely to those arrows with vanishing target(the target is the boundary of ( ̃𝐺, 𝑠 ) in the sense of Definition 4.4.7 below). Since ( 𝐹 , 𝑞 ) and ( 𝐺, 𝑟 ) and theboundary maps in the 𝜌 -construction are Poincaré (see the discussion before Definition 3.6.9) it follows thatthe target in our case already lies in the kernels of both D T 𝑛 ⟶ D 𝐻 𝑛𝑖 and D T 𝑛 ⟶ E T 𝑛 . We claim that the intersection of these kernels is equivalent to
Met( C , Ϙ [1− 𝑛 ] ) as a Poincaré ∞ -category.This is clear on underlying categories, and follows for the hermitian structures from the iterative formulaefor limits of cubical diagrams, i.e. lim T op 𝑛 𝑋 ≃ 𝑋 ({0 , ..., 𝑛 − 1}) × lim T op 𝑛 −1 𝑋 lim T op 𝑛 −1 𝑋 ◦ (− ∪ 𝑛 ) , which is easily verified using [Lur09a, Corollary 4.2.3.10] by decomposing T 𝑛 as the pushout of T 𝑛 −1 and T 𝑛 ⧵ {0 , ..., 𝑛 − 1} over their intersection. We thus find that the cofibre of 𝑠 admits a Lagrangian 𝐿 , sinceobjects in metabolic Poincaré ∞ -categories are canonically metabolic by Remark [I].7.3.23. We can thusperform surgery on (68) with the surgery datum → 𝐿 , see Proposition 2.3.3. The resulting arrow hasvanishing target, and by design the surgery changes neither the image in E T 𝑛 nor the restriction to D 𝐻 𝑛𝑖 .Tranlating back along the algebraic Thom construction thus provides the desired Poincaré lift of ( 𝐹 , 𝑞 ) and ( 𝐺, 𝑞 ) .To deduce the remaining claims, note that the statement about the fibre is immediate from both coten-sors and Pn preserving limits. That L is Verdier-localising now follows, since colimits of simplicial fibresequences with second map a Kan fibration are again fibre sequences, see e.g. [Lur16, Theorem A.5.4.1].To finally obtain bordism invariance, one can either proceed by observing that on account of the Kanproperty the 𝑖 -th homotopy groups of L( C , Ϙ ) = | Pn 𝜌 ( C , Ϙ ) | can be described as the quotient of 𝜋 f ib ( Hom s S (Δ 𝑖 , Pn 𝜌 ( C , Ϙ )) ⟶ Hom s S ( 𝜕 Δ 𝑖 , Pn 𝜌 ( C , Ϙ )) ) by the equivalence relation generated by a pair of such elements admitting an extension to 𝜋 f ib ( Hom s S (Δ × Δ 𝑖 , Pn 𝜌 ( C , Ϙ )) → Hom s S (Δ × 𝜕 Δ 𝑖 , Pn 𝜌 ( C , Ϙ )) ) . This quotient is readily checked to be exactly L ( C , Ϙ [− 𝑖 ] ) . This is the route taken in both [Ran92] and[Lur11].Alternatively, one can employ 3.6.13 to see that L(Met( C , Ϙ )) ≃ | Pn 𝜌 (Met( C , Ϙ )) | is the realisation ofa split simplicial object over , therefore vanishes, and then conclude by 3.5.4. Let us remark that via thealgebraic Thom construction [I].2.3.20 the extra degeneracy of the split simplicial space Pn 𝜌 (Met( C , Ϙ )) ≃Fm 𝜌 ( C , Ϙ [−1] ) attains a particularly easy form: It is simply given by extension-by-zero. We leave the nec-essary unwinding of definitions to the reader. (cid:3) It now follows from Theorem 3.5.9 that L admits an essentially unique lift to a functor with values inspectra.
12 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
Definition.
We define the L -theory spectrum L ∶ Cat p∞ → S 𝑝 by L( C , Ϙ ) = ℂ ob L ( C , Ϙ ) with ( C , Ϙ ) a Poincaré ∞ -category, and denote by L 𝑖 ( C , Ϙ ) its homotopy groups.This definition of the L -groups agrees with Definition 4.2.5, since from Proposition 3.4.5 and Proposi-tion 3.5.8 we obtain:4.4.5. Corollary.
There are canonical equivalences Ω ∞− 𝑖 L( C , Ϙ ) ≃ L ( C , Ϙ [ 𝑖 ] ) for all 𝑖 ∈ ℤ . In particular, there are isomorphisms 𝜋 𝑖 L( C , Ϙ ) ≅ L ( C , Ϙ [− 𝑖 ] ) also for negative 𝑖 . In fact, the definition L( C , Ϙ ) ≃ [ L ( C , Ϙ ) , L ( C , Ϙ [1] ) , L ( C , Ϙ [2] ) , … ] with structure maps arising from Proposition 3.5.8 is a direct generalisation of the classical definition of L -theory spectra due to Ranicki, see for example [Ran92, Section 13], and it is rather more elegant than ourdefinition which iterates the Q -construction on top of the 𝜌 -construction.As an important consequence, weobtain:4.4.6. Corollary.
The functor
L ∶ Cat p∞ → S 𝑝 is bordism invariant and Verdier-localising. One can even directly describe the boundary operator of the long exact sequence on the L -groups of aPoincaré-Verdier sequence.4.4.7. Definition.
Given a Poincaré ∞ -category ( C , Ϙ ) and a hermitian object ( 𝑋, 𝑞 ) ∈ Fm( C , Ϙ ) , the bound-ary of ( 𝑋, 𝑞 ) is the Poincaré object 𝜕 ( 𝑋, 𝑞 ) ∈ Pn( C , Ϙ [1] ) obtained as the result of surgery on ( 𝑋 → , 𝑞 ) ∈Surg ( C , Ϙ [1] ) .Note that by the discussion preceeding Proposition 2.3.3 the object underlying 𝜕 ( 𝑋, 𝑞 ) is given by thecofibre of 𝑞 ♯ ∶ 𝑋 → D Ϙ 𝑋 .4.4.8. Proposition.
Given a Poincaré-Verdier sequence ( C , Ϙ ) 𝑖 ←←←←→ ( D , Φ) 𝑝 ←←←←←←→ ( E , Ψ) the boundary operator L 𝑖 ( E , Ψ) → L 𝑖 −1 ( C , Ϙ ) of the resulting long exact sequence takes a Poincaré object ( 𝑋, 𝑞 ) ∈ Pn( E , Ψ [− 𝑖 ] ) to 𝜕 ( 𝑌 , 𝑞 ′ ) ∈ Pn( C , Ϙ [1− 𝑖 ] ) , where ( 𝑌 , 𝑞 ′ ) ∈ Fm( D , Φ [− 𝑖 ] ) is any lift of ( 𝑋, 𝑞 ) . In particular, the proposition asserts that such a hermitian lift of 𝑋 can always be found, and its imagein L 𝑖 −1 ( C , Ϙ ) is the obstruction against finding a Poincaré lift of 𝑋 . Proof.
From Lemma 3.1.10 we find that the inverse to the boundary isomorphism 𝜋 L( E , Ψ) → 𝜋 L( E , Ψ [−1] ) takes a Poincaré object 𝑋 in the target to the loop 𝑤 represented by ← 𝑋 → 𝜌 ( E , Ψ) . We nowcompute the map L ( E , Ψ) → L ( C , Ϙ ) , the case of general 𝑖 ∈ ℤ follows by shifting the quadratic func-tor. That any Poincaré object ( 𝑋, 𝑞 ) ∈ Pn( E , Ψ [−1] ) can be lifted to some ( 𝑌 , 𝑞 ′ ) ∈ Fm( D , Φ [−1] ) is anapplication of Lemma 4.4.3 (with 𝐾 = ∅ ).Now regarding the map ( 𝑌 → , 𝑞 ′ ) as a surgery datum in Surg ( D , Φ) we can apply Proposition 2.3.3to obtain a cobordism from to the result of surgery, which is 𝜕 ( 𝑌 , 𝑞 ′ ) . We can regard this cobordismas an element of Pn( 𝜌 ( D , Φ)) and thus as a path in L( D , Φ) . By construction this path lifts the loop in L( E , Ψ) defined by 𝑋 via the consideration in the first paragraph. Therefore its endpoint cof ( 𝑌 → D Ϙ [−1] 𝑌 ) represents the image of ( 𝑋, 𝑞 ) under the boundary map as claimed. (cid:3) Remark.
In [Lur11, Lecture 20] Lurie gives yet another definition of the L -theory spectrum, bydirectly constructing an excisive functor S fin∗ → S , whose value on the one point space is L ( C , Ϙ ) . However,while certainly true it is never justified in [Lur11], that the functor constructed evaluates to L ( C , Ϙ [ 𝑖 ] ) on the 𝑖 -sphere. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 113 Now by construction there is a natural transformation Pn ⇒ L ≃ Ω ∞ L , which uniquely extends to atransformation(69) bord∶ GW ⇒ L of functors Cat p∞ → S 𝑝 by Corollary 4.2.2. We record, see Corollary 3.6.19:4.4.10. Corollary.
Under the identifications of Proposition 3.1.9 the map bord∶ 𝜋 GW( C , Ϙ ) → 𝜋 L( C , Ϙ ) becomes the canonical projection GW ( C , Ϙ ) → L ( C , Ϙ ) . Similary, for 𝑖 > the induced map 𝜋 − 𝑖 GW( C , Ϙ ) → 𝜋 − 𝑖 L( C , Ϙ ) is identified with the identity of L ( C , Ϙ [ 𝑖 ] ) by Proposition 3.4.7. Remark.
While the map bord∶ GW( C , Ϙ ) → L( C , Ϙ ) is most easily constructed via the universalproperty of GW , it is also easy to obtain a direct map between these spectra when defining them via the Q - and 𝜌 -constructions: Consider the map of cosimplicial objects 𝜂 ∶ (sd Δ 𝑛 ) op → TwAr(Δ 𝑛 ) , that sendsa non-empty subset 𝑇 ⊆ [ 𝑛 ] to the pair (min 𝑇 ≤ max 𝑇 ) . It is an isomorphism in degrees and and indegree it sends a diagram 𝑣 × 𝑦 𝑤𝑣 𝑤𝑥 𝑦 𝑧 in Q ( C , Ϙ [1] ) to 𝑦𝑥 𝑧𝑣 × 𝑦 𝑤𝑣 𝑤𝑣 × 𝑦 𝑤 in 𝜌 ( C , Ϙ [1] ) . The analogous operation on manifold cobordisms takes two composable cobordisms to the -ad given by the cartesian product of their composition with an interval; the ad-structure is given (aftersmoothing corners) by decomposing the boundary into the original two cobordisms, represented along thediagonal edges, and their composite given by the lower horizontal edge. In general then, the transformation 𝜂 ∶ Q ⇒ 𝜌 regards 𝑛 composable -ads as a special case of an 𝑛 -ad.Now 𝜂 induces a map GW ( C , Ϙ ) = Ω | Pn Q( C , Ϙ [1] ) | Ω | 𝜂 | ←←←←←←←←←←←←←←←←←→ Ω | Pn 𝜌 ( C , Ϙ [1] ) | 𝜕 ←←←←←←→ | Pn 𝜌 ( C , Ϙ ) | = L ( C , Ϙ ) and thus a map 𝜂 ∶ GW = ℂ ob GW ⇒ ℂ ob L = L . Using Lemma 3.1.10 it is not difficult to check, that thismap satisfies the universal property defining bord . Since we shall not have to make use of that statement,we leave the details to the reader.We now turn to the main result of this section:4.4.12. Theorem.
The transformation bord exhibits L as the bordification of GW . In particular, L ∶ Cat p∞ → S 𝑝 is the initial bordism invariant, additive functor equipped with a transformation Pn ⇒ Ω ∞ L of functors Cat p∞ → S . From Theorem 3.5.9 we also find that L ∶ Cat p∞ → S is the initial bordism invariant, additive functorunder either Pn or GW . Proof.
We give two proofs of the first statement. The second is then immediate from Corollary 4.2.2.The first argument employs the formula of Definition 3.6.9 in terms of the ad -construction for bordifica-tions: The natural equivalence of Poincaré ∞ -categories 𝜌 𝑛 Q 𝑚 ( C , Ϙ ) ≃ Q 𝑚 𝜌 𝑛 ( C , Ϙ ) identifies L( C , Ϙ ) withthe geometric realization of the simplicial spectrum GW( 𝜌 ( C , Ϙ )) in the category of prespectra. Since the
14 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE result is already an Ω -spectrum we have that L( C , Ϙ ) is also the geometric realization of GW( 𝜌 ( C , Ϙ )) in S 𝑝 .We obtain a natural identification L ≃ | GW 𝜌 | = ad(GW) , which gives the claim by Corollary 3.6.12.We can also employ the stab -construction: By Proposition 3.5.8, the map bord factors over a map colim 𝑑 𝕊 𝑑 ⊗ GW( C , Ϙ [− 𝑑 ] ) ⟶ L( C , Ϙ ) . But it follows from Corollary 4.4.10 and Corollary 3.6.19 that this map is an isomorphism on homotopygroups. By Corollary 3.6.18 the claim follows a second time. (cid:3)
Remark.
Under the analogy between
GW( C , Ϙ [− 𝑑 ] ) and MTSO( 𝑑 ) (see Remarks 4.2.4 and 4.3.2)the equivalence colim 𝑑 𝕊 𝑑 ⊗ GW( C , Ϙ [− 𝑑 ] ) ≃ L( C , Ϙ ) corresponds to the canonical equivalence colim 𝑑 𝕊 𝑑 ⊗ MTSO( 𝑑 ) ≃ MSO , whose proof is an elementary manipulation of Thom spectra (see [GTMW09, Section 3]). In particular, therole of the spectrum MSO is played by L( C , Ϙ ) in our theory; even its definition in terms of the 𝜌 -constructionis modelled on Quinn’s construction of the ad-spectrum of manifolds Ω SO , whose homotopy groups byconstruction are the cobordism groups. The second proof of the above theorem is then a translation of thewell-known equivalence colim 𝑑 𝕊 𝑑 ⊗ ℂ ob 𝑑 ≃ Ω SO from geometric topology; using the main result of [Ste18] this identification can in fact be achieved withoutreference to Thom spectra whatsoever and therefore used to deduce the equivalence MSO ≃ Ω SO , i.e.the Pontryagin-Thom theorem, from the equivalences ℂ ob 𝑑 ≃ MTSO( 𝑑 ) of Bökstedt, Galatius, Madsen,Tillmann and Weiss.Now since the functor ( C , Ϙ ) ↦ K( C , Ϙ ) tC is bordism invariant by Example 3.5.6 the composite GW fgt ⇒ K hC ⇒ K tC factors uniquely over a map Ξ ∶ L ⇒ K tC and we obtain the main result of this paper:4.4.14. Corollary (The fundamental fiber square) . The natural square (70)
GW( C , Ϙ ) L( C , Ϙ )K( C , Ϙ ) hC K( C , Ϙ ) tC bordfgt Ξ is bicartesian for every Poincaré ∞ -category ( C , Ϙ ) and in particular, there is a natural fibre sequence (71) K( C , Ϙ ) hC hyp ←←←←←←←←←←←←←←→ GW( C , Ϙ ) bord ←←←←←←←←←←←←←←←←←→ L( C , Ϙ ) . Proof.
Apply Corollary 3.6.7 in combination with Corollary 4.3.1 and Theorem 4.4.12. (cid:3)
We will exploit this result to give computations of Grothendieck-Witt groups of discrete rings in Paper[III], and solve the homotopy limit for number rings. For now we record:4.4.15.
Corollary.
The functor
GW ∶ Cat p∞ → S 𝑝 is Verdier-localising.Proof. Given Corollary 4.4.6, we need only recall that K -theory is a Verdier-localising functor K → S 𝑝 (asby Proposition 1.1.4 the underlying sequence of a Poincaré-Verdier sequence is indeed a Verdier sequence).One way to obtain a proof of this from the literature is as the combination ofi) the non-connective K -theory functor 𝕂 taking Karoubi sequences to fibre sequences [BGT13, Section9]ii) the cofinality theorem, i.e. the map K( C ) → K( C idem ) inducing an isomorphism on positive homotopygroups and an injection in degree [Bar16, Theorem 10.19]iii) its consequence Ω ∞ 𝕂 ( C ) ≃ K ( C idem ) , and finally,iv) Thomason’s classification of dense subcategories Theorem A.3.2, i.e. that for a dense stable subcate-gory C ⊆ D , we have 𝑐 ∈ C if and only if [ 𝑐 ] is in the image of K ( C ) → K ( D ) . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 115 Together these statements imply that that for a Verdier-sequence C → D → E the maps K( C ) ⟶ f ib(K( D ) → K( E )) → 𝕂 ( C ) are both isomorphisms in positive degrees, injective in degree and that the images of the right hand mapand the composite in 𝕂 ( C ) agree. Since K( C ) is connective, this gives the claim. (cid:3) Remark.
This circuitous route to the Verdier-localisation property of connective K -theory is ne-cessitated only by the restriction to idempotent complete categories in [BGT13]. In truth, it is a highercategorical version of Waldhausen’s fibration theorem (though of a flavour different from [Bar16, Theorem8.11]) which gives this statement in one fell swoop. We do not, however, know of a reference where this isspelled out.Now, by construction the composite K( C , Ϙ ) hC hyp ←←←←←←←←←←←←←←→ GW( C , Ϙ ) fgt ←←←←←←←←←←←←→ K( C , Ϙ ) hC is the norm map of the C -spectrum K( C , Ϙ ) ∈ Sp hC . In particular, it is split after inverting by thecanonical maps K( C , Ϙ ) hC ⟶ K( C , Ϙ ) ⟶ K( C , Ϙ ) hC divided by . But then also the fibre sequence K( C , Ϙ ) hC hyp ←←←←←←←←←←←←←←→ GW( C , Ϙ ) bord ←←←←←←←←←←←←←←←←←→ L( C , Ϙ ) splits after inverting and we obtain:4.4.17. Corollary.
There is a canonical equivalence
GW( C , Ϙ )[ ] ≃ K( C , Ϙ )[ ] hC ⊕ L( C , Ϙ )[ ] natural in the Poincaré ∞ -category ( C , Ϙ ) and in particular GW 𝑖 ( C , Ϙ )[ ] ≅ K 𝑖 ( C )[ ] C ⊕ L( C , Ϙ )[ ] . Proof.
Only the final statement remains, and this follows immediately from the former and the collapse ofthe homotopy orbit spectral sequence of a C -spectrum in which is invertible to its edge. (cid:3) As a consequence of Corollary 1.4.9, we obtain localisation properties of Grothendieck-Witt spectra,which will form the basis of our analysis of the Grothendieck-Witt groups of Dedekind rings in the thirdpaper of this series, see Corollary [III].2.1.9.From Proposition 1.4.8 we then immediately obtain:4.4.18.
Corollary.
Let 𝐴 be a discrete ring, 𝑀 a discrete invertible module with involution over 𝐴 , c ⊂ K ( 𝐴 ) a subgroup closed under the involution induced by 𝑀 and 𝑆 ⊆ 𝐴 a multiplicative subset compatiblewith 𝑀 , such that ( 𝐴, 𝑆 ) satisfies the left Ore condition. Let, furthermore, D c ( 𝐴 ) 𝑆 denote the full subcat-egory of D c ( 𝐴 ) spanned by the 𝑆 -torsion complexes. Then the inclusion and localisation functors fit intofibre sequences GW( D c ( 𝐴 ) 𝑆 , Ϙ ≥ 𝑚𝑀 ) ⟶ GW( D c ( 𝐴 ) , Ϙ ≥ 𝑚𝑀 ) ⟶ GW( D im(c) ( 𝐴 [ 𝑆 −1 ]) , Ϙ ≥ 𝑚𝑀 [ 𝑆 −1 ] ) for all 𝑚 ∈ ℤ ∪ {±∞} . For the compatibility condition between the multiplicative subset and the invertible module confer Def-inition 1.4.3 and Example 1.4.4.In particular, one obtains a fibre sequence
GW( D f ( 𝐴 ) 𝑆 , Ϙ ≥ 𝑚𝑀 ) ⟶ GW( D f ( 𝐴 ) , Ϙ ≥ 𝑚𝑀 ) ⟶ GW( D f ( 𝐴 [ 𝑆 −1 ]) , Ϙ ≥ 𝑚𝑀 [ 𝑆 −1 ] ) though this generally fails for D p in place of D f , but see Remark 4.4.20 below. Upon taking connectivecovers, the case of commutative 𝐴 is for example also imply by [Sch17]. We similarly obtain fibre sequences L( D c ( 𝐴 ) 𝑆 , Ϙ ≥ 𝑚𝑀 ) ⟶ L( D c ( 𝐴 ) , Ϙ ≥ 𝑚𝑀 ) ⟶ L( D im(c) ( 𝐴 [ 𝑆 −1 ]) , Ϙ ≥ 𝑚𝑀 [ 𝑆 −1 ] ) , which upon investing our identification of the genuine L -spectra in the third installment of this series, seeTheorem [III].1.2.18, recover localisation sequences of Ranicki’s, see [Ran81, Section 3.2].
16 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
Remark.
By Corollary 1.4.6, the quadratic variant of Corollary 4.4.18 actually works for an arbitrary E -ring spectrum 𝐴 and an invertible module 𝑀 with involution over 𝐴 , but for the symmetric and genuinevariants, one has to require further conditions. We leave details to the interested reader, as we shall have noneed for that generality.4.4.20. Remark.
By the cofinality theorem the map K 𝑖 ( D c ( 𝑅 )) → K 𝑖 ( D p ( 𝑅 )) = K 𝑖 ( 𝑅 ) induces an iso-morphism 𝑖 > and is the inclusion c → K ( 𝑅 ) for 𝑖 = 0 . We will show a hermitian analogue in thefourth installment of this series, namely that for any pair of involution-closed subgroups c ⊆ d ⊆ K ( 𝑅 ) the squares GW( D c ( 𝑅 ) , Ϙ ) GW( D d ( 𝑅 ) , Ϙ ) L( D c ( 𝑅 ) , Ϙ ) L( D d ( 𝑅 ) , Ϙ )K( D c ( 𝑅 ) , D Ϙ ) hC K( D d ( 𝑅 ) , D Ϙ ) hC K( D c ( 𝑅 ) , D Ϙ ) tC K( D d ( 𝑅 ) , D Ϙ ) tC are cartesian, see Theorem [IV].2.1.3. It follows that there are fibre sequences GW( D c ( 𝑅 ) , Ϙ ) ⟶ GW( D d ( 𝑅 ) , Ϙ ) ⟶ H(d∕c) hC L( D c ( 𝑅 ) , Ϙ ) ⟶ L( D d ( 𝑅 ) , Ϙ ) ⟶ H(d∕c) tC . In particular, the map GW 𝑖 ( D c ( 𝑅 ) , Ϙ ) ⟶ GW 𝑖 ( D d ( 𝑅 ) , Ϙ ) is an isomorphism for positive 𝑖 and injectivefor 𝑖 = 0 . On the L -theoretic side, we recover Ranicki’s Rothenberg-sequences … ⟶ L 𝑖 ( D c ( 𝑅 ) , Ϙ ) ⟶ L 𝑖 ( D d ( 𝑅 ) , Ϙ ) ⟶ ̂ H − 𝑖 (C ; 𝑑 ∕ 𝑐 ) ⟶ L 𝑖 −1 ( D c ( 𝑅 ) , Ϙ ) ⟶ … [Ran80, Proposition 9.1].In a similar vein, one can compare localisations along a ring homomorphism:4.4.21. Proposition.
Let 𝑝 ∶ 𝐴 → 𝐵 be a homomorphism of discrete rings, 𝑀 and 𝑁 discrete invertiblemodules with involution over 𝐴 and 𝐵 , respectively, 𝜂 ∶ 𝑀 → 𝑁 a group homomorphism that is 𝑝 ⊗ 𝑝 -linear, 𝑆 ⊆ 𝐴 a subset and 𝑚 ∈ ℤ ∪ {±∞} . Then ifi) the map 𝐵 ⊗ 𝐴 𝑀 → 𝑁 induced by 𝜂 is an isomorphism,ii) the subset 𝑆 is compatible with 𝑀 ,iii) for every 𝑠 ∈ 𝑆 the induced map 𝑝 ∶ 𝐴 ⫽ 𝑠 → 𝐵 ⫽ 𝑝 ( 𝑠 ) on cofibres of right multiplication by 𝑠 and 𝑝 ( 𝑠 ) , respectively, is an equivalence in D ( 𝐴 ) ,iv) the pairs ( 𝑆, 𝐴 ) and ( 𝑝 ( 𝑆 ) , 𝐴 ) both satisfy the left Ore condition, andv) the boundary map ̂ H − 𝑚 (C , 𝑁 [ 𝑝 ( 𝑆 ) −1 ]) → ̂ H − 𝑚 +1 (C , 𝑀 ) in Tate cohomology of the short exactsequence 𝑀 (− 𝜂, can) ←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ 𝑁 ⊕ 𝑀 [ 𝑆 −1 ] (can ,𝜂 ) ←←←←←←←←←←←←←←←←←←←←←←←←→ 𝑁 [ 𝑝 ( 𝑆 ) −1 ] vanishes,the square ( D c ( 𝐴 ) , Ϙ ≥ 𝑚𝑀 ) ( D im(c) ( 𝐴 [ 𝑆 −1 ]) , Ϙ ≥ 𝑚𝑀 [ 𝑆 −1 ] )( D 𝑝 (c) ( 𝐵 ) , Ϙ ≥ 𝑚𝑁 ) ( D im( 𝑝 (c)) ( 𝐵 [ 𝑝 ( 𝑆 ) −1 ]) , Ϙ ≥ 𝑚𝑁 [ 𝑝 ( 𝑆 ) −1 ] ) is a Poincaré-Verdier square for every subgroup 𝑐 ⊆ K ( 𝐴 ) stable under the involution induced by 𝑀 , andso in particular becomes cartesian after taking either GW -, K - or L -spectra. Here, condition v) is to be interpreted as vacuous of 𝑚 = ±∞ . Note also that condition iv) is equivalentto requiring that 𝑝 induces an isomorphism on kernels and cokernels of right multiplication by any 𝑠 ∈ 𝑆 .The K -theoretic part is a classical result of Karoubi, Quillen and Vorst, see [Vor79, Proposition 1.5], andinvesting the identification of the L -spectra from the third installment in this series, see Theorem [III].1.2.18,the L -theoretic part recovers analogous result of Ranicki [Ran81, Section 3.6]. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 117 Proof.
Let us start out by observing that the diagram
𝐴 𝐴 [ 𝑆 −1 ] 𝐵 𝐵 [ 𝑝 ( 𝑆 ) −1 ] is cartesian in D ( 𝐴 ) : Denoting the top horizontal fibre by 𝐹 , this is equivalent to the assertion that 𝐹 → 𝐵 ⊗ 𝕃 𝐴 𝐹 is an equivalence in D ( 𝐴 ) , but combining Example 1.4.2 with assumptions iii) and iv) this holdsfor any object of D ( 𝐴 ) 𝑆 . Tensoring the square with 𝑀 (over 𝐴 ) then produces the short exact sequenceappearing in v). Furthermore, from the Ore conditions we also find that the natural map 𝐵 ⊗ 𝕃 𝐴 𝐴 [ 𝑆 −1 ] → 𝐵 [ 𝑝 ( 𝑆 ) −1 ] is an equivalence. It is then readily checked that 𝑝 ( 𝑆 ) is compatible with 𝑁 .Now, the rows of the diagram of Poincaré ∞ -categories ( D c ( 𝐴 ) 𝑆 , Ϙ ≥ 𝑚𝑀 ) ( D c ( 𝐴 ) , Ϙ ≥ 𝑚𝑀 ) ( D im(c) ( 𝐴 [ 𝑆 −1 ]) , Ϙ ≥ 𝑚𝑀 [ 𝑆 −1 ] )( D 𝑝 (c) ( 𝐵 ) 𝑝 ( 𝑆 ) , Ϙ ≥ 𝑚𝑁 ) ( D 𝑝 (c) ( 𝐵 ) , Ϙ ≥ 𝑚𝑁 ) ( D im( 𝑝 (c)) ( 𝐵 [ 𝑝 ( 𝑆 )] −1 ) , Ϙ ≥ 𝑚𝑁 [ 𝑝 ( 𝑆 ) −1 ] ) are Poincaré-Verdier sequences by Proposition 1.4.8, and the vertical maps are Poincaré functors on accountof assumption i), see Lemma [I].3.3.3, and the right hand square is Ind -adjointable: The square formed bythe horizontal right adjoints on inductive completions identifies with the (a priori only lax-commutative)diagram D ( 𝐴 ) D ( 𝐴 [ 𝑆 −1 ]) D ( 𝐵 ) D ( 𝐵 [ 𝑝 ( 𝑆 ) −1 ]) , 𝐵⊗ 𝕃 𝐴 − fgt 𝐵 [ 𝑝 ( 𝑆 ) −1 ] ⊗ 𝕃 𝐴 [ 𝑆 −1] −fgt with structure map given by the natural map 𝐵 ⊗ 𝕃 𝐴 𝑋 → 𝐵 [ 𝑝 ( 𝑆 ) −1 ] ⊗ 𝕃 𝐴 [ 𝑆 −1 ] 𝑋 for 𝑋 ∈ D ( 𝐴 [ 𝑆 −1 ]) . Sinceboth sides commute with colimits it suffices to establish that this map is an equivalence for 𝑋 = 𝐴 [ 𝑆 −1 ] ,which we observed above.We now claim that the left hand vertical map is an equivalence of Poincaré ∞ -categories, whenceLemma 1.5.3 gives the claim. The fact that the underlying functor of stable ∞ -categories is an equiva-lence follows from assumption i): By Example 1.4.2 the categories D p ( 𝐴 ) 𝑆 and D p ( 𝐵 ) 𝑝 ( 𝑠 ) are generated bythe objects 𝐴 ⫽ 𝑠 and 𝐵 ⫽ 𝑝 ( 𝑠 ) under shifts, retracts and finite colimits, so the functor is essentially surjectiveand full faithfulness can be tested on these generators, where we compute Hom 𝐴 ( 𝐴 ⫽ 𝑠, 𝐴 ⫽ 𝑡 ) ≃ Hom 𝐴 ( 𝐴 ⫽ 𝑠, 𝐵 ⫽ 𝑝 ( 𝑡 )) ≃ Hom B ( 𝐵 ⫽ 𝑝 ( 𝑠 ) , 𝐵 ⫽ 𝑝 ( 𝑡 )) . See also [LT19, Proposition 1.17] for an alternative argument that the underlying square of ∞ -categories iscartesian. It remains to check that the natural map Ϙ ≥ 𝑚𝑀 ( 𝑋 ) → Ϙ ≥ 𝑚𝑁 ( 𝑝 ! 𝑋 ) induced by 𝜂 is an equivalence forall 𝑋 ∈ D p ( 𝐴 ) 𝑆 . For 𝑚 = ±∞ this follows from the fact that 𝑝 ! is a Poincaré functor and an equivalenceon underlying ∞ -categories, as this evidently implies that ( 𝑝, 𝜂 ) ! induces an equivalence on bilinear parts.We are thus reduced to considering the linear parts for finite 𝑚 . Using the adjunction 𝑝 ! ⊢ 𝑝 ∗ , we have toshow that for every 𝑆 -torsion perfect complex of 𝐴 -modules 𝑋 , the map hom 𝐴 ( 𝑋, 𝜏 ≥ 𝑚 ( 𝑀 tC )) ⟶ hom 𝐴 ( 𝑋, 𝑝 ∗ 𝜏 ≥ 𝑚 ( 𝑁 tC )) induced by 𝜂 is an equivalence. Since the category D p 𝑆 ( 𝐴 ) in generated under finite colimits and desuspen-sions by objects of the form 𝐴 ⫽ 𝑠 = cof( 𝐴 ⋅ 𝑠 → 𝐴 ) one can equivalently show that every element 𝑠 ∈ 𝑆 actsinvertibly on 𝐹 𝑚 = cof ( 𝜏 ≥ 𝑚 ( 𝑀 tC ) → 𝑓 ∗ 𝜏 ≥ 𝑚 ( 𝑁 tC ) ) , i.e. that the canonical map 𝐹 𝑚 ⟶ 𝐹 𝑚 [ 𝑆 −1 ] is an equivalence. We note that 𝐹 𝑚 → 𝐹 −∞ induces an isomorphism on homology groups in degrees largerthan 𝑚 , and that there is an exact sequence ⟶ H 𝑚 ( 𝐹 𝑚 ) ⟶ H 𝑚 ( 𝐹 −∞ ) ⟶ 𝐾 ⟶ ,
18 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE where 𝐾 = ker ( ̂ H − 𝑚 +1 (C ; 𝑀 ) → ̂ H − 𝑚 +1 (C ; 𝑁 ) ) . From the fact that the bilinear parts of the two functors agree, we find that 𝑆 acts invertibly on 𝐹 −∞ . Hence itremains to show that 𝑆 acts invertibly on H 𝑚 ( 𝐹 𝑚 ) . The above short exact sequence maps into its localisationat 𝑆 . Since this localisation is an exact functor, the snake lemma implies that it suffices to check that themap 𝐾 → 𝐾 [ 𝑆 −1 ] is injective. Writing 𝑀 [ 𝑆 −1 ] as ( 𝑅 [ 𝑆 −1 ] ⊗ 𝑅 [ 𝑆 −1 ]) ⊗ 𝑅⊗𝑅 𝑀 and likewise for 𝑁 ,using assumption ii), we find that 𝐾 [ 𝑆 −1 ] = ker ( ̂ H − 𝑚 +1 (C ; 𝑀 [ 𝑆 −1 ]) → ̂ H − 𝑚 +1 (C ; 𝑁 [ 𝑝 ( 𝑆 −1 )]) ) , since Tate cohomology commutes with filtered colimits in the coefficients (see the discussion in the proofof Proposition 1.4.8). The kernel of 𝐾 → 𝐾 [ 𝑆 −1 ] therefore canonically identifies with the kernel of ̂ H − 𝑚 +1 (C ; 𝑀 ) ⟶ ̂ H − 𝑚 +1 (C ; 𝑁 ⊕ 𝑀 [ 𝑆 −1 ]) which vanishes by assumption v). (cid:3) As the simplest non-trivial special case we for example obtain:4.4.22.
Corollary.
Let 𝑅 be a discrete commutative ring, 𝑀 an invertible 𝑅 -module with an 𝑅 -linear in-volution, 𝑓 , 𝑔 ∈ 𝑅 elements spanning the unit ideal and 𝑐 ⊆ K ( 𝑅 ) closed under the involution associatedto 𝑀 . Then the square GW( D c ( 𝑅 ) , Ϙ ≥ 𝑚𝑀 ) GW( D im(c) ( 𝑅 [ 𝑓 ]) , Ϙ ≥ 𝑚𝑀 [1∕ 𝑓 ] )GW( D im(c) ( 𝑅 [ 𝑔 ]) , Ϙ ≥ 𝑚𝑀 [1∕ 𝑔 ] ) GW( D im(c) ( 𝑅 [ 𝑓𝑔 ]) , Ϙ ≥ 𝑚𝑀 [1∕ 𝑓𝑔 ] ) and the analogous squares in K and L -theory are cartesian.Proof. We verify conditions i) through v) of the previous proposition. The first and fourth are obviousand the second is implied by the two 𝑅 -module structures on 𝑀 agreeing. For the third one simply notethat 𝑔 acts invertibly on 𝑅 ⫽ 𝑓 , since with 𝑓 and 𝑔 also any powers thereof span the unit ideal. To verifythe final condition recall that Tate cohomology groups over C with coefficients in 𝑀 are -periodic withvalues alternating between the kernels of the norm map id 𝑀 ± 𝜎 ∶ 𝑀 C → 𝑀 C . Thus we may check that 𝑀 → 𝑀 [1∕ 𝑓 ] ⊕ 𝑀 [1∕ 𝑔 ] induces injections on both these groups. But taking coinvariants commutes withlocalisation at both 𝑓 and 𝑔 , so the map in question is injective on the entire coinvariants. (cid:3) In completely analogous fashion one can treat the inversion of some prime 𝑙 in 𝑅 → 𝑅 ∧ 𝑙 , leading to alocalisation-completion square we will spell out in Proposition [III].2.1.12, and also the case of localisationof rings with involution at elements invariant under the involution, but let us refrain from spelling this outhere.4.5. The real algebraic K -theory spectrum and Karoubi periodicity. Just as in §3.7, the fundamentalfiber square can be cleanly encapsulated as the isotropy separation square of a genuine C -spectrum:4.5.1. Definition.
We define the real algebraic K -theory spectrum KR( C , Ϙ ) of a Poincaré ∞ -category ( C , Ϙ ) to be the genuine C -spectrum GW ghyp ( C , Ϙ ) .In particular, from Corollary 3.7.4 we obtain:4.5.2. Corollary.
The real algebraic K -theory spectra define an additive functor KR ∶ Cat p∞ ⟶ S 𝑝 gC , such that 𝑢 KR ≃ K , KR gC ≃ GW and KR 𝜑 C ≃ L , where 𝑢 ∶ S 𝑝 gC → Sp hC denotes the functor extracting the underlying C -spectrum, and (−) gC and (−) 𝜑 C ∶ Sp gC → Sp denote the genuine and geometric fixed points, respectively. Furthermore, the isotropyseparation square associated to KR( C , Ϙ ) is naturally equivalent to the fundamental fibre square of ( C , Ϙ ) . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 119 And from Theorem 3.7.7 we succinctly find:4.5.3.
Corollary.
There are canonical equivalences
KR( C , Ϙ [1] ) ≃ 𝕊 𝜎 ⊗ KR( C , Ϙ ) natural in the Poincaré ∞ -category ( C , Ϙ ) . In particular, any equivalence ( C , Ϙ ) → ( C , Ϙ [ 𝑘 ] ) induces aperiodicity equivalence KR( C , Ϙ ) ≃ 𝕊 𝑘 − 𝑘𝜎 KR( C , Ϙ ) . Corollary.
Let 𝑅 be a complex oriented E -ring, for example a discrete ring, 𝑀 an invertible modulewith involution over 𝑅 and c ⊆ K ( 𝑅 ) a subgroup closed under the involution induced by 𝑀 . Then thereare canonical equivalences KR(Mod c 𝑅 , Ϙ s− 𝑀 ) ≃ 𝕊 𝜎 ⊗ KR(Mod c 𝑅 , Ϙ s 𝑀 ) and KR(Mod c 𝑅 , Ϙ q− 𝑀 ) ≃ 𝕊 𝜎 ⊗ KR(Mod c 𝑅 , Ϙ q 𝑀 ) , and if 𝑅 is furthermore connective also KR(Mod c 𝑅 , Ϙ ≥ 𝑚 +1− 𝑀 ) ≃ 𝕊 𝜎 ⊗ KR(Mod c 𝑅 , Ϙ ≥ 𝑚𝑀 ) . In particular, we obtain the following periodicity result:4.5.5.
Corollary (Karoubi periodicity) . Let 𝑅 be a complex oriented E -ring, for example a discrete ring, 𝑀 an invertible module with involution over 𝑅 and c ⊆ K ( 𝑅 ) a subgroup closed under the involutioninduced by 𝑀 . Then the genuine C -spectra KR(Mod c 𝑅 , Ϙ s 𝑀 ) and KR(Mod c 𝑅 , Ϙ q 𝑀 ) are (4 − 4 𝜎 ) -periodic, and even (2 − 2 𝜎 ) -periodic if 𝑅 is real oriented. For connective, complex oriented 𝑅 we, furthermore, have KR(Mod c 𝑅 , Ϙ gq 𝑀 ) ≃ 𝕊 𝜎 ⊗ KR(Mod c 𝑅 , Ϙ gs 𝑀 ) . Passing to geometric fixed points extends the classical periodicity of Ranicki from the case of discreterings:4.5.6.
Corollary (Ranicki periodicity) . Let 𝑅 be a complex oriented E -ring, for example a discrete ring, 𝑀 an invertible module with involution over 𝑅 and c ⊆ K ( 𝑅 ) a subgroup that is closed under the involutioninduced by 𝑀 . Then there are canonical equivalences L(Mod c 𝑅 , Ϙ s− 𝑀 ) ≃ 𝕊 ⊗ L(Mod c 𝑅 , Ϙ s 𝑀 ) and L(Mod c 𝑅 , Ϙ q− 𝑀 ) ≃ 𝕊 ⊗ L(Mod c 𝑅 , Ϙ q 𝑀 ) . In particular,
L(Mod c 𝑅 , Ϙ s 𝑀 ) and L(Mod c 𝑅 , Ϙ q 𝑀 ) are -periodic and if 𝑅 is real orientable, for example a discrete ring of characteristic , they are periodic.Furthermore, for connective complex oriented 𝑅 we have L(Mod c 𝑅 , Ϙ gq 𝑀 ) ≃ 𝕊 ⊗ L(Mod c 𝑅 , Ϙ gs 𝑀 ) . Of course this corollary can also easily be obtained straight from the shifting behaviour of bordisminvariant functors.Let us also mention immediately, that the genuine L -spectra really are not periodic in general, as we willshow in Paper [III] of this series by explicit computation of L(Mod 𝜔 ℤ , Ϙ 𝑔 s ) .Similarly, it follows from [WW14, Theorem 4.5], that L(Mod 𝜔 𝕊 , Ϙ s ) is not periodic, we will explain thisin Remark 4.6.5 below. Consequently, some assumption like complex orientability, or more precisely aThom isomorphism for the vector bundle 𝛾 ⊕𝑘 → BC for some 𝑘 , is a definite requirement for a periodicitystatement even for the symmetric Poincaré structure.
20 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE LA -theory after Weiss and Williams. In this final subsection, we would like to relate the fundamentalfibre square to the LA -spectra arising in the work of Weiss and Williams [WW14]. We start by comparingthe map Ξ ∶ L ⟶ K tC appearing in Corollary 4.4.14 with the map L ⟶ K tC constructed by Weiss andWilliams in [WW98, Section 9]. Translated to our set-up, they consider the map L ( C , Ϙ s ) = | Cr 𝜌 ( C , Ϙ s ) hC | ⟶ Ω | Cr Q 𝜌 ( C , Ϙ s ) hC | ⟶ Ω ∞ ad(K hC )( C , Ϙ s ) , where the second map is a colimit-limit interchange and the first is the realisation (in the 𝜌 -direction) of thestructure maps for the group completions of the additive functor Cr hC ; here Ϙ s denotes the symmetrisationof an hermitian structure Ϙ on C , given by Ϙ s ( 𝑋 ) = B Ϙ ( 𝑋, 𝑋 ) hC as in Example [I].1.1.17. Precomposingwith the composite Pn( C , Ϙ ) ⟶ L ( C , Ϙ ) fgt ←←←←←←←←←←←←→ L ( C , Ϙ s ) and unwinding definitions this is the same as Pn( C , Ϙ ) ⟶ GW ( C , Ϙ ) ⟶ | GW 𝜌 ( C , Ϙ ) | fgt ←←←←←←←←←←←←→ | K 𝜌 ( C , Ϙ s ) hC | ⟶ Ω ∞ ad(K hC )( C , Ϙ s ) . The latter part of this composite can in turn be rewritten as GW ( C , Ϙ ) ≃ Ω ∞ GW( C , Ϙ ) ⟶ Ω ∞ ad GW( C , Ϙ ) fgt ←←←←←←←←←←←←→ Ω ∞ ad(K hC )( C , Ϙ s ) . Now, the canonical map ad(K hC )( C , Ϙ ) → ad(K hC )( C , Ϙ s ) is an equivalence, so the forgetful map is nothingbut Ω ∞ Ξ ∶ Ω ∞ L ( C , Ϙ ) → Ω ∞ K( C , Ϙ ) tC under the identifications of Corollary 3.6.12 and Theorem 4.4.12.By the universal property of L -theory in Theorem 4.4.12, we conclude that the Weiss-Williams map L ⇒ K tC agrees with ours.4.6.1. Corollary.
For a space 𝐵 ∈ S and a stable spherical fibration 𝜉 over 𝐵 the spectrum GW(( S 𝑝 ∕ 𝐵 ) 𝜔 , Ϙ 𝑟𝜉 ) ,identifies with Weiss’ and Williams’ LA 𝑟 ( 𝐵, 𝜉 ) , where 𝑟 ∈ {s , v , q} , i.e. either of symmetric, visible or qua-dratic. In particular, we find equivalences Ω ∞−1 LA 𝑟 ( 𝐵, 𝜉 ) ≃ | Cob(( S 𝑝 ∕ 𝐵 ) 𝜔 , Ϙ 𝑟𝜉 ) | . We think of the displayed equivalence as a cycle model for the left hand object, which seems to be new.In particular, specialising to 𝐵 =∗ we find that the −1 st infinite loop spaces of GW( S 𝑝 𝜔 , Ϙ s ) ≃ LA s (∗) and GW( S 𝑝 𝜔 , Ϙ u ) ≃ LA v (∗) , where Ϙ u ∶ ( S 𝑝 𝜔 ) op → S 𝑝 is the universal hermitian structure of §[I].4.1, are the homotopy types of thecobordism categories of Spanier-Whitehead selfdual spectra, and selfdual spectra equipped with a lift along D 𝕊 𝑋 → (D 𝕊 𝑋 ) ∧2 ≃ hom S 𝑝 ( 𝑋, D 𝕊 𝑋 ) tC of the image of the selfduality map, respectively.4.6.2. Remark.
Here we applied a naming scheme similar to Lurie’s suggestion of writing L q ( 𝑅 ) and L s ( 𝑅 ) instead of Ranicki’s L ∙ ( 𝑅 ) and L ∙ ( 𝑅 ) for what we would systematically call L( D p ( 𝑅 ) , Ϙ q 𝑅 ) and L( D p ( 𝑅 ) , Ϙ s 𝑅 ) .In [WW14] the spectra LA q ( 𝐵, 𝜉 ) , LA v ( 𝐵, 𝜉 ) and LA s ( 𝐵, 𝜉 ) are called LA ∙ ( 𝐵, 𝜉 ⊗ 𝕊 𝑑 , 𝑑 ) , VLA ∙ ( 𝐵, 𝜉 ⊗ 𝕊 𝑑 , 𝑑 ) and LA ∙ ( 𝐵, 𝜉 ⊗ 𝕊 𝑑 , 𝑑 ) , where 𝑑 is the dimension of 𝜉 . Proof.
The spectra LA 𝑟 ( 𝐵, 𝜉 ) are defined by certain pullbacks [WW14, Definition 9.5] LA 𝑟 ( 𝐵, 𝜉 ) L 𝑟 ( 𝐵, 𝜉 ) Ξ A( 𝐵, 𝜉 ) hC A( 𝐵, 𝜉 ) tC , which we claim correspond precisely to our fundamental fibre square Corollary 4.4.14 for (( S 𝑝 ∕ 𝐵 ) 𝜔 , Ϙ 𝑟𝜉 ) .To see this, we first note that Weiss and Williams work in the dual set-up, i.e. they describe Poincaréobjects via their coforms, rather than forms. The translation is achieved via the Costenoble-Waner dualityequivalence D 𝐵 ∶ (( S 𝑝 ∕ 𝐵 ) 𝜔 ) op → ( S 𝑝 ∕ 𝐵 ) 𝜔 , ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 121 which in our setup occurs as the duality associated to Ϙ s 𝕊 𝐵 , see Corollary [I].4.4.3: Indeed, we claim that B Ϙ 𝜉 (D 𝐵 𝐸, D 𝐵 𝐹 ) ≃ M( 𝐸 ⊗ 𝐵 𝐹 ⊗ 𝐵 𝜉 ) , for perfect 𝐸, 𝐹 ∈ S 𝑝 ∕ 𝐵 , where M ∶ S 𝑝 ∕ 𝐵 → S 𝑝 is the Thom spectrum functor (corresponding to colim ∶ Fun( 𝐵, S 𝑝 ) → S 𝑝 ). This most easily follows by observing that both sides are bilinear in ( 𝐸, 𝐹 ) , so it suffices to check theagreeance on ( 𝑖 ! 𝕊 , 𝑗 ! 𝕊 ) for points 𝑖, 𝑗 ∈ 𝐵 (as these form compact generators for S 𝑝 ∕ 𝐵 ). Here both sidesevaluate to M(ev ∗ 𝜉 ) where ev ∶ Ω 𝑖,𝑗 𝐵 → 𝐵 is the evalutation of the path space at the midpoint, say: Theleft hand side evaluated on ( 𝑖 ! 𝕊 , 𝑗 ! 𝕊 ) is by adjunction given by ( 𝑖 × 𝑗 ) ∗ Δ ! ( 𝕊 𝐵 ) , which the Beck-Chevalleyformula for the cartesian diagram Ω 𝑖,𝑗 𝐵 𝐵 ∗ 𝐵 𝑖,𝑗 ) equates with the desired term. For the right hand side, one finds M ( Δ ∗ (( 𝑖 × 𝑗 ) ! 𝕊 ⊠ 𝜉 ) ) , to which one canapply the Beck-Chevalley formula for Ω 𝑖,𝑗 𝐵 𝐵𝐵 𝐵 . evev Δ( 𝑖,𝑗, id 𝐵 ) The result is
M(ev ! ev ∗ 𝜉 ) and since M = 𝑟 ! , where 𝑟 ∶ 𝐵 → ∗ , the claim follows.Now by definition, a symmetric or quadratic Poincaré object in the relevant categories of Weiss andWilliams [WW14, Chapter 8] is given by a pair ( 𝑋, 𝑞 ) , where 𝑋 ∈ ( S 𝑝 ∕ 𝐵 ) 𝜔 and 𝑞 is a point in either the fixedpoints or orbits of Ω ∞ M( 𝑋 ⊗ 𝐵 𝑋 ⊗ 𝐵 𝜉 ) , respectively, that gives rise to an equivalence D Ϙ 𝜉 ( 𝑋 ) → 𝑋 (thesedefinitions are spelled out in [WW98, Definitions 3.6, 9.1 & 11.3]). Thus, the Costenoble-Waner dualityfunctor gives an equivalence between the Poincaré objects in ( C , Ϙ 𝑟𝜉 ) and those occuring in the definitionsof L 𝑟 ( 𝐵, 𝜉 ) given in [WW98, Definitions 9.2 & 11.4]. Thus, we find that our spectra L(( S 𝑝 ∕ 𝐵 ) 𝜔 , Ϙ 𝑟𝜉 ) agreewith the corresponding L -theory spectra of Weiss and Williams. As we identified the map Ξ occuring inthe definition of the LA -spectra with ours above, we obtain the claim in the cases 𝑝 ∈ {q , s} from thewell-known equivalence A( 𝐵 ) ≃ K(( S 𝑝 ∕ 𝐵 ) 𝜔 ) .For the visible refinements, one again computes L Ϙ v 𝜉 (D 𝐵 𝐸 ) ≃ M( 𝐸 ⊗ 𝐵 𝜉 ) . But then [WW14, Definition 3.2 & Corollary 3.5] say, that a visible symmetric structure on 𝐸 , correspondsexactly to an element of Ω ∞ Ϙ v 𝜉 (D 𝐵 𝐸 ) . The claim follows. (cid:3) Remark.
In subsequent work, we will construct for 𝜉 a stable − 𝑑 -dimensional vector bundle over 𝐵 a functor Cob 𝜉𝑑 → Cob(( S 𝑝 ∕ 𝐵 ) 𝜔 , Ϙ v 𝜉 ) from the geometric, normally- 𝜉 oriented cobordism category into algebraic cobordism category of parametrisedspectra over 𝑀 . Through the equivalence Ω ∞ LA v ( 𝐵, 𝜉 ) ≃ Ω | Cob(( S 𝑝 ∕ 𝐵 ) 𝜔 , Ϙ v 𝜉 ) | this provides a factorization of the Weiss-Williams map BTop( 𝑀 ) ⟶ Ω ∞ LA v ( 𝑀, 𝜈 𝑀 ) when 𝑀 is a closed manifold with stable normal bundle 𝜈 𝑀 , through the geometric cobordism category Cob 𝜈 𝑀 𝑑 . Now the homotopy type of the cobordism category is excisive in the bundle data [KGL18]. There-fore we can then follow the strategy developed by Raptis and the 9’th author in the K -theoretic contextfor their proof of the Dwyer-Weiss-Williams index theorem [RS17], as to provide a canonical lift of themap Ω | Cob 𝜈 𝑀 𝑑 | → Ω ∞ LA v ( 𝑀, 𝜈 𝑀 ) into the the source of the assembly map of LA v ; there results a newperspective on substantial parts of [WW14] and Waldhausen’s map ̃ Top( 𝑀 )∕Top( 𝑀 ) ⟶ Wh( 𝑀 ) hC ,
22 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE into the (topological) Whitehead spectrum of 𝑀 .We offer one application of the identification GW( S 𝑝 𝜔 , Ϙ u ) ≃ LA v (∗) . To this end recall that the functors GW and L and K ∶ Cat p∞ → A 𝑏 are compatibly lax symmetric monoidal for the tensor product of Cat p∞ and that ( S 𝑝 𝜔 , Ϙ u ) is the unit of the tensor product on Cat p∞ . Hence there are rings maps K ( S 𝑝 𝜔 ) fgt ←←←←←←←←←←←←← GW ( S 𝑝 𝜔 , Ϙ u ) bord ←←←←←←←←←←←←←←←←←→ L ( S 𝑝 𝜔 , Ϙ u ) . Abbreviating the underlying spectra to K( 𝕊 ) , GW u ( 𝕊 ) and L u ( 𝕊 ) , and similarly their homotopy groups, wehave:4.6.4. Proposition.
There is a commutative diagram with vertical maps isomorphisms
ℤ ℤ [ 𝑒, ℎ ]∕ 𝐼 ℤ [ 𝑒 ]∕( 𝑒 − 8 𝑒 )K ( 𝕊 ) GW u0 ( 𝕊 ) L u0 ( 𝕊 ) ↤ 𝑒, ↤ ℎ 𝑒 ↦ 𝑒, ℎ ↦ where 𝑒 and ℎ denote the classes of the spherical 𝐸 -lattice and hyp( 𝕊 ) , respectively, and 𝐼 is the idealgenerated by 𝑒 − 8 𝑒, ℎ − 2 ℎ and 𝑒ℎ − 8 ℎ .Furthermore, there are canonical isomorphisms GW u− 𝑖 ( 𝕊 ) ≅ L u− 𝑖 ( 𝕊 ) ≅ L q− 𝑖 ( ℤ ) for 𝑖 > induced by the comparison maps with quadratic L -theory of the sphere spectrum, whereas GW u−1 ( 𝕊 ) and GW u−2 ( 𝕊 ) both vanish. The calculation of K ( 𝕊 ) = 𝜋 A(∗) is of course due to Waldhausen and the calculation of L u0 ( 𝕊 ) is dueto Weiss-Williams (due to the identification L u ( 𝕊 ) = L( S 𝑝 𝜔 , Ϙ v 𝕊 ) ).Without multiplicative structures the result says that GW u0 ( 𝕊 ) is free of rank generated by the Poincaréspectra hyp( 𝕊 ) , ( 𝕊 , id 𝕊 ) and the spherical lift of the 𝐸 -lattice. In particular, as already observed by Weissand Williams, the equality [ 𝐸 ] = 8[ ℤ , id ℤ ] in the symmetric (Grothendieck-)Witt-group of the integers (aconsequence of the classification of indefinite forms over ℤ through rank and signature) does not lift to thesphere spectrum. Proof.
We first identify the underlying abelian groups in all cases. From Corollary 4.4.14 we have a fibresequence K( 𝕊 ) hC ⟶ GW u ( 𝕊 ) ⟶ L u ( 𝕊 ) , which we identified with A(∗) hC ⟶ LA v (∗) ⟶ L v (∗) above. Using the former naming, Weiss and Williams constructed a fibre sequence L q ( 𝕊 ) ⟶ L u ( 𝕊 ) ⟶ 𝕊 ⊕ MTO(1) , by identifying the latter term with visible, normal (or hyperquadratic) L -theory of the sphere in [WW14,Theorem 4.3]. By the algebraic 𝜋 - 𝜋 -theorem the base change map L q ( 𝕊 ) ⟶ L q ( ℤ ) is an equivalence; this appears for example as [WW89, Proposition 6.2], a proof in the present language isgiven in [Lur11, Lecture 14] and we will also derive it in the third installement of this series, see Corol-lary [III].1.2.24. We thus obtain an exact sequence ⟶ L u1 ( 𝕊 ) ⟶ 𝜋 ( 𝕊 ⊕ MTO(1)) ⟶ ℤ ⟶ L u0 ( 𝕊 ) ⟶ 𝜋 ( 𝕊 ⊕ MTO(1)) ⟶ , since the odd quadratic L -groups of the integers vanish, whereas L q0 ( ℤ ) = ℤ , spanned by the 𝐸 -lattice.Thus we also find that L q0 ( 𝕊 ) is spanned by a spherical lift of 𝐸 (note that 𝜋 Ϙ q ( 𝕊 ⊕𝑖 ) → 𝜋 Ϙ q ( ℤ 𝑖 ) is alwaysan isomorphism, so this lift is unique up to homotopy). Now to obtain the homotopy groups of MTO(1) ,recall from [GTMW09, Section 3] the fibre sequence
MTO(1) ⟶ 𝕊 [BO(1)] ⟶ MTO(0) , ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 123 the latter term being equivalent to the sphere 𝕊 . Now the first nine (reduced) homotopy groups of 𝕊 [BO(1)] were computed by Liulevicius in [Liu63, Theorem II.6], and the map 𝕊 [BO(1)] → 𝕊 is easily checked tobe the transfer map for the canonical double cover of BO(1) . Therefore it is -locally surjective on positivehomotopy groups by the Kahn-Priddy theorem [KP78] and given by multiplication by on 𝜋 . We obtain 𝜋 𝑖 ( 𝕊 ⊕ MTO(1)) = ⎧⎪⎪⎨⎪⎪⎩ 𝑖 < −1 ℤ ∕2 𝑖 = −1 ℤ 𝑖 = 0( ℤ ∕2) 𝑖 = 1 Thus we find L u0 ( 𝕊 ) ≅ ℤ generated by the spherical 𝐸 -lattice and ( 𝕊 , id 𝕊 ) , compare the discussion fol-lowing [WW14, Theorem 4.3]. Furthermore, we also find L u1 ( 𝕊 ) = ( ℤ ∕2) , so obtain an exact sequence ⟶ K ( 𝕊 ) C hyp ←←←←←←←←←←←←←←→ GW u0 ( 𝕊 ) ⟶ ℤ ⟶ , because the first term is torsionfree, since the involution D Ϙ u evidently acts trivially on K ( 𝕊 ) ≅ ℤ . Thisgives the first claim.The second claim also follows, as 𝕊 ⊕ MTO(1) is −2 -connected, so the maps GW u ( 𝕊 ) ⟶ L u ( 𝕊 ) ⟵ L q ( 𝕊 ) are isomorphisms on homotopy groups from degree −3 on. For degrees −1 and −2 we find an exact sequence ⟶ L u−1 ( 𝕊 ) ⟶ 𝜋 −1 ( 𝕊 ⊕ MTO(1)) ⟶ L q−2 ( 𝕊 ) ⟶ L u−2 ( 𝕊 ) ⟶ with both middle terms isomorphic to ℤ ∕2 . We now claim that the right map vanishes, forcing the middleone to be an isomorphism completing the computation of the additive structure (see also Corollary [III].1.2.24iv) for a more direct proof that the outer terms vanish).For this we first note that the canonical map L q−2 ( ℤ ) → L s−2 ( ℤ ) vanishes; indeed, the source is spannedby the standard unimodular skew-quadratic form of Arf-invariant on ℤ (regarded as a chain complexconcentrated in degree ), given by the matrix ( ) , whose underlying anti-symmetric bilinear form ( ) admits the Lagrangian ℤ ⊕ . But then L q−2 ( 𝕊 ) = 𝜋 L( S 𝑝 p , Ϙ q[2] ) is spanned by the lift of this quadraticform to 𝕊 ⊕ 𝕊 and we claim that the Lagrangian lifts as well: Decoding this is implied by 𝜋 ( Ϙ u ) [2] ( 𝕊 ) = 𝜋 −2 Ϙ u ( 𝕊 ) , which gives the vanishing of the underlying form 𝑞 ∈ Ω ∞ ( Ϙ u ) [2] ( 𝕊 ⊕ 𝕊 ) restricted to one of the summands;the resulting object of Fm(Met( S 𝑝 p , ( Ϙ u ) [2] )) is automatically Poincaré, as this can be checked after basechange to the integers by Whitehead’s theorem, where it reduces to the computation above.To see the vanishing, consider the square Ϙ u ( 𝕊 ) hom 𝕊 ( 𝕊 , 𝕊 )hom 𝕊 ( 𝕊 ⊗ 𝕊 , 𝕊 ) hC hom 𝕊 ( 𝕊 ⊗ 𝕊 , 𝕊 ) tC
24 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE from the definition of Ϙ u (with the C -action flipping the 𝕊 -factors). Since hom 𝕊 ( 𝕊 ⊗ 𝕊 , 𝕊 ) ≃ 𝕊 −1− 𝜎 itgives rise to a diagram 𝜋 −1 𝕊 −1 𝜋 −2 𝕊 −1− 𝜎 hC 𝜋 −2 Ϙ u ( 𝕊 ) 𝜋 −2 𝕊 −1 𝜋 −1 ( 𝕊 −1− 𝜎 ) tC 𝜋 −2 𝕊 −1− 𝜎 hC 𝜋 −2 ( 𝕊 −1− 𝜎 ) hC 𝜋 −2 ( 𝕊 −1− 𝜎 ) tC . with exact rows. Now, the top right corner vanishes and the homotopy orbit terms evaluate to ℤ ∕2 . Thuswe will be done if we show that the homotopy fixed point term vanishes as well, since then the lower lefthorizontal map is surjective and by Lin’s theorem [Lin80] (identifying the left vertical map as ℤ → ℤ ∧2 ),so is the upper left horizontal map. But dualising the fibre sequence 𝕊 ⊗ C → 𝕊 → 𝕊 𝜎 and applyinghomotopy fixed points yields a fibre sequence ( 𝕊 − 𝜎 ) hC ⟶ 𝕊 hC ⟶ 𝕊 with the right hand map the forgetful one. This map is split, since the C -action on 𝕊 is trivial, and so wefind that the negative homotopy groups of the left and middle term agree. But in the exact sequence 𝜋 −1 𝕊 hC ⟶ 𝜋 −1 𝕊 hC ⟶ 𝜋 −1 𝕊 tC both outer terms vanish (by connectivity on the left, and Lin’s theorem on the right). The claim follows.We are left to calculate the ring structures on GW u0 ( 𝕊 ) and L u0 ( 𝕊 ) . We start with the latter. By Exam-ple [I].5.4.10 the map L q0 ( 𝕊 ) ⟶ L u0 ( 𝕊 ) is an L u0 ( 𝕊 ) -module map, so [ 𝐸 ] = 𝑛 [ 𝐸 ] for some 𝑛 ∈ ℤ . Mapping to the integers shows that 𝑛 = 8 ,giving the claim. For the ring structure of GW u0 ( 𝕊 ) we similarly observe that the exact sequence K ( 𝕊 ) hyp ←←←←←←←←←←←←←←→ GW u0 ( 𝕊 ) ⟶ L u0 ( 𝕊 ) , consists of GW u0 ( 𝕊 ) -modules by Corollary [I].7.5.13. This immediately gives 𝑒ℎ = 8 ℎ and ℎ = 2 ℎ , andalso that 𝑒 = 8 𝑒 + 𝑘ℎ for some 𝑘 ∈ ℤ . But then we find ℎ = 8 ℎ𝑒 = ℎ𝑒 = ℎ (8 𝑒 + 𝑘ℎ ) = 16 ℎ + 2 𝑘ℎ which forces 𝑘 = 0 . (cid:3) Remark.
Similar to the sequence used in the previous proof, Weiss and Williams produce a fibresequence L q ( ℤ ) ⟶ L s ( 𝕊 ) ⟶ ( 𝕊 ∧2 ⊗ 𝕊 ∧2 ) ⊕ MTO(1) , in [WW14, Theorem 4.5], which rules out any sort of periodicity for L s ( 𝕊 ) .Finally, we use Proposition 4.6.4 to determine the automorphisms of the Grothendieck-Witt and L -theory functors. Yoneda’s lemma, the universal property of the Grothendieck-Witt spectrum and Propo-sition [I].4.1.3 provide equivalences Nat(GW , GW) ≃ Nat(Pn , Ω ∞ GW) ≃ Nat(Hom
Cat p∞ (( S 𝑝 p , Ϙ u ) , −) , GW ) ≃ GW u ( 𝕊 ) . Similarly,
Nat(L , L) ≃ L u ( 𝕊 ) , while bordification induces a map Nat(GW , GW) ⟶ Nat(L , L) , which identifies with GW u ( 𝕊 ) bord ←←←←←←←←←←←←←←←←←→ L u ( 𝕊 ) , giving in particular E -structures to these spaces. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 125 Corollary.
These identifications provide isomorphisms 𝜋 Nat(GW , GW) ≅ ℤ [ 𝑒, ℎ ]∕ 𝐼 and 𝜋 Nat(L , L) ≅ ℤ [ 𝑒 ]∕( 𝑒 − 8 𝑒 ) with 𝐼 = ( 𝑒 − 8 𝑒, ℎ𝑒 − 8 ℎ, ℎ − 2 ℎ ) as before.In particular, we have 𝜋 Aut(GW) = {±1 , ±(1 − ℎ )} ≅ (C ) and 𝜋 Aut(L) = {±id} ≅ C . Proof.
It only remains to show that the identifications GW u ( 𝕊 ) ≃ Nat(GW , GW) are compatible with the multiplicative structures present on their -th homotopy groups, and similarly in L -theory. This will immediately follow from our work in Paper [IV], where we show that both GW and L carry lax symmetric monoidal structures. But we can also argue more directly:The spaces Nat(GW , GW) and
Nat(L , L) receive compatible E -maps from Nat(Pn , Pn) and from Yoneda’slemma we find
Nat(Pn , Pn) ≃ Hom
Cat p∞ (( S 𝑝 𝜔 , Ϙ u ) , ( S 𝑝 p , Ϙ u )) ≃ Pn( S 𝑝 𝜔 , Ϙ u ) . Since ( S 𝑝 𝜔 , Ϙ u ) is the unit of the symmetric monoidal structure on Cat p∞ , the functor Pn ≃ Hom
Cat p∞ (( S 𝑝 𝜔 , Ϙ u ) , −) inherits a lax symmetric monoidal structure. The left hand equivalence is then a map of E -spaces using thecomposition, and the right hand map refines to one of E ∞ -space for the multiplication induced by the tensorproduct of Poincaré ∞ -categories. But on the middle term this E ∞ -structure restricts to the compositionproduct by naturality. In total then, we obtain an E -refinement of the canonical map Pn( S 𝑝 𝜔 , Ϙ u ) ⟶ GW ( S 𝑝 𝜔 , Ϙ u ) ≃ Nat(GW , GW) . Since the map 𝜋 Pn( S 𝑝 𝜔 , Ϙ u ) → 𝜋 GW ( S 𝑝 𝜔 , Ϙ u ) = GW u0 ( 𝕊 ) is surjective, this shows that the isomorphism GW u0 ( 𝕊 ) ≃ 𝜋 Nat(GW , GW) is multiplicative and similarly in L -theory. The claims then follow from Proposition 4.6.4 and a quickcalculation of the units in the displayed rings. (cid:3) A PPENDIX
A. V
ERDIER SEQUENCES , K
AROUBI SEQUENCES AND STABLE RECOLLEMENTS
In this appendix we investigate in detail the ∞ -categorical variants of the notion of Verdier sequence, i.e.fibre-cofibre sequences in Cat ex∞ and the same notion up to idempotent completion, called Karoubi sequence.The results are mostly well-known and various parts can be found in the literature, but we do not know of acoherent account at the level of detail we need. In the hope that it can serve as a general reference for thismaterial, we have kept this appendix self-contained.
Remark.
For the reader familiar with [BGT13], here is a comparison of terminology: A Karoubi sequence iscalled an exact sequence in [BGT13], while our notion of a Verdier sequence corresponds to that of a strict-exact sequence in [BGT13]; this follows from Proposition A.1.6, Proposition A.1.9 and Proposition A.3.7.Our notion of a split-exact sequence is however stricter than the corresponding notion of split-exact se-quence in [BGT13], since we require the projection to have both adjoints (in which case these adjoints areautomatically fully-faithful, and the injection has both adjoints as well, see Proposition A.2.10), while inthe corresponding notion in [BGT13] only the right adjoints are assumed to exist.A.1.
Verdier sequences.
We start out by analysing in detail the notion of a Verdier sequence. We recallthe definition:A.1.1.
Definition.
Let(72) C 𝑓 ←←←←←←←→ D 𝑝 ←←←←←←→ E be a sequence in Cat ex∞ with vanishing composite. We will say that (72) is a
Verdier sequence if it is botha fibre and a cofibre sequence in
Cat ex∞ . In this case we will refer to 𝑓 as a Verdier inclusion and to 𝑝 as a Verdier projection .
26 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
A.1.2.
Remark.
The condition that the composite of the sequence (72) vanishes, simply means that it sendsevery object of C to a zero object in E . Equivalently, the exact functor 𝑝 ◦ 𝑓 ∶ C → E is a zero object inthe stable ∞ -category Fun ex ( C , E ) . Since the full subcategory of Fun ex ( C , E ) spanned by zero objects iscontractible we may identify 𝑝 ◦ 𝑓 in this case with a composite functor of the form C → {0} ⊆ E in anessentially unique manner. Thus, (72) refines to a commutative square C D {0} E 𝑓 𝑝 in an essentially unique manner, and the condition of being a fibre or cofibre sequence refers to this diagrambeing cartesian or cocartesian, respectively.Let us recall how to compute fibres and cofibres in Cat ex∞ : The fibre of an exact functor 𝑓 ∶ C → D iscomputed in Cat ∞ and given by the the kernel ker( 𝑓 ) , which is the full subcategory of C on all objectsmapping to a zero object in D . Cofibres, in turn, are described by Verdier quotients:A.1.3. Definition.
Let 𝑓 ∶ C → D be an exact functor between stable ∞ -categories. We say that a map in D is an equivalence modulo C if its fibre (equivalently, its cofibre) lies in the smallest stable subcategoryspanned by the essential image of 𝑓 . We write D ∕ C for the localisation of D with respect to the collection 𝑊 of equivalences modulo C and refer to D ∕ C as the Verdier quotient of D by C .A.1.4. Remark.
Let us stress that we differ in our use of the term localisation from Lurie’s: For us, thelocalisation of an ∞ -category D at a set 𝑊 of morphisms is the essentially unique functor D → D [ 𝑊 −1 ] such that for any ∞ -category D ′ , the pull-back functor Fun( D [ 𝑊 −1 ] , D ′ ) → Fun( D , D ′ ) is fully-faithful, with essential image the functors sending the morphisms from 𝑊 to equivalences. Werefer to localisations which are left or right adjoints as left and right Bousfield localisations , respectively.(See Lemma A.2.3 below for the precise relation between the two notions.)The following result is proven in [NS18, Theorem I.3.3(i)] (at least in the case where 𝑓 is fully-faithful,but the general case follows at once).A.1.5. Proposition.
Let 𝑓 ∶ C → D be an exact functor between stable ∞ -categories. Then:i) The ∞ -category D ∕ C is stable and the localisation functor D → D ∕ C is exact.ii) For every stable ∞ -category E the restriction functor Fun ex ( D ∕ C , E ) → Fun ex ( D , E ) is fully-faithful,and its essential image is spanned by those functors which vanish after precomposition with 𝑓 . Inparticular, the sequence C → D → D ∕ C is a cofibre sequence in Cat ex∞ . A.1.6.
Proposition.
Let 𝑝 ∶ D → E be an exact functor between stable ∞ -categories. Then the followingare equivalent:i) 𝑝 is a Verdier projection.ii) 𝑝 is the canonical map into a Verdier quotient of D .iii) 𝑝 is a localisation (at the maps it takes to equivalences).Proof. If 𝑝 is a Verdier projection, then it is a cofibre in Cat ex∞ . So i) ⇒ ii) follows from Proposition A.1.5;and ii) ⇒ iii) holds by definition of Verdier quotient. Finally, assume that iii) holds. Since 𝑝 is exact, amorphism in D maps to an equivalence in E if and only if its cofibre lies in the kernel of 𝑝 . Therefore 𝑝 isindeed the localisation at the class of equivalences modulo ker( 𝑝 ) , and therefore the cofibre of the inclusion ker( 𝑝 ) → D . Thus, the sequence ker( 𝑝 ) → C → D is both a fibre sequence and a cofibre sequence in Cat ex∞ ,so that i) holds. (cid:3)
A.1.7.
Corollary.
Every Verdier projection is essentially surjective.
We now examine the notion of a Verdier inclusion. For this, we need the following result:A.1.8.
Lemma.
The kernel of the canonical map 𝑝 ∶ D → D ∕ C consists of all objects of D which areretracts of objects in C . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 127 Proof.
Clearly, any retract of an object in C lies in the kernel of 𝑝 . For the converse inclusion, let 𝑥 be anobject of ker( 𝑝 ) . We note that by Proposition A.1.5 every exact functor D → S 𝑝 that vanishes on C alsovanishes on ker( 𝑝 ) . In particular, we may consider the exact functor 𝜑 𝑥 ∶ D → S 𝑝 given by the formula 𝜑 𝑥 ( 𝑦 ) = colim [ 𝛽 ∶ 𝑧 → 𝑦 ]∈ C ∕ 𝑦 hom D ( 𝑥, cof( 𝛽 )) where C ∕ 𝑦 ∶= C × D D ∕ 𝑦 is the associated comma ∞ -category. Then 𝜑 𝑥 vanishes on C : indeed, for 𝑦 ∈ C we have that C ∕ 𝑦 has a final object given by the identity id ∶ 𝑦 → 𝑦 , and hom D ( 𝑥, cof(id)) = 0 , which meansthat 𝜑 𝑥 ( 𝑦 ) = 0 .By the above we then get that 𝜑 𝑥 vanishes on ker( 𝑝 ) . In particular 𝜑 𝑥 vanishes on 𝑥 ∈ ker( 𝑝 ) itself,which implies the existence of a map 𝛽 ∶ 𝑧 → 𝑥 for some 𝑧 ∈ C such that id ∶ 𝑥 → 𝑥 is in the kernel of thecomposed map 𝜋 hom D ( 𝑥, cof(0 → 𝑥 )) → 𝜋 hom D ( 𝑥, cof( 𝛽 )) . We may then conclude that id ∶ 𝑥 → 𝑥 factors through 𝑧 and hence 𝑥 is retract of 𝑧 , as desired. (cid:3) A.1.9.
Proposition.
Let 𝑓 ∶ C → D be an exact functor between stable ∞ -categories. Then the followingare equivalent:i) 𝑓 is a Verdier inclusion.ii) 𝑓 is fully-faithful and its essential image is closed under retracts in D .Proof. If 𝑓 is a Verdier inclusion, then it is a kernel so that ii) holds. On the other hand, if ii) holds, then 𝑓 extends to a cofibre sequence C → D → D ∕ C , and by Lemma A.1.8 this is also a fibre sequence. (cid:3) Summarizing our discussion, we obtain:A.1.10.
Corollary.
For a sequence C 𝑓 ←←←←←←←→ D 𝑝 ←←←←←←→ E in Cat ex∞ with vanishing composite, the following areequivalent:i) The sequence is a Verdier sequence.ii) 𝑓 is fully-faithful with essential image closed under retracts in D , and 𝑝 exhibits E as the Verdierquotient of D by C .iii) 𝑝 is a localisation, and 𝑓 exhibits C as the kernel of 𝑝 . Finally, we record:A.1.11.
Lemma.
Any pullback of a Verdier projection is again a Verdier projection.Proof.
So consider a cartesian diagram
D D ′ E E ′ 𝑘𝑝 𝑝 ′ 𝑙 in Cat ex∞ with 𝑝 ′ a Verdier projection and C the common vertical fibre. Then we claim that the canonical map 𝑝 ∶ D ∕ C → E is an equivalence, which gives the claim. Since 𝑝 ′ is essentially surjective by Corollary A.1.7,so is 𝑝 by inspection, so we are left to check full faithfulness of 𝑝 . Using [NS18, Theorem I.3.3 (ii)] twicewe find Hom D ∕ C ( 𝑑, 𝑑 ′ ) ≃ colim 𝑐 ∈ C ∕ 𝑑 ′ Hom D ( 𝑑, cof( 𝑐 → 𝑑 ))≃ colim 𝑐 ∈ C ∕ 𝑑 ′ Hom D ′ ( 𝑘 ( 𝑑 ) , 𝑘 (cof( 𝑐 → 𝑑 ′ ))) × Hom E ′ ( 𝑙𝑝 ( 𝑑 ) ,𝑙𝑝 (cof( 𝑐 → 𝑑 ′ ))) Hom E ( 𝑝 ( 𝑑 ) , 𝑝 (cof( 𝑐 → 𝑑 ′ )))≃ colim 𝑐 ∈ C ∕ 𝑑 ′ Hom D ′ ( 𝑘 ( 𝑑 ) , cof( 𝑐 → 𝑘 ( 𝑑 ′ ))) × Hom E ′ ( 𝑙𝑝 ( 𝑑 ) ,𝑙𝑝 ( 𝑑 ′ )) Hom E ( 𝑝 ( 𝑑 ) , 𝑝 ( 𝑑 ′ ))≃ colim 𝑐 ∈ C ∕ 𝑘 ( 𝑑 ′) Hom D ′ ( 𝑘 ( 𝑑 ) , cof( 𝑐 → 𝑘 ( 𝑑 ′ ))) × Hom D ′∕ C ( 𝑘 ( 𝑑 ) ,𝑘 ( 𝑑 ′ )) Hom E ( 𝑝 ( 𝑑 ) , 𝑝 ( 𝑑 ′ ))≃ Hom E ( 𝑝 ( 𝑑 ) , 𝑝 ( 𝑑 ′ )) , where we have invested C ∕ 𝑑 ′ ≃ C ∕ 𝑘 ( 𝑑 ′ ) into the fourth step; this equivalence is immediate by regarding C ∕ 𝑑 ′ as the pullback of C × { 𝑑 ′ } → D × D ← Ar( D ) , and then commuting the pullback defining D out. (cid:3)
28 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
A.2.
Split Verdier sequences, Bousfield localisations and stable recollements.
We now discuss the ex-istence of adjoints to the inclusion and projection in a Verdier sequence. It leads to the central theme of thissection, the notion of split Verdier sequence (Definition A.2.4), and its relationship with stable recollements(Definition A.2.9 and Proposition A.2.10).To obtain criteria similar to Propositions A.1.6 and A.1.9 for exact functors fitting into split Verdiersequences, we first recall the relationship between two notions of localisation: the universal one we haveused so far and the notion of Bousfield localisation, compare Remark A.1.4.A.2.1.
Lemma.
Let C be a small ∞ -category and 𝑊 a collection of morphisms in C . Then the localisation 𝑝 ∶ C → C [ 𝑊 −1 ] admits a left or right adjoint, if and only if for every 𝑋 ∈ C there exists a 𝑌 ∈ C and anequivalence 𝑝𝑋 → 𝑝𝑌 in C [ 𝑊 −1 ] , such that the functors Hom C ( 𝑌 , −) or Hom C (− , 𝑌 ) , send all morphisms in 𝑊 to equivalences in S , respectively.In either case, the Yoneda lemma assembles such choices of objects 𝑌 for all 𝑋 ∈ C into the requisiteadjoint to the localisation functor, which is automatically fully faithful, and therefore renders 𝑝 into a rightor left Bousfield localisation, respectively. A.2.2.
Lemma.
If a functor 𝐹 ∶ C → D admits a fully faithful left adjoint 𝐿 , i.e. 𝐹 is a right Bousfieldlocalisation, then it is a localisation at those maps 𝑓 ∶ 𝑋 → 𝑌 in C , for which the induced map Hom C ( 𝐿 − , 𝑋 ) → Hom C ( 𝐿 − , 𝑌 ) is natural equivalence of functors D → S . The same of course holds mutatis mutandis for left Bousfield localisations.
Proof of Lemma A.2.1.
We prove the left adjoint variant. Since 𝑝 ∶ C → C [ 𝑊 −1 ] is essentially surjective,it admits a left adjoint if and only if, for each 𝑋 ∈ C the functor Hom C [ 𝑊 −1 ] ( 𝑝𝑋, 𝑝 −) ∶ C → S is representable. We claim that a representing object is precisely an object 𝑌 ∈ C as in the statement.To see this let us note generally, that for any 𝑌 ∈ C such that Hom C ( 𝑌 , −) inverts the morphisms in 𝑊 ,then 𝑝 provides a natural equivalence Hom C ( 𝑌 , −) ≃ Hom C [ 𝑊 −1 ] ( 𝑝𝑌 , 𝑝 −) . To see this, descend
Hom C ( 𝑌 , −) to a functor 𝐹 𝑌 ∶ C [ 𝑊 −1 ] → S and compute Nat( 𝐹 𝑌 , 𝐺 ) ≃ Nat( 𝐹 𝑌 𝑝, 𝐺𝑝 )≃ Nat(Hom C ( 𝑌 , −) , 𝐺𝑝 )≃ 𝐺 ( 𝑝𝑌 )≃ Nat(Hom C [ 𝑊 −1 ] ( 𝑝𝑌 , −) , 𝐺 ) for an arbitrary 𝐺 ∶ C [ 𝑊 −1 ] → S ; the first equivalence arising from the definition of localisations. But thenYoneda’s lemma implies that 𝐹 𝑌 ≃ Hom C [ 𝑊 −1 ] ( 𝑝𝑌 , −) and precomposing with 𝑝 gives the claim.Therefore a 𝑌 ∈ C as in the statement represents the functor Hom C [ 𝑊 −1 ] ( 𝑝𝑋, 𝑝 −) .If, on the other hand, 𝑝 admits a left adjoint 𝐿 , and 𝑋 ∈ C , then one can take 𝐿𝑝𝑋 for 𝑌 : By adjunction Hom C ( 𝐿𝑝𝑋, −) ≃ Hom C [ 𝑊 −1 ] ( 𝑝𝑋, 𝑝 −) inverts the morphisms in 𝑊 , and by the previous consideration we then find Hom C ( 𝐿𝑝𝑋, −) ≃ Hom C [ 𝑊 −1 ] ( 𝑝𝐿𝑝𝑋, 𝑝 −) which gives 𝑝𝐿𝑝𝑋 ≃ 𝑝𝑋 via the adjunction unit, since 𝑝 is essentially surjective.The adjunction unit being an equivalence also implies that 𝐿 is automatically fully faithful. (cid:3) Proof of Lemma A.2.2.
The proof that Bousfield localisations are indeed localisations in our sense is [Lur09a,Proposition 5.2.7.12] and the characterisation of the morphisms that are inverted is immediate from Yoneda’slemma. (cid:3)
ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 129 Let us apply this to give a criterion to recognize Verdier projections with a one-sided adjoint. In whatfollows, given a stable ∞ -category D and a full subcategory C ⊆ D , let us say that an object 𝑦 ∈ D is rightorthogonal to C if hom D ( 𝑥, 𝑦 ) ≃ 0 for every 𝑥 ∈ C and that 𝑦 is left orthogonal to C if hom D ( 𝑦, 𝑥 ) ≃ 0 forevery 𝑥 ∈ C .Let us write, C 𝑟 and C 𝑙 for the subcategories spanned by these objects.A.2.3. Lemma.
Let 𝑝 ∶ D → E be an exact functor of stable ∞ -categories. Then the following are equiva-lent:i) 𝑝 is a Verdier projection and admits a right (left) adjoint.ii) 𝑝 is a localisation, and ker( 𝑝 ) 𝑟 (or ker( 𝑝 ) 𝑙 ) projects essentially surjectively to E via 𝑝 .iii) 𝑝 is a localisation, and its restriction to ker( 𝑝 ) 𝑟 (or ker( 𝑝 ) 𝑙 ) is an equivalence.iv) 𝑝 admits a fully-faithful right (left) adjoint, i.e. is a left (or right) Bousfield localisation.In this situation, ker( 𝑝 ) 𝑟 (or ker( 𝑝 ) 𝑙 ) agrees with the essential image of the right (or left) adjoint of 𝑝 .Proof of Lemma A.2.3. Let us treat the non-parenthesised variants. Recalling from Proposition A.1.6 thatVerdier projections are localisations, the implications between i) and iv) are proven in Lemmas A.2.1 andA.2.2.Now suppose that 𝑝 admits a fully faithful right adjoint 𝑅 , then 𝑝 and 𝑅 determine mutually inverseequivalences between E and the essential image of 𝑅 and it follows from Lemma A.2.1 that this essentialimage of 𝑅 agrees with C . Together with Lemma A.2.2 this proves the implication iv) ⇒ iii) and the lastclaim. The implication iii) ⇒ ii) is trivial. Finally, if ii) holds, then preimages under 𝑝 ∶ ker( 𝑝 ) 𝑟 → E yieldexactly the desired objects to obtain a right adjoint via Lemma A.2.1. (cid:3) A.2.4.
Definition.
A Verdier sequence C 𝑓 ←←←←←←←→ D 𝑝 ←←←←←←→ E is split if 𝑝 admits both a left and a right adjoint.In this definition, we might just as well require that 𝑓 admit both adjoints, by the following result:A.2.5. Lemma.
Let (73) C 𝑓 ←←←←←←←→ D 𝑝 ←←←←←←→ E be a sequence in Cat ex∞ with vanishing composite. Then the following are equivalent:i) (73) is a fibre sequence, and 𝑝 admits a fully-faithful left (right) adjoint 𝑞 .ii) (73) is a cofibre sequence, and 𝑓 is fully-faithful and admits a left (right) adjoint 𝑔 .Furthermore, if i) and ii) hold, then both sequences C 𝑓 ←←←←←←←→ D 𝑝 ←←←←←←→ E and E 𝑞 ←←←←←←→ D 𝑔 ←←←←←←→ C are Verdier sequences. Explicitly, in the case of left adjoints 𝑔 is described as the cofibre of the counit 𝑞𝑝 → id D , thought of as afunctor D → D that vanishes after projection to E and therefore uniquely lifts to C . Similarly, the adjoint 𝑞 is described as the fibre of the unit id D → 𝑓 𝑔 , thought of as a functor D → D that vanishes after restrictionto C and therefore uniquely factors through E .A.2.6. Corollary.
An exact functor 𝑝 ∶ D → E is a split Verdier projection if and only if it admits fullyfaithful left and right adjoints. An exact functor 𝑓 ∶ C → D is a split Verdier inclusion if and only if it isfully faithful and admits left and right adjoints.Proof of Lemma A.2.5. We prove the claim for left adjoints. The claim for right adjoints follows by the dualargument (or by replacing all ∞ -categories by their opposites).Suppose first that i) holds. Then we obtain a left adjoint 𝑔 of 𝑓 by considering the exact functor ̃𝑔 = cof [ 𝑞𝑝 → id] ∶ D → D given by the cofibre of the counit. Since 𝑞 is fully-faithful, the unit map id → 𝑞𝑝 is an equivalence, fromwhich we can conclude that 𝑝 ◦ ̃𝑔 vanishes. Thus, ̃𝑔 factors uniquely through 𝑓 , giving rise to a functor
30 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE 𝑔 ∶ D → C . We now claim that the canonical transformation id → ̃𝑔 = 𝑓 ◦ 𝑔 acts as a unit exhibiting 𝑔 asleft adjoint to 𝑓 . Given objects 𝑥 ∈ D and 𝑦 ∈ C it will suffice to check that the composite map hom C ( 𝑔 ( 𝑥 ) , 𝑦 ) → hom D ( 𝑓 𝑔 ( 𝑥 ) , 𝑓 ( 𝑦 )) → hom D ( 𝑥, 𝑓 ( 𝑦 )) is an equivalence of spectra. Indeed, the first map is an equivalence since 𝑓 is fully-faithful and the secondmap is an equivalence because its cofibre is hom D ( 𝑞𝑝 ( 𝑥 ) , 𝑓 ( 𝑦 )) ≃ hom E ( 𝑝 ( 𝑥 ) , 𝑝𝑓 ( 𝑦 )) ≃ 0 .In this situation, 𝑝 is a localisation by Lemma A.2.2, so the sequence formed by 𝑓 and 𝑝 is a Verdiersequence by Corollary A.1.10, in particular a cofibre sequence. Also, the kernel of 𝑔 consists, by theadjunction rule, of those objects that are left orthogonal to C , and by Lemma A.2.3 this agrees with theessential image of 𝑞 . So the sequence formed by the adjoints satisfies i) (in the version with right adjoints),and is therefore also a Verdier sequence by what we have just shown.On the other hand, suppose that ii) holds. Then 𝑔 is a localisation by Lemma A.2.1 and thus the essentialimage of 𝑓 is given by the right orthogonal of ker( 𝑔 ) . It is therefore, in particular, closed under retracts in D . But according to Proposition A.1.6, 𝑝 exhibits E as the Verdier quotient of D by this image so it equals ker( 𝑝 ) by Lemma A.1.8. This shows that (73) is a fibre sequence. To see that 𝑝 admits a left adjoint we canappeal to Lemma A.2.1: For 𝑥 ∈ D the fibre of the unit map 𝑥 → 𝑓 𝑔 ( 𝑥 ) clearly projects to 𝑝 ( 𝑥 ) under 𝑝 ,and for 𝑐 ∈ C we have Hom D ( f ib( 𝑥 → 𝑓 𝑔 ( 𝑥 )) , 𝑓 ( 𝑐 ) ) ≃ cof [ Hom D ( 𝑥, 𝑓 ( 𝑐 )) → Hom D ( 𝑓 𝑔 ( 𝑥 ) , 𝑓 ( 𝑐 )) ] and since 𝑓 is fully faithful the latter term is also given by Hom C ( 𝑔 ( 𝑥 ) , 𝑐 ) , which identifies the map on theright as the adjunction equivalence. (cid:3) As a straight-forward consequence of Corollary A.2.6 we record:A.2.7.
Corollary.
A pullback of a split Verdier projection is again a split Verdier projection.Proof.
Using the universal property of the pullback one readily constructs the requisite functors from theoriginal adjoints (using the fact that these are fully faithful, and therefore sections of the original Verdierprojection). That these are again fully faithful adjoints follows immediately from the description of mappingspaces in pullbacks of ∞ -categories as pullbacks of mapping spaces. (cid:3) One might call Verdier sequences as in Lemma A.2.5 left-split and right-split , respectively. We willnot invest too much in this terminology, mostly since in the Poincaré context, the existence of one adjointimplies the existence of both, see Proposition 1.2.2. We do, however, take this opportunity to frame thefollowing corollary, which shows that the scenario of a left-split/right-split Verdier sequence as above canbe recognized in several ways (we make use of this in Section 3.2). For the statement recall that we denoteby C 𝑟 and C 𝑙 for the left and right orthogonal to a full subcategory C ⊆ D .A.2.8. Corollary.
Let D be a stable ∞ -category and C , E ⊆ D two full stable subcategories such that hom D ( 𝑥, 𝑦 ) ≃ 0 for every 𝑥 ∈ C , 𝑦 ∈ E . Then the following are equivalent:i) C ⊆ D admits a right adjoint 𝑝 ∶ D → C and the inclusion E ⊆ C 𝑟 is an equivalence.ii) E ⊆ D is a Verdier inclusion and the projection C → D ∕ E is an equivalence.iii) E ⊆ D admits a left adjoint 𝑞 ∶ D → E and the inclusion C ⊆ E 𝑙 is an equivalence.iv) C ⊆ D is a Verdier inclusion and the projection E → D ∕ C is an equivalence.Furthermore, when either of these equivalent conditions holds, the resulting sequences C → D → E and E → D → C formed by the inclusions and their adjoints are right-split and left-split Verdier sequences, respectively.Proof. The implications i) ⇒ ii) and iii) ⇒ iv) are dual to each other, and the same for the implications ii) ⇒ iii) and iv) ⇒ i). It will hence suffice to show i) ⇒ ii) ⇒ iii), along with the last claim.To prove the first of these implications, suppose that 𝑖 ∶ C ⊆ D admits a right adjoint 𝑝 ∶ D → C andthat E ⊆ C 𝑟 is an equivalence. By the adjunction rule, C 𝑟 agrees with the kernel of 𝑝 so we have a right-splitVerdier sequence E → D 𝑝 ←←←←←←→ C from which we conclude that the map D ∕ E → C induced by 𝑝 is an equivalence. The projection C → D ∕ E is a one-sided inverse and therefore also an equivalence. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 131 On the other hand, if ii) holds then by Lemma A.2.3, the projection D → D ∕ E has a left adjoint, and theinclusion of C into E 𝑙 is an equivalence (since both project to D ∕ E by an equivalence); the existence of theleft adjoint 𝑞 follows from Lemma A.2.5.The first Verdier sequence follows by duality. (cid:3) We now come back to the notion of a split Verdier sequence and show that it is essentially equivalent tothat of a recollement in the sense of [Lur17, Section A.8] in the setting of stable ∞ -categories. Specialisingthe definition to this case, we have:A.2.9. Definition.
A stable ∞ -category D is a stable recollement of a pair of stable subcategories C and E if i) the inclusions of both C and E admit left adjoints 𝐿 C and 𝐿 E ,ii) the composite C → D 𝐿 E ←←←←←←←←←←←←←→ E vanishes, andiii) 𝐿 E and 𝐿 C are jointly conservative.A.2.10. Proposition. If D is a stable recollement of C and E , then the sequence C → D 𝐿 E ←←←←←←←←←←←←←→ E is a splitVerdier sequence.Conversely, if C 𝑓 ←←←←←←←→ D 𝑝 ←←←←←←→ E is a split Verdier sequence, then D is a stable recollement of the essentialimages 𝑓 ( C ) and 𝑞 ( E ) , where 𝑞 denotes the right adjoint of 𝑝 .Proof. Consider the first statement. We claim that the sequence under consideration is a fibre sequence,so that it is split Verdier by Lemma A.2.5. Since the composite is zero by assumption, we are left to showthat every object 𝑥 in ker( 𝐿 E ) already belongs to the essential image of C . Denoting by 𝐿 C the left adjointof the inclusion of C , then the unit 𝑥 → 𝐿 C ( 𝑥 ) is mapped to an equivalence under both 𝐿 E and 𝐿 C . Byassumption 𝐿 E and 𝐿 C are jointly conservative, so the unit 𝑥 → 𝐿 C ( 𝑥 ) is an equivalence and therefore 𝑥 lies indeed in the essential image of C .For the second statement 𝑓 admits a left adjoint 𝑔 by Lemma A.2.5 , since 𝑝 does and it remains tosee that 𝑝 and 𝑔 are jointly conservative. Since we are in the stable setting it will suffice to show that thefunctors 𝑝 and 𝑔 together detect zero objects. Indeed, if 𝑥 ∈ D is such that 𝑝 ( 𝑥 ) ≃ 0 then 𝑥 belongs tothe essential image of 𝑓 . In this case, if 𝑔 ( 𝑥 ) is zero as well then 𝑥 ≃ 0 because the counit of 𝑔 ⊣ 𝑓 is anequivalence. (cid:3) In pictures, a stable recollement is given by
C D E , 𝐿 E 𝐿 C ⟂ ⟂ a split Verdier sequence is as left of the following diagram and in [BG16] Barwick and Glasman considereddiagrams as on the right: C D E 𝑓 ⟂⟂ and C D E . 𝑝 ⟂⟂ Here the non-curved maps form a Verdier sequence and left adjoints are on top. Our results above showthat all of these types of diagrams can be completed to the full
C D E , 𝑔 ′ ⟂ 𝑔 ⟂ 𝑞 ′ ⟂ 𝑞 ⟂
32 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE in which both the top and the bottom left pointing maps also form Verdier sequences, and whose maps arerelated by the bifibre sequences 𝑞𝑝𝑓 𝑔 ′ id D 𝑞 ′ 𝑝.𝑓 𝑔 From this data, one obtains a canonical transformation 𝑔 ′ ⇒ 𝑔 whose (co)fibre descends to a functor E → C ,and another transformation 𝑞 ⇒ 𝑞 ′ , whose (co)fibre also lifts to a functor E → C . We then have 𝑔𝑞 ′ ≃ cof( 𝑞 ⇒ 𝑞 ′ ) ≃ cof( 𝑔 ′ ⇒ 𝑔 ) ≃ Σ C 𝑔 ′ 𝑞, where the middle equivalence comes from the cofibre sequence describing the cofibre of a composition interms of the cofibres of the constituents. The functor 𝑐 ∶ E → C specified by any of the formulae above issaid to classify the recollement, as it participates in the following result:A.2.11. Proposition.
Given a split Verdier sequence in the notation above, the diagram D Ar( C ) E C 𝑔 → 𝑐𝑝𝑝 t 𝑐 is cartesian, where t is the target projection. Moreover, for any object 𝑥 ∈ D there is a cartesian diagram 𝑥 𝑓 𝑔 ( 𝑥 ) 𝑞 ′ 𝑝 ( 𝑥 ) 𝑓 𝑔𝑞 ′ 𝑝 ( 𝑥 ) with all maps induced by the units of the respective adjunctions. Let us remark that the sequence C Ar( C ) C r tsfib q 𝛿 is indeed a split Verdier sequence, where r( 𝑥 ) = ( 𝑥 → , with left and right adjoints being s( 𝑥 → 𝑦 ) = 𝑥 and f ib( 𝑥 → 𝑦 ) , respectively, while t( 𝑥 → 𝑦 ) = 𝑦 with left and right adjoints q( 𝑦 ) = (0 → 𝑦 ) and 𝛿 ( 𝑦 ) = id 𝑦 ,respectively. It underlies the metabolic sequence of Example 1.2.5 which plays fundamental role in ourresults. Proof.
The inverse functor from the pullback to D is given by sending a pair ( 𝑒, 𝑎 → 𝑐 ( 𝑒 )) to the pullback 𝑞 ′ ( 𝑒 ) × 𝑓𝑐 ( 𝑒 ) 𝑓 ( 𝑎 ) , with the left structure map coming from the definition of 𝑐 . That the composite on thepullback E × C Ar( C ) is equivalent to the identity follows from unwinding the definitions, whereas for thecomposite on D it is precisely the cartesianness of the diagram from the statement. But the induced mapon its vertical fibres is the unit map of 𝑓 𝑔 ′ ( 𝑥 ) → 𝑓 𝑔𝑓 𝑔 ′ ( 𝑥 ) of the adjunction 𝑓 𝑔 which is an equivalencesince 𝑓 is fully faithful, together with the triangle identity. (cid:3) A.2.12.
Remark.
A monoidal refinement of this result was recently given in [QS19, Section 1].Finally, we characterise the horizontal maps appearing in Proposition A.2.11. To this end consider acommutative diagram
D D ′ E E ′ 𝑝 𝑝 ′ ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 133 with vertical split Verdier projections. Such a diagram gives rise to two new (not necessarily commutative)diagrams of the shape D D ′ E E ′ by passing to either left or right adjoints in the vertical direction. The original square is called adjointable if both squares of adjoints do in fact commute, i.e. if the Beck-Chevalley transformations connecting thecomposites are equivalences, see [Lur09a, Section 7.3.1] for details. It is readily checked that cartesiansquares as above are adjointable.A.2.13. Proposition.
Given a split Verdier sequence C → D → E and another stable ∞ -category C ′ thefull subcategory of Fun ex ( D , Ar( C ′ )) spanned by the functors 𝜑 that give rise to adjointable squares D Ar( C ′ ) E C ′ 𝜑 𝑡𝜑 is equivalent to Fun ex ( C , C ′ ) via restriction to horizontal fibres. In particular, the classifying functor in Proposition A.2.11 is uniquely determined by yielding a cartesiandiagram and inducing the identity on fibres, so 𝑡 ∶ Ar( C ) → C really is the universal split Verdier projectionwith fibre C . Similarly, we find that for a cartesian square D D ′ E E ′ 𝑝 𝑝 ′ with common fibre C the classifying functor E → C of 𝑝 is the composite of that for 𝑝 ′ and the given map E → E ′ . We shall make use of the functoriality of the classifying map in adjointable (and not just cartesian)squares arising from Proposition A.2.13 in Lemma 1.5.3. Proof.
Using the fibre sequences connecting the various adjoints one readily checks that generally ad-jointability of the two squares
D D ′ C C ′ E E ′ D D ′ 𝜑𝑝 𝑝 ′ 𝑓 𝑓 ′ 𝜑 are equivalent conditions for two (vertical) Verdier sequences. We will use the latter description in the caseat hand to see that the restriction functor in the statement is fully faithful: Rewriting then Fun ex ( D , Ar( C ′ )) =Ar(Fun ex ( D , C ′ )) we compute for 𝜑, 𝜓 ∶ D → Ar( C ′ ) that nat( 𝜑, 𝜓 ) ≃ nat( 𝑠𝜑, 𝑠𝜓 ) × nat( 𝑠𝜑,𝑡𝜓 ) nat( 𝑡𝜑, 𝑡𝜓 ) . Using the fact that 𝑠, f ib ∶ Ar( C ′ ) → C are the left and right adjoint to the Verdier inclusion C ′ → Ar( C ′ ) we find 𝑠𝜑 ≃ 𝜑 | 𝐹 ◦ 𝑔 and 𝑡𝜑 ≃ 𝜑 | 𝐹 ◦ 𝑐𝑝 from adjointability of 𝜑 and similarly for 𝜓 . Thus the above can berewritten as nat( 𝜑 | 𝐹 ◦ 𝑔, 𝜓 | 𝐹 ◦ 𝑔 ) × nat( 𝜑 | 𝐹 ◦ 𝑔,𝜓 | 𝐹 ◦ 𝑐𝑝 ) nat( 𝜑 | 𝐹 ◦ 𝑐𝑝, 𝜓 | 𝐹 ◦ 𝑐𝑝 ) . But 𝑔 ∶ D → C is a localisation (since it has 𝑓 as a fully faithful right adjoint) so nat( 𝜑 | 𝐹 ◦ 𝑔, 𝜓 | 𝐹 ◦ 𝑔 ) ≃ nat( 𝜑 | 𝐹 , 𝜓 | 𝐹 ) and we claim that the restriction map nat( 𝜑 | 𝐹 ◦ 𝑐𝑝, 𝜓 | 𝐹 ◦ 𝑐𝑝 ) ⟶ nat( 𝜑 | 𝐹 ◦ 𝑔, 𝜓 | 𝐹 ◦ 𝑐𝑝 )
34 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE is an equivalence, which gives fully faithfullness. To see this consider its fibre nat( 𝜑 | 𝐹 ◦ 𝑔 ′ , 𝜓 | 𝐹 ◦ 𝑐𝑝 ) andrecall that ( 𝑔 ′ ) ∗ ∶ Fun( C , C ′ ) ⟂ Fun( D , C ′ ) ∶ 𝑓 ∗ is also an adjunction, so nat( 𝜑 | 𝐹 ◦ 𝑔 ′ , 𝜓 | 𝐹 ◦ 𝑐𝑝 ) ≃ nat( 𝜑 | 𝐹 , 𝜓 | 𝐹 ◦ 𝑐𝑝𝑓 ) ≃ 0 as desired.We are left to show that the restriction functor is essentially surjective, but this is obvious by followingthe classification arrow from Proposition A.2.11 with the one induced by given functor C → C ′ on arrowcategories. (cid:3) A.2.14.
Remark.
Proposition A.2.11 and the entire discussion preceding it apply equally well to stable ∞ -categories that are not small, and for example recover the observation of Barwick and Glasman [BG16,Proposition 7], that the left and right orthogonal to the inclusion of C in a stable recollement are canonicallyequivalent.One example we established in Paper [I] is given by Fun ex ( C op , S 𝑝 ) Fun q ( C op , S 𝑝 ) Fun s ( C op , Sp) , L ⟂⟂ (−) s (−) q ⟂⟂ whose fracture square gives exactly the classification of quadratic functors in Corollary [I].1.3.12.Another standard example is the case where D = S 𝑝 and 𝑓 is the inclusion of those spectra on which aprime 𝑙 acts invertibly: S 𝑝 [ 𝑙 ] S 𝑝 S 𝑝 [ l-adic equiv’s −1 ] , div 𝑙 (−)[ 𝑙 ] ⟂⟂ (−) ∧ 𝑙 (−)[ 𝑙 ∞ ] ⟂⟂ where div 𝑙 ( 𝑋 ) = lim − ⋅ 𝑙 𝑋 is the 𝑙 -divisible part of 𝑋 , together with the fibre sequences div 𝑙 ( 𝑋 ) ⟶ 𝑋 ⟶ 𝑋 ∧ 𝑙 and 𝑋 [ 𝑙 ∞ ] ⟶ 𝑋 ⟶ 𝑋 [ 𝑙 ] , classifying functor 𝑋 ↦ 𝑋 ∧ 𝑙 [ 𝑙 ] ≃ Ωdiv 𝑙 ( 𝑋 [ 𝑙 ∞ ]) and fracture square 𝑋 𝑋 [ 𝑙 ] 𝑋 ∧ 𝑙 𝑋 ∧ 𝑙 [ 𝑙 ] . A.3.
Karoubi sequences.
We now move to the more general notion of Karoubi sequences, which are aversion of Verdier sequences invariant under the addition of direct summands in the categories at hand.Let us briefly record some basic statements:A.3.1.
Definition.
We call an exact functor C → D between stable ∞ -categories a Karoubi equivalence ifit is fully faithful and has dense image, in the sense that every object of D is a retract of an object in theessential image.The most important example of Karoubi equivalences are of course idempotent completions C → C ♮ .When fixing the target Karoubi equivalences can be entirely classified, see [Tho97, Theorem 2.1]:A.3.2. Theorem (Thomason) . Karoubi equivalences induce injections on K ∶ Cat ex∞ → A 𝑏 , and Karoubiequivalences to a fixed small stable ∞ -category C (up to equivalence over C ) are in bijection with subgroupsof K ( C ) by taking the image of their induced map. Note that the statement in [Tho97] is for triangulated categories, but the proof works verbatim in thesetting of stable ∞ -categories. ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 135 A.3.3.
Proposition.
The localisation of
Cat ex∞ at the Karoubi equivalences is both a left and a right Bousfieldlocalisation. The right adjoint is given by C ↦ C ♮ , and the left adjoint takes C to C min , the full subcategoryspanned by the objects 𝑥 ∈ C with 𝑥 ] ∈ K ( C ) .Furthermore, an exact functor is a Karoubi equivalence if and only if it induces an equivalence onminimalisations or equivalently idempotent completions. Denoting by
Cat ex∞ , idem the full subcategory of Cat ex∞ spanned by the small, idempotent complete stable ∞ -categories, we in particular find that (−) ♮ ∶ Cat ex∞ → Cat ex∞ , idem preserves both limits and colimits.A.3.4. Definition.
Small stable ∞ -categories C with the property that K ( C ) vanishes we will call minimal and refer to the assignment C ↦ C min as minimalisation . Proof of Proposition A.3.3.
It is an exercise in pasting retract diagrams to check that Karoubi equivalencesare closed under 2-out-of-3. The characterisation in the last statement then follows immediately from thefact that both inclusions C min ⊆ C ⊆ C ♮ are Karoubi equivalences, the former for example by Thomason’sresult. Furthermore, [Lur09a, Lemma 5.1.4.7] then implies that given a Karoubi equivalence 𝑖 ∶ C → D and a functor 𝑓 ∶ D → E the exactness of 𝑓 is equivalent to that of 𝑓 𝑖 .The statement about the adjoints now follows from Lemma A.2.1: That idempotent completion satisfiesthe requisite conditions is [Lur09a, Proposition 5.1.4.9] and that minimalisations do is immediate from thefunctoriality of K . (cid:3) Let us now define our main object of study in this section.A.3.5.
Definition.
A sequence C 𝑓 ←←←←←←←→ D 𝑝 ←←←←←←→ E of exact functors with vanishing composite is a Karoubi sequence if the sequence C ♮ → D ♮ → E ♮ is both a fibre and cofibre sequence in Cat ex∞ , idem . In this case we refer to 𝑓 as a Karoubi inclusion and to 𝑝 as a Karoubi projection .A.3.6.
Remark.
Equivalently, by Proposition A.3.3, we might ask the sequence C min → D min → E min to be both a fibre and a cofibre sequence in the full subcategory of Cat ex∞ spanned by the minimal stable ∞ -categories, or more symmetrically that the original sequence give a fibre and cofibre sequence in thelocalisation of Cat ex∞ at the Karoubi equivalences.We have chosen the present formulation as the idempotent completion plays a disproportionally moreimportant role, both in the detection of Karoubi sequences and in applications.We also have a concrete characterisation of Karoubi sequences, analogous to the one for Verdier se-quences Corollary A.1.10.A.3.7.
Proposition.
Let C 𝑓 ←←←←←←←→ D 𝑝 ←←←←←←→ E be a sequence of exact functors between small stable ∞ -categorieswith vanishing composite. Theni) the sequence C ♮ 𝑓 ♮ ←←←←←←←←←←←→ D ♮ 𝑝 ♮ ←←←←←←←←←→ E ♮ is a fibre sequence in Cat ex∞ , idem if and only if 𝑓 becomes a Karoubiequivalence when regarded as a functor C → ker( 𝑝 ) .ii) the sequence C ♮ 𝑓 ♮ ←←←←←←←←←←←→ D ♮ 𝑝 ♮ ←←←←←←←←←→ E ♮ is a cofibre sequence in Cat ex∞ , idem if and only if the induced functorfrom the Verdier quotient of D by the stable subcategory generated by the image of 𝑓 is a Karoubiequivalence to E .iii) the sequence C 𝑓 ←←←←←←←→ D 𝑝 ←←←←←←→ E is a Karoubi sequence if and only if 𝑓 is fully-faithful and the induced map D ∕ C → E is a Karoubi equivalence.In particular, every Verdier sequence is a Karoubi sequence. Let us explicitely warn the reader, however, that the Verdier quotient of two idempotent complete, stable ∞ -categories need not be idempotent complete.
36 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
Proof.
By Proposition A.3.3 the functor (−) ♮ ∶ Cat ex∞ → Cat ex∞ , idem preserves both limits and colimits, and Cat ex∞ , idem is closed under limits in Cat ex∞ . This yields an equivalence ker( 𝑝 ♮ ) ≃ ker( 𝑝 ) ♮ , which proves i).Similarly, ii) follows from the description of cofibres in Cat ex∞ as Verdier quotients together with thepreservation of cofibres under idempotent completion.Finally, the forwards direction of iii) follows directly from the previous two statements. On the otherhand, if 𝑓 is fully faithful, and D ∕ C → E is a Karoubi equivalence, then the kernel of 𝑝 agrees with thekernel of the projection 𝑞 ∶ D → D ∕ C . Thus, by Lemma A.1.8, the map 𝑓 ∶ C → ker( 𝑞 ) has denseessential image and therefore is a Karoubi equivalence. The reverse claim thus also follows from the firsttwo statements. (cid:3) A.3.8.
Corollary.
An exact functor 𝑓 ∶ C → D is a Karoubi inclusion if and only if it is fully-faithful. Itis a Karoubi projection if and only if it has dense essential image 𝑓 ( C ) ⊆ D , and the induced functor 𝑓 ∶ C → 𝑓 ( C ) is Verdier projection. Combining this statement with Thomason’s result above, we find:A.3.9.
Corollary.
Let 𝑝 ∶ D → E be a Karoubi projection. Then the following are equivalent:i) 𝑝 is a Verdier projection.ii) 𝑝 is essentially surjective.iii) The induced group homomorphism K ( D ) → K ( E ) is surjective. We also note:A.3.10.
Lemma.
Any pullback of a Karoubi projection is again a Karoubi projection.Proof.
Given Lemma A.1.11 and the characterisation of Karoubi projections in Corollary A.3.8 it sufficesto show that the pullback D → D ′ of a Karoubi equivalence E → E ′ along 𝑖 ∶ D ′ → E ′ is again one such.But one readily checks that this pullback is given by the full subcategory { 𝑥 ∈ E ′ ∣ 𝑖 [ 𝑥 ] ∈ K ( E )} of E ′ ,whence Thomason’s theorem A.3.2 gives the claim. (cid:3) Next, we record the following detection criterion for Karoubi-sequences, often called the Thomason-Neeman localisation theorem in the context of triangulated categories, see [Nee92, Theorem 2.1]. To stateit, we need to extend the notion of Verdier sequences to non-small stable ∞ -categories. This is achievedfor example by Corollary A.1.10 which does not require any smallness assumption.A.3.11. Theorem.
A sequence C → D → E of small stable ∞ -categories and exact functors with vanishingcomposite is a Karoubi sequence if and only if the induced sequence Ind( C ) ⟶ Ind( D ) ⟶ Ind( E ) is a Verdier sequence (of not necessarily small ∞ -categories). Here
Ind denotes the inductive completion of a small category, characterised for example as the smallestsubcategory of
Fun( C op , S ) stable under filtered colimits and containing all representable functors. Proof.
First of all, note that inductive completion preserves both stability of ∞ -categories and exactnessof functors for example as a consequence of [Lur09a, Proposition 5.3.5.10]: The colimit preserving exten-sion of suspension is suspension, and the extension of loops is its inverse. Furthermore, it preserves fullfaithfullness by [Lur09a, 5.3.5.11], commutes with Verdier quotients by [NS18, Proposition I.3.5] and by[Lur09a, Lemma 5.4.2.4] the compact objects in Ind( C ) form an idempotent completion of C . Combiningthese statements it follows that an exact functor is a Karoubi equivalence if and only if it induces an equiv-alence on inductive completions: The backwards direction is immediate, and given a Karoubi equivalence C → D we find Ind( C ) the kernel of Ind( D ) → Ind( D ∕ C ) ≃ 0 by Lemma A.1.8, since cocomplete categoriesare in particular idempotent complete by [Lur09a, Corollary 4.4.5.16].Reusing the three statements, the claim now follows from our characterisation of Verdier and Karoubisequences, Corollary A.1.10 and Proposition A.3.7. (cid:3) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 137 In fact, given a Karoubi sequence C 𝑖 ←←←←→ D 𝑝 ←←←←←←→ E the sequence Ind( C ) → Ind( D ) → Ind( E ) consists of theleft adjoints in a stable recollement (note the order reversal) Ind( E ) Ind( D ) Ind( C ) Ind( 𝑝 ) ⟂⟂ Ind( 𝑖 ) ⟂⟂ It follows immediately from [NS18, Proposition I.3.5], that
Ind( 𝑝 ) admits a fully faithful right adjoint whichpreserves colimits. By [Lur09a, Corollary 5.5.2.9] it then follows that this functor has a further right adjoint,whence the results of the previous section give the adjoint to Ind( 𝑖 ) and its adjoint.The other functors in this recollement do not, however, in general preserve compact objects, so onecannot pass to them to obtain further Karoubi sequences.A.3.12. Remark.
By [NS18, Theorem I.3.3] the adjoint on inductive completions may be explicitely de-scribed as taking 𝑝 ( 𝑥 ) to colim 𝑧 ∈ C ∕ 𝑥 cof( 𝑧 → 𝑥 ) , and dually the right adjoint on projective completions isgiven by taking 𝑥 to lim 𝑧 ∈ C 𝑥 ∕ f ib( 𝑥 → 𝑦 ) . As adjoints to localisations these are fully faithful and via theinclusions Ind( C ) ⊆ Fun( C op , S ) and Pro( C ) ⊆ Fun( C , S ) op they give a concrete way of constructing theVerdier quotient.A.3.13. Remark.
Let us also warn the reader of the following asymmetry: Suppose given compactly gener-ated stable ∞ -categories C and D (i.e. cocomplete, stable ∞ -categories that admit a set of compact objectswhich jointly detect equivalences) and a functor 𝐹 ∶ C → D which preserves colimits and compact objects.If such 𝐹 is a Verdier inclusion (of non-small ∞ -categories) its restriction 𝑓 to compact objects is auto-matically a Karoubi inclusion, since full faithfulness is clearly retained. In fact, such an 𝐹 is automaticallyof the form Ind( 𝑓 ) by [Lur09a, Propositions 5.4.2.17 & 5.4.2.19], and another application of [NS18, Propo-sition I.3.5] exhibits the Verdier quotient of 𝐹 as the inductive completion of that of 𝑓 .Conversely, however, if 𝐹 is a Verdier projection (of non-small ∞ -categories), it needs not follow thatits restriction to compact objects is a Karoubi projection, as the kernel of 𝐹 may fail to be compactlygenerated; in fact ker( 𝐹 ) need not have any non-trivial compact at all. The first example of such a situationwas exhibited by Keller in [Kel94], we recall it in Example A.4.6 below.The fibre of a Verdier projection between compactly generated categories is, however, automatically du-alisable in the symmetric monoidal category of stable presentable ∞ -categories. In as of now unpublishedwork Efimov constructed an extension of any localising invariants Cat ex∞ , idem → Sp , such as non-connective K -theory, to such dualisable categories. This allows one to circumvent the difficulties for localisation se-quences caused by the failure of compact generation, see [Hoy18] or [Efi18] for an account.Finally, we extend the classification result Proposition A.2.11 for split Verdier sequences to the non-splitcase. To this end consider a Verdier sequence C → D → E . Recalling Pro( C ) = Ind( C op ) op we obtain fromTheorem A.3.11 and the discussion thereafter a split Verdier sequence Pro Ind( C ) Pro Ind( D ) Pro Ind( E ) 𝑔 ′ 𝑔 ⟂⟂ 𝑞 ′ 𝑞 ⟂⟂ of fairly large categories, together with a classifying functor 𝑐 ∶ Pro Ind( E ) → Pro Ind( C ) . Now considerthe categories Tate( C ) and Latt( C ) of (elementary) Tate objects and their lattices from [Hen17] (though wewarn the reader that Hennion denotes by Tate( C ) the idempotent completion of the category we considerhere): Tate( C ) is the smallest stable subcategory of Pro Ind( C ) spanned by its full subcategories Pro( C ) and Ind( C ) , and Latt( C ) is the full subcategory of Ar(Pro Ind( C )) spanned by the arrows with source in Ind( C ) and target in Pro( C ) . We obtain a commutative square(74) Pro Ind( D ) Ar(Pro Ind( C ))Pro Ind( E ) Pro Ind( C ) , 𝑔 ′ → 𝑔 cof 𝑐
38 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE which is a pullback by (a rotation in the top right corner of) Proposition A.2.11. By direct inspection itrestricts to a commutative diagram D Latt( C ) E Tate( C ) 𝑔 ′ → 𝑔 cof 𝑐 and we find:A.3.14. Proposition.
For any stable ∞ -category C the map cof ∶ Latt( C ) → Tate( C ) is a Verdier projectionwith fibre C ♮ and for a Verdier sequence C → D → E with C idempotent complete the diagram above iscartesian. The first part of this result is a special case of Clausen’s discussion of cone categories in [Cla17, Sec-tion 3.1], particularly [Cla17, Remark 3.23], whereas the second part along with the uniqueness statementin Proposition A.3.15 below was first observed by the eighth author in [Nik20], which also discusses amonoidal version.Combined they imply that the functor cof ∶ Latt( C ) → Tate( C ) is the universal Verdier projection withfibre C though it does not run between small categories. One also readily checks, that a Verdier projection D → E is right or left split if and only if the functor E → Tate( C ) takes values in Pro( C ) or Ind( C ) ,respectively, so that the pullbacks to these categories give the universal right or left split Verdier sequences.For consistency note also that Ind( C ) ∩ Pro( C ) = C ♮ : Write 𝑋 ∈ Ind( C ) ∩ Pro( C ) both as the limit of aprojective system 𝑃 𝑖 and as the colimit of an inductive system 𝐼 𝑗 in C . Then by the computation of mappingspaces in Ind - and
Pro -categories in [Lur09a, Section 5.3] the identity of 𝑋 factors as 𝑋 → 𝑃 𝑖 → 𝐼 𝑗 → 𝑋 for some 𝑖 and 𝑗 , making 𝑋 a retract of either object. Thus Proposition A.3.14 specialises back to thesplit case Proposition A.2.11 (under the additional assumption that C be idempotent complete). Similarly, Latt( C ) → Tate( C ) ♮ is the universal Karoubi projection with fibre C ♮ , and one readily checks that Verdierprojections are characterised among Karoubi projections by the property that the classifying functor factorsthrough Tate( C ) ⊆ Tate( C ) ♮ . Proof.
We start with the first claim: Evidently the kernel of the functor cof consists exactly of the equiv-alences from an inductive to a projective object in C . This forces both to be constant (by the argument wegave before the proof), whence the kernel is the full subcategory of Ar( C ♮ ) spanned by the equivalences,which is equivalent to C ♮ itself. Consider then the natural functor Latt( C )∕ C ♮ → Tate( C ) which we have to show is an equivalence. We start with full faithfulness. On the one hand, using Ω cof ≃ f ib the space
Hom
Tate( C ) (cof( 𝑖 → 𝑝 ) , cof( 𝑖 ′ → 𝑝 ′ )) can be described as Ω ∞−1 of the total fibre of the square hom Tate( C ) ( 𝑝, 𝑖 ′ ) hom Tate( C ) ( 𝑖, 𝑖 ′ )hom Tate( C ) ( 𝑝, 𝑝 ′ ) hom Tate( C ) ( 𝑖, 𝑝 ′ ) using the evident maps. On the other hand using [NS18, Theorem I.3.3 (ii)] we have hom Latt( C )∕ C ♮ ( 𝑖 → 𝑝, 𝑖 ′ → 𝑝 ′ ) ≃ colim 𝑐 ∈ C ♮ ∕ 𝑖 ′ hom Latt( C ) ( 𝑖 → 𝑝, cof( 𝑐 → 𝑖 ′ ) → cof( 𝑐 → 𝑝 ′ ))≃ colim 𝑐 ∈ C ♮ ∕ 𝑖 ′ hom Tate( C ) ( 𝑖, cof( 𝑐 → 𝑖 ′ )) × hom Tate( C ) ( 𝑖, cof( 𝑐 → 𝑝 ′ )) hom Tate( C ) ( 𝑝, cof( 𝑐 → 𝑝 ′ )) Now the total fibre above is invariant under replacing 𝑖 ′ and 𝑝 ′ by cof( 𝑐 → 𝑖 ′ ) and cof( 𝑐 → 𝑝 ′ ) , respectively,so straight from the definition of total fibres we find the fibre of hom Latt( C )∕ C ♮ ( 𝑖 → 𝑝, 𝑖 ′ → 𝑝 ′ ) ⟶ hom Tate( C ) (cof( 𝑖 → 𝑝 ) , cof( 𝑖 ′ → 𝑝 ′ )) . ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 139 given by colim 𝑐 ∈ C ♮ ∕ 𝑖 ′ hom Tate( C ) ( 𝑝, cof( 𝑐 → 𝑖 ′ )) . We claim that this term vanishes. For writing 𝑝 = lim 𝑘 ∈ 𝐾 𝑝 𝑘 for some 𝐾 → C , we find from the computationof mapping spaces in categories of projective systems in [Lur09a, Section 5.3], that colim 𝑐 ∈ C ♮ ∕ 𝑖 ′ hom Tate( C ) ( 𝑝, cof( 𝑐 → 𝑖 ′ )) ≃ colim 𝑐 ∈ C ♮ ∕ 𝑖 ′ colim 𝑘 ∈ 𝐾 hom Ind( C ) ( 𝑝 𝑘 , cof( 𝑐 → 𝑖 ′ ))≃ colim 𝑘 ∈ 𝐾 hom Ind( C )∕ C ♮ ( 𝑝 𝑘 , 𝑖 ′ ) and the last term clearly vanishes.Finally, we note that the image of the functor Latt( C )∕ C ♮ in Tate( C ) is a stable subcategory containingboth Ind( C ) and Pro( C ) so is essentially surjective by definition of Tate( C ) .We thus turn to the cartesianness of the square involving the Verdier projection D → E . We will reducethe statement to the split case by means of the embedding into (74), see [Nik20] for a more direct argument.Let 𝑃 denote the pullback of E → Tate( C ) ← Latt( C ) . Then the induced functor D → 𝑃 is fully faithful,since the square in question fully faithfully embeds into the right hand square before the proposition, whichis cartesian. It remains to check that D → 𝑃 is essentially surjective. But by Proposition A.2.11 any 𝑒 ∈ E ,together with 𝑖 → 𝑝 ∈ Latt( C ) and an equivalence cof( 𝑖 → 𝑝 ) ≃ 𝑐 ( 𝑒 ) , determines an essentially uniqueobject 𝑑 ∈ Pro Ind( D ) , namely 𝑞 ′ ( 𝑒 ) × 𝑐 ( 𝑒 ) 𝑝 . This object lies in D ♮ = Pro( D ) ∩ Ind( D ) ⊆ Pro Ind( D ) , sinceby construction there are fibre sequences 𝑞 ( 𝑒 ) → 𝑑 → 𝑝 and 𝑖 → 𝑑 → 𝑞 ′ ( 𝑒 ) as 𝑐 ( 𝑒 ) ≃ cof ( 𝑞 ( 𝑒 ) → 𝑞 ′ ( 𝑒 )) and the outer terms on the left are projective systems, whereas those on theright are inductive ones. We claim that D D ♮ E E ♮𝑝 𝑝 ♮ is cartesian, whence 𝑑 actually defines an object of D , which one readily checks to be a preimage of thedesired sort. For this final claim it is clearly necessary that C be idempotent complete, but this also suffices:The functor from the pullback 𝑃 of the remaining diagram (with D removed) to D ♮ is clearly fully faithful,thus so is D → 𝑃 . It remains to show that this functor is essentially surjective. Pick then an 𝑑 ∈ D ♮ with 𝑝 ♮ ( 𝑑 ) ∈ E and a witnessing retract diagram 𝑑 ⟶ 𝑑 ′ ⟶ 𝑑 with 𝑑 ′ ∈ D . By yet another application of [NS18, Theorem I.3.3 (ii)] we can find an 𝑥 ∈ D together witha map 𝑑 ′ → 𝑥 covering the projection 𝑝 ( 𝑑 ′ ) → 𝑝 ♮ ( 𝑑 ) . But then the fibre of the composite 𝑑 → 𝑑 ′ → 𝑥 liesin C ♮ = C ⊆ D , and thus so does 𝑑 ∈ D as desired. (cid:3) Regarding the uniqueness of the classifying map in Proposition A.3.14, we extend the notion of ad-jointability to commutative squares
D D ′ E E ′ 𝑖𝑝 𝑝 ′ 𝑗 with vertical Verdier projections by requiring their inductive and projective completions to be right and leftadjointable, respectively. Then we find:
40 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
A.3.15.
Proposition.
Given a Verdier sequence C → D → E and another stable ∞ -category C ′ , the fullsubcategory of Fun ex ( D , Ar( C ′ )) spanned by the functors 𝜑 that give rise to adjointable squares D Latt( C ′ ) E Tate( C ′ ) 𝜑 cof 𝜑 in the sense just described is equivalent to Fun ex ( C , C ′ ) via restriction to vertical fibres. Furthermore, anycartesian square whose vertical maps are Verdier projections is adjointable.Proof. The first part follows from Proposition A.2.13 by unwinding definitions. The argument that induc-tive completions of cartesian squares are adjointable is, however, more subtle (the case of the projectivecompletion is dual): To see that
Ind( D ) Ind( D ′ )Ind( E ) Ind( E ′ ) 𝜑 ! 𝜑 ! 𝑝 ∗ ( 𝑝 ′ ) ∗ commutes, note first that by the universal property of inductive completions it suffices to check this af-ter restriction to E ⊆ Ind( E ) . Next note that the statement becomes true after postcomposition with 𝑝 ′! ∶ Ind( D ′ ) → Ind( E ′ ) , since 𝑝 ′! 𝜑 ! 𝑝 ∗ ≃ 𝜑 ! 𝑝 ! 𝑝 ∗ ≃ 𝜑 ! ≃ 𝑝 ′! ( 𝑝 ′ ) ∗ 𝜑 ! via the canonical maps, since 𝑝 ∗ and ( 𝑝 ′ ) ∗ are fully faithful by assumption. It therefore only remains tocheck that the composite 𝜑 ! 𝑝 ∗ takes values in the image of ( 𝑝 ′ ) ∗ , since 𝑝 ′! restricts to an equivalence onthis part on account of being a localisation. Using the standard embedding Ind( D ′ ) ⊆ Fun ex (( D ′ ) op , S 𝑝 ) ,this image unwinds to exactly those functors ( D ′ ) op → S 𝑝 that vanish on C ′ , the kernel of 𝑝 ′ . Under thisembedding 𝜑 ! 𝑝 ∗ ( 𝑒 ) unwinds to the left Kan extension of hom E ( 𝑝 − , 𝑒 ) ∶ D op → S 𝑝 along 𝜑 op ∶ D op → E op .Evaluating at some 𝑐 ′ ∈ C ′ using the pointwise formula yields [ 𝜑 ! 𝑝 ∗ ( 𝑒 )]( 𝑐 ′ ) ≃ colim 𝑑 ∈ D ∕ 𝑐 ′ hom E ( 𝑝 ( 𝑑 ) , 𝑒 ) . But since we started with a cartesian square, picking a preimage 𝑐 ∈ C = ker( 𝑝 ) of 𝑐 ′ yields an equivalence D ∕ 𝑐 → D ∕ 𝑐 ′ , which shows that ( 𝑐, 𝜑 ( 𝑐 ) → 𝑐 ′ ) is a terminal object in D ∕ 𝑐 ′ , so [ 𝜑 ! 𝑝 ∗ ( 𝑒 )]( 𝑐 ′ ) ≃ hom E ( 𝑝 ( 𝑐 ) , 𝑒 ) ≃ 0 as desired. (cid:3) A.3.16.
Example.
Let C and E be stable ∞ -categories, with C idempotent complete, and let B ∶ C op × E → S 𝑝 be a bilinear functor. Interpreting B as a functor E → Fun ex ( C op , S 𝑝 ) ≃ Ind( C ) ⊂ Tate( C ) , we can pull backthe universal Verdier sequence with fibre C along B as to obtain a Verdier sequence C → D → E (whichautomatically has a left adjoint, since the restriction of the universal Verdier sequence to Ind( C ) ⊂ Tate( C ) does). Then, this Verdier sequence is the sequence C 𝑓 ←←←←←←←→ Pair( C , E , B) 𝑝 ←←←←←←→ E obtained from the pairings construction from § [I].7, where the first map includes C as objects of the form ( 𝑐, , and the second map projects ( 𝑐, 𝑒, 𝛽 ) to 𝑒 .Indeed, the classifying map of the latter sequence is given by the suspension of the composite E 𝑞 ←←←←←←→ Pair( C , E , B) 𝑔 ′ ←←←←←←←←←→ Ind( C ) ⊂ Tate( C ) where 𝑞 is the left adjoint of 𝑝 (given by the inclusion as objects of the form (0 , 𝑒, ), and 𝑔 ′ is the rightadjoint of Ind( 𝑓 ) . Identifying Ind( C ) with Fun ex ( C op , S 𝑝 ) , this right adjoint is given by the formula 𝑋 ↦ Hom
Pair( C , E , B) ( 𝑓 (−) , 𝑋 ) , so the above composite corresponds to the bilinear functor C op × E → S 𝑝, ( 𝑐, 𝑒 ) ↦ Hom
Pair( C , E , B) ( 𝑓 ( 𝑐 ) , 𝑞 ( 𝑒 )) , ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 141 whose suspension agrees with B( 𝑐, 𝑒 ) by the formula for mapping spaces in pairing categories, [I].(171).A.4. Verdier and Karoubi sequences among module categories.
Let 𝜙 ∶ 𝐴 → 𝐵 be a map of E -ringspectra. Extension of scalars induces an exact induction functor 𝜙 ! ∶ Mod 𝐴 → Mod 𝐵 , 𝑀 ↦ 𝐵 ⊗ 𝐴 𝑀 on the categories of (left) modules, which is left adjoint to the restriction of scalars functor 𝜙 ∗ ∶ Mod( 𝐵 ) → Mod( 𝐴 ) . Induction restricts to functors 𝜙 ! ∶ Mod 𝜔𝐴 → Mod 𝜔𝐵 and 𝜙 ! ∶ Mod c 𝐴 → Mod 𝜙 (c) 𝐵 , where c ⊆ K ( 𝐴 ) is a subgroup and Mod c 𝐴 the full subcategory of Mod 𝜔𝐴 spanned by those 𝐴 -modules 𝑋 with [ 𝑋 ] ∈ c ⊆ K ( 𝐴 ) . The most important special case of the latter construction is the case where c is theimage of the canonical map ℤ → K ( 𝐴 ) , ↦ 𝐴 , in which case Mod c 𝐴 = Mod f 𝐴 is the stable subcategory of Mod 𝜔𝐴 generated by 𝐴 . In this section we analyse when these functors are Verdier or Karoubi projections. Remark.
We remind the reader mainly interested in the classical case of discrete rings of the followingdictionary: The Eilenberg-Mac Lane spectrum of a discrete ring 𝐴 is an E -ring spectrum, which we denoteby H 𝐴 . The category Mod(H 𝐴 ) of H 𝐴 -module spectra is then equivalent to the (unbounded) derived ∞ -category of 𝐴 , that is, the ∞ -categorical localisation of the category of 𝐴 -chain complexes at the class ofhomology equivalences, see [Lur17, Remark 7.1.1.16].The reader should be aware that under this equivalence, H 𝑀 ⊗ H 𝐴 H 𝑁 corresponds to the derived tensorproduct 𝑀 ⊗ 𝕃 𝐴 𝑁 of 𝑀 and 𝑁 which may be non-discrete, even if 𝑀 and 𝑁 are discrete; in this case thederived tensor product is connective and we have 𝜋 𝑖 (H 𝑀 ⊗ H 𝐴 H 𝑁 ) ≅ Tor 𝐴𝑖 ( 𝑀, 𝑁 ) , ( 𝑖 ≥ . Now let
Mod( 𝐴 ) 𝐵 ⊆ Mod( 𝐴 ) denote the kernel of the induction functor 𝜙 ! ∶ Mod( 𝐴 ) → Mod( 𝐵 ) .A.4.1. Lemma.
Let 𝜙 ∶ 𝐴 → 𝐵 be a map of E -ring spectra and denote by 𝐼 the fibre of 𝜙 , considered asan 𝐴 -bimodule. Then the following are equivalent:i) The multiplication 𝐵 ⊗ 𝐴 B → 𝐵 is an equivalence.ii) We have 𝐵 ⊗ 𝐴 𝐼 ≃ 0 .iii) The diagram Mod( 𝐵 ) Mod( 𝐴 ) Mod( 𝐴 ) 𝐵𝜙 ∗ 𝜙 ! ⟂⟂ Hom 𝐴 ( 𝐼, −)inc ⟂⟂ with the right pointing arrows given by 𝜙 ∗ and 𝐼 ⊗ 𝐴 − , respectively, is a stable recollement . Note that iii) in particular contains the statement that
𝐼 ⊗ 𝐴 − ∶ Mod( 𝐴 ) → Mod( 𝐴 ) has image in Mod( 𝐴 ) 𝐵 as indicated. Proof.
For the equivalence between the first two items simply note that 𝐵 ≃ 𝐵 ⊗ 𝐴 𝐴 id ⊗𝜙 ←←←←←←←←←←←←←←←←←←←←→ 𝐵 ⊗ 𝐴 𝐵 is always a right inverse to the multiplication map of 𝐵 . So the latter is an equivalence if the fibre of theformer vanishes. The statement of iii) contains ii), since 𝐼 = 𝐼 ⊗ 𝐴 𝐴 ∈ Mod( 𝐴 ) 𝐵 . Finally, assuming thefirst two items, we first find find that 𝐵 ⊗ 𝐴 𝐼 ⊗ 𝐴 𝑋 ≃ 0 so that 𝐼 ⊗ 𝐴 𝑋 ∈ Mod( 𝐴 ) 𝐵 for all 𝑋 ∈ Mod( 𝐴 ) and the diagram in iii) is well-defined. Furthermore, itfollows that 𝜙 ∗ is fully faithful: For this one needs to check that the counit transformation 𝐵 ⊗ 𝐴 𝑌 → 𝑌 isan equivalence for every 𝐵 -module 𝑌 . But as both sides preserve colimits and 𝐵 generates Mod( 𝐵 ) undercolimits, it suffices to check this for 𝑌 = 𝐵 where we have assumed it. It then follows from the discussionafter Proposition A.2.10 that the diagram Mod( 𝐵 ) Mod( 𝐴 ) 𝜙 ∗ 𝜙 ! ⟂⟂
42 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE can be completed to a stable recollement and the fibre sequences connecting the various adjoints are easilychecked to give the formulae from the statement. (cid:3)
A.4.2.
Definition.
We will call a map 𝜙 ∶ 𝐴 → 𝐵 of E -ring spectra satisfying the equivalent conditions ofthe previous lemma a localisation .A map 𝑅 → 𝑆 between discrete rings, will be called a derived localisation if the associated map H 𝑅 → H 𝑆 is a localisation in the sense above.A.4.3. Remark. i) We warn the reader that it is not true, that a localisation of discrete rings 𝐴 → 𝐵 isgenerally a derived localisation in the sense of Definition A.4.2. The latter condition additionally en-tails that Tor 𝐴𝑖 ( 𝐵, 𝐵 ) = 0 for all 𝑖 > . This is automatic if 𝐴 and 𝐵 are commutative or more generallyif the localisation satisfies an Ore condition, see Corollary A.4.5 below, but can fail in general.ii) The discrete counterpart of Definition A.4.2 for ordinary rings and ordinary tensor products was stud-ied by Bousfield and Kan in [BK72], where they have classified all commutative rings 𝑅 whose mul-tiplication 𝑅 ⊗ ℤ 𝑅 → 𝑅 is an isomorphism, a property which is called solid in [BK72]. We note thatfor a map of connective E -rings 𝐴 → 𝐵 being a localisation implies the solidity of 𝜋 𝐴 → 𝜋 𝐵 buteven for discrete 𝐴 and 𝐵 the converse is not true.iii) Besides localisations, there is another common source of solid ring maps, namely quotients. Evenamong commutative rings these are, however, rarely derived localisations: For example, if 𝐴 is com-mutative then Tor 𝐴 ( 𝐴 ∕ 𝐼, 𝐴 ∕ 𝐼 ) ≅ 𝐼 ⊗ 𝐴 𝐴 ∕ 𝐼 ≅ 𝐼 ∕ 𝐼 . If 𝐼 is finitely generated and 𝐼 ∕ 𝐼 = 0 ,Nakayama’s lemma implies that 𝐼 is principal on an idempotent element in 𝐴 . Thus, if 𝐴 has nonon-trivial idempotents, either 𝐼 = 0 or 𝐼 = 𝐴 . For an example of a quotient map that is a derivedlocalisation, see Example A.4.6 below.iv) Finally, let us mention that any for any open embedding of affine schemes 𝑋 → 𝑌 the restriction map O ( 𝑌 ) → O ( 𝑋 ) is a derived localisation and under mild finiteness assumption this in fact characterisesopen embeddings among affines by [TV07, Lemma 2.1.4].Now, to state the main result of this section we need a bit of terminology. By Lemma A.4.1 a map 𝜙 ∶ 𝐴 → 𝐵 is a localisation if and only if 𝐼 = f ib( 𝐴 → 𝐵 ) ∈ Mod( 𝐴 ) 𝐵 . Given a full subcategory C ⊆ Mod( 𝐴 ) let us write C 𝐵 = C ∩ Mod( 𝐴 ) 𝐵 and say that 𝜙 has perfectly generated fibre, if 𝐼 lies in thesmallest subcategory of Mod( 𝐴 ) 𝐵 containing (Mod 𝜔𝐴 ) 𝐵 and closed under colimits.A.4.4. Proposition.
Let 𝜙 ∶ 𝐴 → 𝐵 be a localisation of E -rings with perfectly generated fibre. Then (Mod 𝜔𝐴 ) 𝐵 ⟶ Mod 𝜔𝐴 𝜙 ! ←←←←←←←←←←→ Mod 𝜔𝐵 is a Karoubi sequence and (Mod c 𝐴 ) 𝐵 ⟶ Mod c 𝐴 𝜙 ! ←←←←←←←←←←→ Mod 𝜙 (c) 𝐵 is a Verdier sequence for every c ⊆ K ( 𝐴 ) .Proof. Combining Theorem A.3.11 and Lemma A.4.1 it only remains to show that
Ind((Mod 𝜔𝐴 ) 𝐵 ) ≃ Mod( 𝐴 ) 𝐵 ) to obtain the first claim. But by [Lur09a, Proposition 5.3.5.11] the former term is equivalent to the smallestsubcategory of Mod( 𝐴 ) 𝐵 containing (Mod 𝜔𝐴 ) 𝐵 closed under colimits, so by assumption it contains 𝐼 . Butthe smallest stable subcategory of Mod( 𝐴 ) containing 𝐼 and closed under colimits is Mod( 𝐴 ) 𝐵 as followsimmediately from the stable recollement of iii) (since 𝐴 generates Mod( 𝐴 ) under colimits).Since the inclusion Mod c 𝐴 → Mod 𝜔𝐴 is a Karoubi equivalence and similarly for 𝐵 , it follows that also 𝜙 ! ∶ Mod c 𝐴 → Mod 𝜙 (c) 𝐵 is a Karoubi projection. But the essential image of this functor is then the Verdierquotient by its kernel, see Corollary A.3.8, and therefore a dense stable subcategory of Mod 𝜔𝐵 . The secondclaim follows from the classification of dense subcategories A.3.2 and Proposition A.3.9. (cid:3) A.4.5.
Corollary.
Given an E -ring 𝐴 and a subset 𝑆 ∈ 𝜋 ∗ ( 𝐴 ) of homogeneous elements satisfying the leftOre condition, for example 𝜋 ∗ ( 𝐴 ) could be (skew-)commutative, then (Mod 𝜔𝐴 ) 𝑆 ⟶ Mod 𝜔𝐴 −[ 𝑆 −1 ] ←←←←←←←←←←←←←←←←←←←←←←←←←←←→ Mod 𝜔𝐴 [ 𝑆 −1 ] is a Karoubi sequence and for every 𝑐 ⊆ K ( 𝐴 )(Mod c 𝐴 ) 𝑆 ⟶ Mod c 𝐴 −[ 𝑆 −1 ] ←←←←←←←←←←←←←←←←←←←←←←←←←←←→ Mod im(c) 𝐴 [ 𝑆 −1 ] ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 143 is a Verdier sequence. Here we have abbreviated (Mod 𝜔𝐴 ) 𝐴 [ 𝑆 −1 ] to (Mod 𝜔𝐴 ) 𝑆 and similarly in the case of finitely presented mod-ule spectra. Proof.
Under the Ore condition the
Mod( 𝐴 ) 𝑆 is generated under colimits by the perfect modules 𝐴 ∕ 𝑠 =cof[ 𝐴 𝑠 ←←←←←→ 𝐴 ] for 𝑠 ∈ 𝑆 , see [Lur17, Lemma 7.2.3.13]. Thus Proposition A.4.4 applies. (cid:3) While the theorem of Thomason-Trobaugh for example implies that for an open embedding 𝑋 → 𝑌 of affine schemes the map O ( 𝑌 ) → O ( 𝑋 ) has perfectly generated fibre, this condition is unfortunately notautomatic for a general localisation, and we do not know of a reformulation in purely ring theoretic terms,even when all constituents rings are discrete. The following counter-example is due to Keller [Kel94], wethank Akhil Mathew for pointing it out to us:A.4.6. Example.
Let 𝑘 be a field and 𝐴 ∶= 𝑘 [ 𝑡, 𝑡 , 𝑡 , ... ] the commutative 𝑘 -algebra obtained from thepolynomial algebra 𝑘 [ 𝑡 ] by adding a formal 𝑖 -order root 𝑡 𝑖 of 𝑡 for every 𝑖 ≥ . Let 𝐼 ⊆ 𝐴 be theideal generated by the 𝑡 𝑖 for 𝑖 ≥ and 𝜙 ∶ 𝐴 → 𝐴 ∕ 𝐼 = 𝑘 the quotient map. We then claim that 𝜙 is alocalisation. To see this, note first that by Lemma A.4.1 it will suffice to show that 𝐼 ⊗ 𝐴 𝑘 ≃ 0 . Now theascending filtration of 𝐼 by the free cyclic submodules 𝑡 𝑖 𝐴 ⊆ 𝐼 gives a presentation of 𝐼 as a filteredcolimit 𝐼 = colim[ 𝐴 𝑡 ←←←←←←←←←←←←←←←→ 𝐴 𝑡 ←←←←←←←←←←←←←←←→ 𝐴 𝑡 ←←←←←←←←←←←←←←←→ ... ] and so 𝐼 ⊗ 𝐴 𝑘 ≃ colim[ 𝑘 ←←←←←←→ 𝑘 ←←←←←←→ ... ] ≃ 0 , cf. Wodzicki [Wod89, Example 4.7(3)].But the fibre of 𝜙 is not perfectly generated. To see this, let 𝑆 ⊆ 𝐴 be the multiplicative set of allelements which are not in 𝐼 and let 𝐴 [ 𝑆 −1 ] be the localisation of 𝐴 at 𝑆 , so that 𝐴 [ 𝑆 −1 ] is a local 𝑘 -algebrawith maximal ideal 𝐼 [ 𝑆 −1 ] . By (a derived version of) Nakayama’s lemma every perfect 𝐴 -module 𝑀 suchthat 𝑀 ⊗ 𝐴 𝑘 ≃ 0 will also satisfy 𝑀 [ 𝑆 −1 ] ≃ 0 . It then follows that also the colimit closure of ( D p ( 𝐴 )) 𝐴 ∕ 𝐼 is contained in D ( 𝐴 ) 𝑆 . But 𝐼 𝑆 ≠ and so these do not contain 𝐼 .A.4.7. Example.
In the situation of Proposition A.4.4
Mod 𝜔𝐴 → Mod 𝜔𝐵 can easily fail to be a Verdier projec-tion. For example, suppose that 𝑘 is a field and 𝐴 = 𝑘 [ 𝑥, 𝑦 ] is the (ordinary) commutative polynomial ringin two variables over 𝑘 . Let 𝑝 ∈ 𝐴 be an element such that spec( 𝐴 ∕ 𝑝 ) ⊆ spec( 𝐴 ) = 𝔸 𝑘 is a reduced andgeometrically irreducible affine curve with a unique a singular point which is a node (e.g., 𝑝 = 𝑦 − 𝑥 − 𝑥 ).Set 𝐵 = 𝐴 [ 𝑝 ] . Then the fibre of the inclusion 𝐴 → 𝐵 is generated by 𝐵 ∕ 𝑝 and so(75) D p ( 𝐴 ) → D p ( 𝐵 ) is a Karoubi projection. However, since spec( 𝐴 ) is smooth one has that K −1 ( 𝐴 ) = 0 while by [Wei01,Lemma 2.3] one has K −1 ( 𝐴 ∕ 𝑝 ) ≅ ℤ . By the localisation sequence in algebraic K -theory it then followsthat coker(K ( 𝐴 ) → K ( 𝐵 )) ≅ ℤ . In particular, the functor (75) is not essentially surjective and hence nota Verdier projection, see Corollary A.1.7.A PPENDIX
B. C
OMPARISONS TO PREVIOUS WORK
In this appendix we compare our construction of Grothendieck-Witt spectra to two constructions in theliterature: Schlichting’s definition of Grothendieck-Witt spectra of rings with invertible [Sch10a], andSpitzweck’s definition of Grothendieck-Witt spaces for stable ∞ -categories with duality [Spi16]. In ourlanguage both cases pertain solely to symmetric Poincaré structures: In the case of Spitzweck’s work,this largely consists of unfolding the definitions, whereas for exact categories, this is enforced by beinginvertible, which makes the quadratic and symmetric Poincaré structures, and also their variants such asthe genuine ones, agree. Spitzweck already gave a comparison between his definition and Schlichting’swhen applied to categories of chain complexes over a ring in which is invertible, and our proof is astraightforward generalisation of his.From Schlichting’s work we then also obtain, that for a ring with involution 𝑅 , in which is invertible,and an invertible 𝑅 -module 𝑀 the canonical map Unimod(
𝑅, 𝑀 ) grp → GW ( D p ( 𝑅 ) , Ϙ s 𝑀 )
44 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE is an equivalence; here,
Unimod(
𝑅, 𝑀 ) denotes the groupoid of unimodular, 𝑀 -valued symmetric bilinearforms on finitely generated projective 𝑅 -modules, symmetric monoidal under orthogonal sum. As explainedin the introduction, this statement is no longer true if is not invertible in 𝑅 : The target has to be replacedby the Grothendieck-Witt space associated to the genuine Poincaré structure, and furthermore, one needsto distinguish between symmetric and quadratic forms. A proof of this more general statement, entirelyindependent from the discussion here, will be given in [HS20] by adapting the parametrised surgery methodsof Galatius and Randal-Williams from [GRW14] to the present setting. Remark.
We do not attempt here a comparison of our work to the recent definitions in [HSV19, Sch19a].For the latter, it requires a more detailed discussion of the genuine quadratic functors and the result is aconsequence of [HS20]; for the former we note that Poincaré ∞ -categories provide examples of Waldhausencategories with genuine duality and that we expect our definition of the KR -functor to coincide with therestriction of that from [HSV19, Corollary 5.18].B.1. Spitzweck’s Grothendieck-Witt space of a stable ∞ -category with duality. We start by comparingour definition to Spitzweck’s from [Spi16]. To this end recall from Section [I].7.2 the forgetful functor
Cat p∞ ⟶ Cat ps∞ , where an object in the target consists of a stable ∞ -category equipped with a perfect biliear functor C op × C op → S 𝑝 . Informally, the functor is given by taking a Poincaré ∞ -category ( C , Ϙ ) to ( C , B Ϙ ) . This functorhas fully faithful left and a right adjoints informally given by taking ( C , B) to ( C , Ϙ qB ) and ( C , Ϙ sB ) , respectively,see Proposition [I].7.2.17. Extracting the duality from a perfect symmetric bilinear functor results in anequivalence Cat ps∞ ⟶ (Cat ex∞ ) hC , where C acts on Cat ex∞ by taking opposites, see Corollary [I].7.2.15. We will use this equivalence and theright adjoint above to regard a stable ∞ -category with duality as a Poincaré ∞ -category throughout thissection.Let us denote by GW ( C , D) Spitzweck’s Grothendieck-Witt space from [Spi16, Definition 3.4], we recallthe definition below. We purpose of this section is to show:B.1.1.
Proposition.
For any perfect symmetric bilinear functor B on a small stable ∞ -category there is acanonical equivalence GW ( C , D B ) ≃ GW ( C , Ϙ sB ) of E ∞ -groups natural in the input. For the definition of GW ( C , D) Spitzweck employs the edgewise subdivision of Segal’s S -construction:Recall the usual S -construction Cat ex∞ → sCat ex∞ given degreewise as the full subcategory of Fun(Ar(Δ 𝑛 ) , C ) spanned by those diagrams 𝜑 with 𝜑 ( 𝑖 ≤ 𝑖 ) ≃ 0 and having the squares 𝜑 ( 𝑖 ≤ 𝑘 ) 𝜑 ( 𝑖 ≤ 𝑙 ) 𝜑 ( 𝑗 ≤ 𝑘 ) 𝜑 ( 𝑖 ≤ 𝑙 ) bicartesian for every set of numbers 𝑖 ≤ 𝑗 ≤ 𝑘 ≤ 𝑙 . The edgewise subdivision S 𝑒 ( C ) of S( C ) is then givenby precomposing this simplicial category with the functor Δ op → Δ op , sending [ 𝑛 ] to [ 𝑛 ] ∗ [ 𝑛 ] op . NowSpitzweck equips the categories Fun(Ar(Δ 𝑛 ∗ (Δ 𝑛 ) op ) , C ) with the duality D 𝑛 induced by conjugation withrespect to flipping the join factors in the source and the given duality D on C ; more formally, let us denotethe internal mapping objects of the cartesian closed category Cat hC ∞ by Fun hC . Then the arrow categoriesinherit dualities via Ar( C , D) = Fun hC ((Δ , f l) , ( C , D)) and S 𝑒𝑛 ( C , D) is defined as the full subcategory of Fun hC ( (Ar(Δ 𝑛 ∗ (Δ 𝑛 ) op ) , f l) , ( C , D) ) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 145 spanned by the diagrams in S 𝑒𝑛 ( C ) = S 𝑛 +1 ( C ) , which is meaningful since the duality carries this subcategoryinto itself. Naturality in 𝑛 then assembles S 𝑒 ( C , D) into a simplicial category with duality, i.e. a functor Δ op → (Cat ex∞ ) hC . Spitzweck sets GW ( C , D) = f ib( | CrS 𝑒 ( C ) hC | → | CrS 𝑒 ( C ) | ) . To start the comparison, we also recall that S 𝑒 ( C ) is canonically equivalent to Q( C ) : There is a canonicalmap TwAr(Δ 𝑛 ) → Ar(Δ 𝑛 ∗ (Δ 𝑛 ) op ) natural in 𝑛 , taking ( 𝑖 ≤ 𝑗 ) to ( 𝑖 ≤ 𝑗 ) , where the subscript indicatesthe join factor. Pullback along this map is easily checked to give an equivalence S 𝑒 ( C ) ⟶ Q( C ) . In degree for example it takes 𝜙 (0 ≤ ) 𝜙 (0 ≤ ) 𝜙 (0 ≤ )0 𝜙 (1 ≤ ) 𝜙 (1 ≤ )0 𝜙 (1 ≤ )0 to 𝜙 (0 ≤ ) 𝜙 (0 ≤ ) 𝜙 (1 ≤ ) . Now we claim that the duality described above associated to D B corresponds exactly to that induced by ( Ϙ sB ) 𝑛 ∶ Q 𝑛 ( C ) op → S 𝑝 . To see this recall that Q 𝑛 ( C , Ϙ ) is a full subcategory of the cotensoring ( C , Ϙ ) TwAr(Δ 𝑛 ) ,and thus to refine the equivalence between the Q - and S -constructions to a hermitian functor it suffices togive a functor 𝑞 𝑛 ∶ TwAr(Δ 𝑛 ) ⟶ Fun h ( (S 𝑒𝑛 ( C ) , Ϙ sD 𝑛 ) , ( C , Ϙ sD ) ) refining the one on underlying categories described above. To this end, we note that assigning to ( 𝑖 ≤ 𝑗 ) ∈TwAr(Δ 𝑛 ) the arrow ( 𝑖 ≤ 𝑗 ) → ( 𝑗 ≤ 𝑖 ) in Ar(Δ 𝑛 ∗ (Δ 𝑛 ) op ) gives a functor 𝑝 𝑛 ∶ TwAr(Δ 𝑛 ) ⟶ Ar(Ar(Δ 𝑛 ∗ (Δ 𝑛 ) op , f l)) C , natural for 𝑛 ∈ Δ ; here the superscript indicates functors strictly commuting the with identifications of theinput categories with their opposites. Pullback along this equivariant functor produces a map TwAr(Δ 𝑛 ) ⟶ Fun hC ( (S 𝑒𝑛 ( C ) , D 𝑛 ) , Fun hC ((Δ , f lip) , ( C , D)) ) . Now, by Remark [I].7.3.4
Ar( C , D) is the underlying ∞ -category with duality of the Poincaré ∞ -category Ar( C , Ϙ sD ) and since the functor (Cat ex∞ ) hC ≃ Cat ps∞ → Cat p∞ , ( C , D) ↦ ( C , Ϙ sD ) is fully faithful by Proposi-tion [I].7.2.17 and preserves products (as a right adjoint), we get a canonical equivalence Fun hC ((S 𝑒𝑛 ( C ) , D 𝑛 ) , Ar( C , D)) ≃ Fun p ((S 𝑒𝑛 ( C ) , Ϙ sD 𝑛 ) , Ar( C , Ϙ sD )) . But by construction of the Poincaré structure on
Ar( C , Ϙ ) evaluation at the source defines a hermitian functor(that is not usually Poincaré) Ar( C , Ϙ ) → ( C , Ϙ ) . In total, we obtain the desired functor 𝑞 𝑛 by composing thethree steps just described. To see that its adjoint (S 𝑒𝑛 ( C ) , Ϙ sD 𝑛 ) → Q 𝑛 ( C , Ϙ sD ) is Poincaré (and thus in fact anequivalence of Poincaré ∞ -categories) it suffices to check this after postcomposition with the Segal maps Q 𝑛 ( C , Ϙ ) → Q ( C , Ϙ ) by Lemma 2.1.5, where it is a simple application of the formula for the duality incotensor categories Proposition [I].6.3.2.The proof of Proposition B.1.1 is now simple: Proof.
The natural equivalence (S 𝑒𝑛 ( C ) , Ϙ sD 𝑛 ) ≃ Q 𝑛 ( C , Ϙ sD )
46 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE constructed above implies that (CrS 𝑒 ( C )) hC ≃ (Cr Q 𝑛 ( C )) hC ≃ Pn Q( C , Ϙ sB ) by Proposition [I].2.2.11 and therefore one obtains GW ( C , D B ) = f ib( | Pn Q( C , Ϙ sB ) | → | Cr Q( C ) | ) . The proposition then follows from the metabolic fibre sequence Corollary 4.1.5. (cid:3)
B.2.
Schlichting’s Grothendieck-Witt-spectrum of a ring with invertible. We now turn to the moredelicate comparison to the classical set-up of exact categories with duality from [Sch10a] and [Sch10b]. Itconsists of an additive (ordinary) category E , equipped with three special types of arrows, namely, inflations,deflations and weak equivalences, satisfying suitable properties, as well as a duality D ∶ E → E op , whichswitches between the inflations and deflations and preserves weak equivalences. A symmetric object in E is then an object 𝑋 ∈ E equipped with a self-dual map 𝜙 ∶ 𝑋 → D 𝑋 . Such a symmetric object issaid to be Poincaré 𝜙 is a weak equivalence. Let us denote by Poi( E , 𝑊 , D) the category whose objectsare the Poincaré objects ( 𝑋, 𝜙 ) in E and the morphisms are the weak equivalences 𝑓 ∶ 𝑋 → 𝑋 ′ such that D( 𝑓 ) 𝜙 ′ 𝑓 = 𝜙 . Similarly, let cor( E , 𝑊 ) denote the subcategory of E containing only the weak equivalencesas morphisms. In both cases we may suppress 𝑊 and D to declutter the notation.To define a Grothendieck-Witt space in this context one again uses the edgewise subdivision S 𝑒𝑛 E ofthe S -construction (see the previous section for a recollection), which in the case at hand inherits an exactstructure with pointwise weak equivalences, and a duality which is defined on objects by sending a diagram 𝑋 to (D 𝑋 )( 𝑖 𝜖 ≤ 𝑗 𝛿 ) = D( 𝑋 ( 𝑗 𝛿 ≤ 𝑖 𝜖 )) . One then defines the associated Grothendieck-Witt space GW ( E , 𝑊 , D ) as the fiber of the map | Poi(S 𝑒 E ) | → | cor(S 𝑒 E ) | . For a ring 𝑅 and a (discrete) invertible 𝑅 -module with involution 𝑀 one can consider the category Proj( 𝑅 ) of finitely generated projective 𝑅 -modules as an exact category (with inflations the split injections and con-flations the split surjections), with the duality D 𝑀 ∶ Proj( 𝑅 ) → Proj( 𝑅 ) op given by 𝑋 ↦ Hom 𝑅 ( 𝑋, 𝑀 ) andweak equivalences the isomorphisms. Under the assumption that is invertible in 𝑅 Schlichting then provesthat GW (Proj( 𝑅 ) , Iso , D 𝑀 ) is naturally equivalent to the group completion of the symmetric monoidal E ∞ -space | Unimod(
𝑅, 𝑀 ) | = | Poi(Proj( 𝑅 ) , Iso , D 𝑀 ) | , see [Sch17, Appendix A].On the other hand, one may also consider the exact category Ch b ( 𝑅 ) of bounded chain complexes in Proj( 𝑅 ) with weak equivalences being the quasi-isomorphisms and with the exact structure and dual-ity induced by those of Proj( 𝑅 ) . Schlichting then shows that the natural map GW (Proj( 𝑅 ) , Iso , D 𝑀 ) → GW (Ch b ( 𝑅 ) , qIso , D 𝑀 ) is an equivalence, see [Sch10b, Proposition 6]; this does not require being invert-ible in 𝑅 .The advantage of working with Ch b ( 𝑅 ) instead of Proj( 𝑅 ) is that it enables one to refine the above def-inition into a Grothendieck-Witt spectrum. For this one considers the shifted duality D [ 𝑛 ] 𝑀 ∶ Ch b ( 𝑅 ) → Ch b ( 𝑅 ) op obtained by post-composing D 𝑀 with the 𝑛 ’th suspension functor sending 𝐶 to the shifted com-plex 𝐶 [ 𝑛 ] defined by 𝐶 [ 𝑛 ] 𝑖 = 𝐶 𝑖 − 𝑛 . Schlichting’s Grothendieck-Witt (pre-)spectrum GW(
𝑅, 𝑀 ) is thendefined as the sequence of spaces GW(
𝑅, 𝑀 ) = (| Poi(Ch b ( 𝑅 ) , qIso , D 𝑀 ) | , | Poi S 𝑒 (Ch b ( 𝑅 ) , qIso , D [1] 𝑀 ) | , | Poi ( (S 𝑒 ) (2) (Ch b ( 𝑅 ) , qIso , D [2] 𝑀 ) ) , … ) with bonding maps induced by the map into the -simplices Poi ( Ch b ( 𝑅 ) , qIso , D [ 𝑛 ] 𝑀 ) ⟶ Poi ( S 𝑒 (Ch b ( 𝑅 ) , qIso , D [ 𝑛 +1] 𝑀 ) ) ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 147 given by the duality preserving functor Ch b ( 𝑅 ) → S 𝑒 (Ch b ( 𝑅 )) that sends a chain complex 𝐶 to the diagram 𝐶 𝐶
00 0 𝐶 [1]0 𝐶 [1]0 of S 𝑒 Ch b ( 𝑅 ) = S Ch b ( 𝑅 ) .B.2.1. Remark.
The Grothendieck-Witt spectrum of [Sch17, Section 5] is defined more generally for dg -categories with duality, in which case the shifted duality requires a more careful construction. When appliedto Ch b ( 𝑅 ) , this construction yields a different, but equivalent model for (Ch b ( 𝑅 ) , D [ 𝑛 ] ) , see the remarksimmediately following [Sch17, (5.1)].We refrain from carrying out the necessarily more elaborate comparison at this level of generality, as thepresent one suffices for our applications in Paper [III].Now recall that D p ( 𝑅 ) is the ∞ -categorical localisation of Ch b ( 𝑅 ) with respect to quasi-isomorphisms.The duality D [ 𝑛 ] 𝑀 then induces the duality on D p ( 𝑅 ) associated to ( Ϙ s 𝑀 ) [ 𝑛 ] , which we will also denote by D [ 𝑛 ] 𝑀 .The localisation functor Ch b ( 𝑅 ) → D p ( 𝑅 ) then determines a compatible collection of duality preservingfunctors (S 𝑒 ) ( 𝑛 ) (Ch b ( 𝑅 )) (S 𝑒 ) ( 𝑛 ) ( D p ( 𝑅 )) Q ( 𝑛 ) ( D p ( 𝑅 )) , see the discussion after Proposition B.1.1. Using Proposition [I].2.2.11 it induces a compatible collectionof maps(76) | Poi((S 𝑒 ) ( 𝑛 ) (Ch b ( 𝑅 ) , qIso , D [ 𝑛 ] 𝑀 )) | Pn(Q ( 𝑛 ) ( D p ( 𝑅 ) , ( Ϙ 𝑠𝑀 ) [ 𝑛 ] )) , which fit together to give a natural map of spectra(77) GW(
𝑅, 𝑀 ) GW( D p ( 𝑅 ) , Ϙ s 𝑀 ) . Our goal in this subsection is to prove:B.2.2.
Proposition.
Let 𝑀 be a (discrete) invertible 𝑅 -module with involution, such that is invertible in 𝑅 . Then the map (77) is an equivalence of spectra. We will also show, that Schlichting’s definition of the Grothendieck-Witt space of an exact category inwhich is invertible agrees with ours. To this end let E be an exact category with duality D and weak equiv-alences 𝑊 . We will say that E is homotopically sound if the collection of deflations and weak equivalenceson E exhibits it as a category of fibrant objects in the sense of [Cis19, Definition 7.5.7]. Since the duality D preserves weak equivalences and switches between inflations and deflations this is also equivalent to sayingthat the collection of inflations and weak equivalences on E exhibits it as a category of cofibrant objects.In this case we will denote by E [ 𝑊 −1 ] the ∞ -categorical localisation of E with respect to the collection ofweak equivalences.B.2.3. Proposition.
Suppose that E is a homotopically sound, exact category with duality D and weakequivalences 𝑊 , in which is invertible. Suppose further that E [ 𝑊 −1 ] is stable. Then the natural map | Poi( E , 𝑊 , D) | → Pn( E [ 𝑊 −1 ] , Ϙ sD ) is an equivalence.
48 CALMÈS, DOTTO, HARPAZ, HEBESTREIT, LAND, MOI, NARDIN, NIKOLAUS, AND STEIMLE
Proof.
Put E [ 𝑊 −1 ] = E ∞ for readibility. Consider then the commutative square(78) Poi( E , 𝑊 , D) Pn( E ∞ , Ϙ 𝑠 D ) 𝑊 Cr E ∞ where the vertical functors are forgetful. By [Cis19, Corollary 7.6.9] the bottom horizontal map becomesan equivalence upon realisation. By Quillen’s theorem B it will hence suffice to show that for every 𝑋 ∈ E the map(79) Poi( E , 𝑊 , D) × 𝑊 𝑊 ∕ 𝑋 Pn( E ∞ , Ϙ sD ) × Cr E ∞ (Cr E ∞ ) ∕ 𝑋 is an equivalence after realisation. Let I 𝑋 ⊆ 𝑊 ∕ 𝑋 be the full subcategory spanned by the deflations 𝑌 ↠ 𝑋 that are also weak equivalences. We claim that the map(80) 𝑖 ∶ Poi( E , 𝑊 , D) × 𝑊 I 𝑋 ⟶ Poi( E , 𝑊 , D) × 𝑊 𝑊 ∕ 𝑋 induces an equivalence on realisations. To see this, let 𝑋 → 𝑋 𝐼 → 𝑋 × 𝑋 be a path object for 𝑋 , whoseexistence is guaranteed by our assumption that E is a category of fibrant objects with respect to deflations.Construct a functor 𝑞 ∶ Poi( E , 𝑊 , D) × 𝑊 𝑊 ∕ 𝑋 → Poi( E , 𝑊 , D) × 𝑊 I 𝑋 by sending ( 𝑞 ∶ 𝑌 → D 𝑌 , 𝑌 → 𝑋 ) to (Dpr ◦ 𝑞 ◦ pr , 𝑌 × 𝑋 𝑋 𝐼 ↠ 𝑋, )) , where pr ∶ 𝑌 × 𝑋 𝑋 𝐼 → 𝑌 is the projectionto the first component. The natural map ( 𝑞, 𝑌 → 𝑋 ) → (Dpr ◦ 𝑞 ◦ pr , 𝑌 × 𝑋 𝑋 𝐼 ) induced by the structuremap 𝑋 → 𝑋 𝐼 then determines natural transformations id ⇒ 𝑞 ◦ 𝑖 and id ⇒ 𝑖 ◦ 𝑞 , showing that (80) is anequivalence after realisation. It will hence suffice to show that the map(81) | Poi( E , 𝑊 , D) × 𝑊 I 𝑋 | Pn( E ∞ , Ϙ 𝑠 ) × Cr E ∞ (Cr E ∞ ) ∕ 𝑋𝑖 is an equivalence. We now observe that the left vertical map in (78) is a right fibration classified by thefunctor 𝑋 ↦ Hom 𝑊 ( 𝑋, D 𝑋 ) 𝐶 ≃ Hom 𝑊 ( 𝑋, D 𝑋 ) hC , recall that Set ⊂ S is closed under limits. Similarly,the right vertical map is classified by 𝑋 ↦ Map Cr E ∞ ( 𝑋, D 𝑋 ) hC ; since Cr( E ∞ ) ∕ 𝑋 is contractible we willnot need the full statement here, but rather only that the fibre of Pn( E ∞ , Ϙ 𝑠 D ) → Cr( E ∞ ) over a point 𝑋 isgiven by Map Cr E ∞ ( 𝑋, D 𝑋 ) hC . This follows from the general fact that for a C -space 𝑋 ∈ S hC the fibreof 𝑋 hC → 𝑋 over some 𝑥 ∈ 𝑋 may be computed as Map 𝑥 (S 𝜎 , 𝑋 ) hC from the fibre sequence Map 𝑥 (S 𝜎 , 𝑋 ) ⟶ Map(∗ , 𝑋 ) ⟶ Map(C , 𝑋 ) . Since total spaces of right fibrations are given as the opposites of the colimits in
Cat ∞ of their classifiedfunctors by [Lur09a, Corollary 3.3.4.6], and thus their realisation as the colimits in S , we may identify (81)with the natural map(82) colim [ 𝑌 ↠ 𝑋 ]∈ I op 𝑋 Hom 𝑊 ( 𝑌 , D 𝑌 ) hC ⟶ Hom Cr E ∞ ( 𝑋, D 𝑋 ) hC in S . Now, since is assumed invertible in E , multiplication by acts invertibly on the E ∞ -groups Hom E ( 𝑌 , D 𝑌 ) ,which is of course an ordinary abelian group, and Hom E ∞ ( 𝑌 , D 𝑌 ) . It follows that the norm map identifiestheir homotopy fixed points with their homotopy orbits (in E ∞ -groups). In particular, the homotopy fixedpoint functor commutes with colimits of E ∞ -groups in which is invertible. Now note that the category I 𝑋 admits products (given by fibre products in E over 𝑋 ), and so I op 𝑋 is sifted in the ∞ -categorical sense. Sincethe forgetful functor from E ∞ -groups to spaces preserves sifted colimits by [Lur17, Proposition 1.4.3.9],we conclude that colim [ 𝑌 ↠ 𝑋 ]∈ I op 𝑋 Hom 𝑊 ( 𝑌 , D 𝑌 ) hC ≃ [ colim [ 𝑌 ↠ 𝑋 ]∈ I op 𝑋 Hom 𝑊 ( 𝑌 , D 𝑌 ) ] hC , ERMITIAN K-THEORY FOR STABLE ∞ -CATEGORIES II: COBORDISM CATEGORIES AND ADDITIVITY 149 so it suffices to establish that(83) colim [ 𝑌 → 𝑋 ]∈ I op 𝑋 Hom 𝑊 ( 𝑌 , D 𝑌 ) Hom Cr E ∞ ( 𝑋, D 𝑋 ) is an equivalence. Since I op 𝑋 is sifted the map induced by the diagonal colim [ 𝑌 ↠ 𝑋 ]∈ I op 𝑋 Hom 𝑊 ( 𝑌 , D 𝑌 ) colim [ 𝑌 ↠ 𝑋,𝑍 ↠ 𝑋 ]∈ I op 𝑋 × I op 𝑋 Hom 𝑊 ( 𝑌 , D 𝑍 ) is an equivalence. We have thus reduced to showing that the natural map(84) colim [ 𝑌 ↠ 𝑋,𝑍 ↠ 𝑋 ]∈ I op 𝑋 × I op 𝑋 Hom 𝑊 ( 𝑌 , D 𝑍 ) Hom Cr E ∞ ( 𝑋, D 𝑋 ) is an equivalence. Since the duality switches inflations and deflations we may rewrite this as(85) colim [ 𝑌 ↠ 𝑋, D 𝑋 ↪ 𝑍 ′ ]∈ I op 𝑋 × J D 𝑋 Hom 𝑊 ( 𝑌 , 𝑍 ′ ) Hom Cr E ∞ ( 𝑋, D 𝑋 ) , where J D 𝑋 denotes the subcategory of 𝑊 D 𝑋 ∕ spanned by the inflations. Now this last map is an equiva-lence on general grounds; it is one formula for derived mapping spaces in categories of fibrant/cofibrantobjects [Cis10, Proposition 3.23]. (cid:3) B.2.4.
Lemma.
Suppose that E is homotopically sound. Then for every 𝑛 ≥ the exact category withweak equivalences S 𝑛 E is homotopically sound and the natural functor S 𝑛 E → S 𝑛 ( E [ 𝑊 −1 ]) exhibits the ∞ -category S 𝑛 ( E [ 𝑊 −1 ]) as the localisation of S 𝑛 E with respect to the pointwise weak equivalences.Proof. We note that S 𝑛 E is equivalent to the category of sequences of inflations 𝑋 ↪ 𝑋 ↪ ... ↪ 𝑋 𝑛 with the inflations in S 𝑛 E being the Reedy inflations. It is then standard that if E is category of cofi-brant objects then the collection of Reedy inflations exhibit S 𝑛 E as a category of cofibrant objects, seee.g. [Cis19, Theorem 7.4.20 & Example 7.5.8]. On the other hand, S 𝑛 ( E [ 𝑊 −1 ]) is equivalent to the ∞ -category Fun(Δ 𝑛 , E [ 𝑊 −1 ]) of sequences of 𝑛 −1 composable maps in E [ 𝑊 −1 ] . The fact that Fun(Δ 𝑛 , E [ 𝑊 −1 ]) is the ∞ -categorical localisation of the category of Reedy sequences of inflations then follows from [Cis19,Theorems 7.5.18 & 7.6.17]. (cid:3) Proof of Proposition B.2.2.
Let us denote the duality induced by 𝑀 simply by D , and the induced dual-ity on the 𝑟 -fold S -construction by D ( 𝑟 ) . Applying Proposition B.2.3 to the levels of the multisimplicialexact category with duality (S 𝑒 ) ( 𝑟 ) (Ch b ( 𝑅 ) , qIso , D [ 𝑟 ] 𝑀 ) , which is possible by Lemma B.2.4, we obtain anequivalence of Schlichting’s GW(
𝑅, 𝑀 ) to the (pre-)spectrum formed by the sequence ( Pn( D p ( 𝑅 ) , Ϙ sD ) , | Pn(S 𝑒 ( D p ( 𝑅 )) , Ϙ sD [1](1) ) | , | Pn((S 𝑒 ) (2) ( D p ( 𝑅 )) , Ϙ sD [2](2) ) | , … ) But the latter agrees with ( Pn( D p ( 𝑅 ) , Ϙ sD ) , | Pn Q( D p ( 𝑅 ) , ( Ϙ sD ) [1] ) | , | Pn Q (2) ( D p ( 𝑅 ) , ( Ϙ sD ) [2] ) | , … ) = GW( D p ( 𝑅 ) , Ϙ sD ) termwise by the discussion following Proposition B.1.1, and one readily checks that also the bonding mapscorrespond. (cid:3) B.2.5.
Corollary.
Let E be a homotopically sound, exact category with duality D and weak equivalences 𝑊 , in which is invertible, and such that E [ 𝑊 −1 ] is stable. Then there is a canonical equivalence GW ( E , 𝑊 , D) ≃ GW ( E [ 𝑊 −1 ] , Ϙ sD ) . Proof.
From Proposition B.2.3 and Lemma B.2.4 we find that the defining map | Poi(S 𝑒 E ) | → | cor(S 𝑒 E ) | is equivalently given by | Pn(S 𝑒 ( E [ 𝑊 −1 ]) , Ϙ sD (1) ) | → | Cr(S 𝑒 ( E [ 𝑊 −1 ]) | . But the discussion following Proposition B.1.1 identifies this further with | Pn Q( E [ 𝑊 −1 ] , Ϙ sD ) | ⟶ | Cr Q( E [ 𝑊 −1 ]) | and so Corollary 4.1.5 gives the claim. (cid:3)
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