Noncommutative CW-spectra as enriched presheaves on matrix algebras
aa r X i v : . [ m a t h . A T ] J a n Noncommutative CW-spectra as enrichedpresheaves on matrix algebras
Gregory Arone ∗ Stockholm [email protected] Ilan Barnea † Haifa [email protected] M. Schlank ‡ Hebrew [email protected] 27, 2021
Abstract
Motivated by the philosophy that C ∗ -algebras reflect noncommutativetopology, we investigate the stable homotopy theory of the (opposite) cate-gory of C ∗ -algebras. We focus on C ∗ -algebras which are non-commutativeCW-complexes in the sense of [ELP]. We construct the stable ∞ -categoryof noncommutative CW-spectra, which we denote by NSp . Let M be thefull spectral subcategory of NSp spanned by “noncommutative suspensionspectra” of matrix algebras. Our main result is that
NSp is equivalent tothe ∞ -category of spectral presheaves on M .To prove this we first prove a general result which states that anycompactly generated stable ∞ -category is naturally equivalent to the ∞ -category of spectral presheaves on a full spectral subcategory spanned bya set of compact generators. This is an ∞ -categorical version of a resultby Schwede and Shipley [ScSh1]. In proving this we use the language ofenriched ∞ -categories as developed by Hinich [Hin2, Hin3].We end by presenting a “strict” model for M . That is, we define acategory M s strictly enriched in a certain monoidal model category ofspectra Sp M . We give a direct proof that the category of Sp M -enrichedpresheaves M op s → Sp M with the projective model structure models NSp and conclude that M s is a strict model for M . Contents ∗ Supported in part by the Swedish Research Council, grant number 2016-05440 † Supported by ISF 786/19 ‡ Supported by ISF 1588/18 and BSF 2018389 The ∞ -category of noncommutative CW-complexes 83 Enriched infinity categories 104 Stable ∞ -categories and spectral presheaves 175 Stabilization of categories 196 The ∞ -category of noncommutative CW-spectra 25 The celebrated Gelfand theorem gives a contravariant equivalence between thecategories of locally compact Hausdorff spaces and commutative C ∗ -algebras.This correspondence led to the point of view that C ∗ -algebras are noncommu-tative generalizations of topological spaces. The study of C ∗ -algebras from thisperspective is the subject of noncommutative geometry and topology. In thispaper we study noncommutative stable homotopy theory, i.e., the stable homo-topy category of the opposite of the category of C ∗ -algebras. In doing this weare continuing the investigations of Østvær [Ost] and Mahanta [Mah1], amongothers.To be a little more specific, in this paper we construct the ∞ -category ofnoncommutative CW-spectra, which we denote NSp , and show that
NSp is equiv-alent to the category of spectral presheaves over a spectrally enriched category M . The objects of M are noncommutative suspension spectra of matrix al-gebras, and its morphisms are mapping spectra between matrix algebras. In acompanion paper [ABS2] we analyze the category M in considerable detail. Inthat paper we introduce a rank filtration of M , describe the subquotients ofthe rank filtration, and use it, in particular, to give an explicit model for therationalization of NSp .We construct the ∞ -category of noncommutative CW-spectra, as the stabi-lization of the ∞ -category of noncommutative CW-complexes. Our constructionof the ∞ -category of noncommutative CW-complexes mimics Lurie’s construc-tion of the ∞ -category of “ordinary” CW-complexes as the ind-completion ofthe ∞ -category of finite CW-complexes [Lur1]. The latter is considered an ∞ -category by first viewing it as a topological category in the obvious way andthen taking the topological nerve [Lur1, Section 1.1.5]. (By a topological cat-egory in this paper we mean a category enriched in the category of compactlygenerated weak Hausdorff spaces.)In more detail, consider the class of C ∗ -algebras called “noncommutativeCW-complexes” in [ELP, Section 2.4]. These are algebras generated from thefinite dimensional matrix algebras by a finite inductive procedure, generalizingthe construction of finite CW-complexes from S in the commutative case. Wewill therefore call them finite noncommutative CW-complexes in this paper.Finite noncommutative CW-complexes have been studied in several places(for instance [Ped] and [Die]). In Section 2 we define the topological category f finite noncommutative CW-complexes to be the opposite of the topologicalcategory whose objects are the C ∗ -algebras which are noncommutative CW-complexes and whose hom-spaces are given by taking the topology of pointwisenorm convergence on the sets of ∗ -homomorphisms. We define the ∞ -categoryof finite noncommutative CW-complexes by taking the topological nerve of thistopological category. We denote both versions of this category by NCW f . Wenow define the ∞ -category of noncommutative CW-complexes to be the ind-completion of NCW f . This will be our generalization of the ∞ -category of spacesand we denote it by NCW . It can be shown that the ∞ -category NCW f is pointed,essentially small and admits finite colimits, so NCW is a pointed compactly gen-erated ∞ -category.Let NSp := Sp ( NCW ) be the ∞ -category of noncommutative CW-spectra , i.e.,the stabilization of NCW . By construction
NSp is a stable ∞ -category. In partic-ular, it is enriched and tensored over the ∞ -category of “ordinary” spectra Sp .There is a suspension-spectrum functor from noncommutative spaces to non-commutative spectra, which we denote Σ ∞ NC : NCW → NSp . It can be shown (seeSection 2) that the (maximal) tensor product of C ∗ -algebras induces a closedsymmetric monoidal structure on both NCW and
NSp , such that Σ ∞ NC : NCW −→ NSp is symmetric monoidal. Our main result is a presentation of
NSp as a categoryof spectral presheaves over a full spectral subcategory spanned by an explicitset of generators.In order to prove this we first prove a general result about presenting acompactly generated stable ∞ -category as a category of spectral presheavesover a full spectral subcategory spanned by a set of compact generators. Such aresult was proven by Schwede and Shipley [ScSh1] using model categories (seealso [GM] for the same result under more general hypotheses). However, in thispaper we need a more general result formed in the language of ∞ -categories. Toobtain this we use the formalism of enriched ∞ -categories developed by Hinich[Hin2, Hin3] and reviewed in Section 3. We can formulate our result as follows: Theorem 1.1 (Theorem 4.1) . Let D be a cocomplete stable ∞ -category. Sup-pose that there is a small set C of compact objects in D , that generates D undercolimits and desuspensions. Thinking of D as left-tensored over the ∞ -categoryof spectra Sp , we let C be the full Sp -enriched subcategory of D spanned by C .Then D is naturally equivalent to the ∞ -category of spectral presheaves on C ,denoted P Sp ( C ) . There is also a monoidal version of the Theorem 1.1, given in Theorem 4.3.The classical stable infinity category of spectra Sp is generated by a singleobject, the sphere spectrum S , and this is closely related to the fact that Sp canbe identified with the category of S -modules. By contrast NSp requires infinitelymany generators. Let M n be the algebra of n × n matrices over C . The algebras { M n | n = 1 , , . . . } are the finite-dimensional simple C ∗ -algebras. Collectively,they play the same role in NCW as S in the usual category of CW-complexes.The suspension spectra { Σ ∞ NC M n | n = 1 , , . . . } are compact objects of NSp ,and they generate
NSp under ∞ -colimits and desuspensions. Let M be the full Sp -enriched subcategory of NSp spanned by { Σ ∞ NC M n | n = 1 , , . . . } .3or every n, m ≥ M n ⊗ M m ≃ M n × m , so the set { M n | n =1 , , . . . } is closed under the tensor product. Since Σ ∞ NC is monoidal, we see that { Σ ∞ NC M n | n = 1 , , . . . } is also closed under the tensor product. It follows thatthe Sp -enriched category M acquires a symmetric monoidal structure from NSp .This monoidal structure induces a symmetric monoidal structure on the ∞ -category of spectral presheaves P Sp ( M ) (Day convolution). Using the monoidalversion of Theorem 1.1 we obtain Theorem 1.2 (Theorem 6.2) . The symmetric monoidal ∞ -category NSp isnaturally equivalent to the symmetric monoidal ∞ -category P Sp ( M ) of spectralpresheaves on M . Thus, understanding the spectral ∞ -category M should help us understandthe ∞ -category NSp . The objects of M are in one-one correspondence withnatural numbers and the monoidal product acts as multiplication. Given naturalnumbers k, l , we denote the corresponding mapping spectrum by S k,l S k,l := Hom NSp (Σ ∞ NC M k , Σ ∞ NC M l ) . One can describe S k,l explicitly as follows. First, let us define a functor G k,l from finite pointed spaces to pointed spaces by the formula G k,l ( X ) = Map NCW f ( M k , X ∧ M l ) . Since the pointed ∞ -category NCW f has finite colimits, it is tensored over finitespaces and enriched over spaces. The spectrum S k,l is the stabilization of G k,l ,i.e., S k,l is the spectrum given by the sequence { G k,l ( S ) , G k,l ( S ) , . . . } . In thecompanion paper [ABS2] we undertake a detailed study of the spectra S k,l andthe structure of M .We end the paper by constructing a “strict” version of M . Namely, let Sp M be the category of continuous pointed functors from finite pointed CW-complexes to topological spaces, endowed with the stable model structure. Thisis a symmetric monoidal model category, that models the ∞ -category of spectra[Lyd, MMSS]. In Definition 6.3, we define a symmetric monoidal category,strictly enriched in Sp M , denoted M s . We give a direct proof of the following,which can be considered a strict version of Theorem 1.2: Theorem 1.3 (Theorem 6.7) . The category of Sp M -enriched functors M op s → Sp M with the projective model structure and Day convolution is a symmetricmonoidal model category that models the symmetric monoidal ∞ -category NSp . In Definition 3.6 we define the notion of enriched ∞ -localization. This takesa category strictly enriched in a monoidal model category, and produces an ∞ -category enriched in the ∞ -localization of this model category (see also Remark2.1). A consequence of Theorem 1.3 is that the enriched ∞ -localization of M s is equivalent to M . Remark . Using the work of Blom and Moerdijk [BM], it is possible to definea model category structure on the opposite of the pro-category of separable C ∗ -algebras, that models NCW . This model structure is a right Bousfield localization4f the model structure presented in [BJM]. We might then be able to use knownresults on stable model categories to prove a similar result to Theorem 1.3. Wedid not pursue this approach in this paper.
An alternative definition, via nonabelian derived categories
We will now digress to describe another natural way of defining a noncommu-tative analogue to the ∞ -category of pointed spaces of Lurie. It relies evenmore on ∞ -categorical constructions, and we do not develop it in this paper.Namely, we can do so using the concept of a nonabelian derived category (see[Lur1, Section 5.5.8]). If C is a small ∞ -category with finite coproducts, Luriedefines the nonabelian derived category of C , denoted P Σ ( C ), as the ∞ -categoryobtained from C by formally adjoining colimits of sifted diagrams. Looselyspeaking sifted diagrams are generated by filtered diagrams and the simplicialdiagram ∆ op . Taking C = Fin ∗ to be the category of finite pointed sets, weobtain the ∞ -category of pointed spaces, that is, we have a natural equivalence P Σ ( Fin ∗ ) ≃ Top.Under the Gelfand correspondence, the finite pointed sets correspond to thefinite dimensional commutative C ∗ -algebras. We thus denote by NFin ∗ the fullsubcategory of NCW f spanned by the finite dimensional C ∗ -algebras (which arejust finite products of matrix algebras). We can now define a noncommutativeanalogue to the ∞ -category of pointed spaces to be the nonabelian derivedcategory of NFin ∗ : NCW := P Σ ( NFin ∗ ) . We have natural inclusions
NFin ∗ ֒ → NCW f ֒ → Ind(
NCW f ) = NCW and the ∞ -category NCW admits sifted colimits, so by the universal property wehave an induced functor
NCW = P Σ ( NFin ∗ ) → NCW , that commutes with sifted colimits. We do not know if this functor is an equiv-alence. This is true iff for every n ≥ X in NCW f the natural mapcolim ∆ op Map
NCW ( M n , X ) → Map
NCW ( M n , colim NCW ∆ op X )is an equivalence. We know how to prove this last assertion when X is a simpli-cial object in NFin ∗ or X has the form Y ⊗ M k for k ≥ Y is a simplicialobject in NCW f composed of commutative algebras.What we do know is that the induced map on stabilizations: Sp ( NCW ) → Sp ( NCW ) =
NSp is an equivalence. To see this note that, by a similar reasoning as in the be-ginning of Section 6, we have that M := { Σ ∞ M i | i ∈ N } generates Sp ( NCW )5nder small colimits. Thus, by Theorem 6.2, it is enough to show that for every k, l ≥ Sp ( NCW ) (Σ ∞ M k , Σ ∞ M l ) → Hom
NSp (Σ ∞ M k , Σ ∞ M l )is an equivalence. We can define the functor G k,l from finite pointed spaces topointed spaces by G k,l ( X ) := Map NCW ( M k , X ∧ M l ) . The stabilization of G k,l is the mapping spectrumMap Sp ( NCW ) (Σ ∞ M k , Σ ∞ M l ) . It is thus enough to show that the induced natural transformation G k,l → G k,l is an equivalence. The functor G k,l clearly commutes with sifted colimits andtherefore it is equivalent to the (derived) left Kan extension of G k,l | Fin ∗ alongthe inclusion Fin ∗ ⊆ S ∗ . We prove in [ABS2] that the same holds for the functor G k,l . Therefore it is enough to show that the restriction G k,l | Fin ∗ → G k,l | Fin ∗ isan equivalence. But for every [ t ] ∈ Fin ∗ we have G k,l ([ t ]) = Map P Σ ( NFin ∗ ) ( M k , [ t ] ∧ M l ) ≃ Map
NFin ∗ ( M k , M tl ) =Map NCW f ( M k , M tl ) ≃ Map
Ind(
NCW f ) ( M k , [ t ] ∧ M l ) = G k,l ([ t ]) , so we are done. Note that since the main results in this paper (and in [ABS2])concern the stabilization of NCW , they apply equally well to the stabilization ofthe alternative model
NCW . Comparison with previous work
We end the introduction by relating the ∞ -categories NCW f and NCW constructedhere with different ∞ -categories constructed in [Mah1]. For more detail see Sec-tion 2. In [Mah1], Mahanta constructed the ∞ -category SC ∗∞ as the topologicalnerve of the topological category of all separable C ∗ -algebras, with the map-ping spaces given by the topology of pointwise norm convergence on the setsof ∗ -homomorphisms. He called ( SC ∗∞ ) op the ∞ -category of pointed compactmetrizable noncommutatives spaces . He then defined the ∞ -category N S ∗ as theind-completion of ( SC ∗∞ ) op , and called it the ∞ -category of pointed noncommu-tative spaces .It can be shown that our ∞ -category NCW f is a full subcategory of ( SC ∗∞ ) op and the inclusion commutes with finite colimits. It follows that our ∞ -category NCW is a coreflective full subcategory of N S ∗ , and thus the inclusion admits aright adjoint: i : NCW ⇄ N S ∗ : R. We call a morphism g : X → Y in N S ∗ a weak homotopy equivalence if for every n ≥ g ∗ : Map N S ∗ ( M n , X ) → Map N S ∗ ( M n , Y )6s an equivalence of spaces. This is analogous to weak homotopy equivalencesbetween topological spaces. The counit of the adjunction above i ◦ R → Id N S ∗ isa levelwise weak equivalence, and thus can be thought of as a CW approximationto elements in N S ∗ . If X and Y are noncommutative CW-complexes then g is aweak equivalence iff it is an equivalence in N S ∗ .Informally speaking, since the equivalences in ( SC ∗∞ ) op are homotopy equiva-lences of C ∗ -algebras, the category N S ∗ = Ind(( SC ∗∞ ) op ) is somewhat analogousto the infinity category modeled by the Strøm model structure on topologicalspaces [Str], in which the weak equivalences are the homotopy equivalences.The category NCW constructed here is analogous to the infinity category mod-eled by the Quillen model structure on topological spaces, in which the weakequivalences are the weak homotopy equivalences.
Section by section outline of the paper
In Section 2 we define the ∞ -category NCW : a noncommutative analogue of the ∞ -category of pointed spaces.In Section 3 we review the theory of enriched ∞ -categories, as developed byHinich [Hin2, Hin3]. In particular we state the enriched Yoneda lemma for ∞ -categories. We also present a way to pass from model categories to ∞ -categoriesis the enriched setting.In Section 4 we review the notion of a stable ∞ -category and show that acompactly generated stable ∞ -category is equivalent to the category of spectralpresheaves on a full subcategory spanned by a set of compact generators.In Section 5 we review the process of stabilizing an ∞ -category. We use theframework established by Lurie in [Lur2, Section 1.4]. We also review how asimilar procedure can be applied to an ordinary topologically enriched category,and compare the strict and the ∞ -categorical versions of stabilization.In Section 6 we define the category of non-commutative CW-spectra NSp as the stabilization of the category
NCW . We identify the suspension spectraof matrix algebras as an explicit set of generators of
NSp . Letting M be thefull subcategory of NSp spanned by matrix algebras, we conclude that
NSp is(monoidally) equivalent to P Sp ( M ), the category of spectral presheaves on M .We give a strict model for M , denoted M s , as a category enriched over a Quillenmodel category of spectra Sp M . We also show that the category of Sp M -enrichedfunctors M op s → Sp M with the projective model structure models the ∞ -category NSp and conclude that M is equivalent to the enriched ∞ -localization of M s . Acknowledgements
We are grateful to Vladimir Hinich for explaining to us his theory of enrichedinfinity categories and its relevance to our work.7
The ∞ -category of noncommutative CW-complexes In this section we define a noncommutative analogue of the ∞ -category ofpointed spaces defined by Lurie [Lur1].Let SC ∗ (resp. CSC ∗ ) denote the category of all (resp. commutative) sepa-rable C ∗ -algebras and ∗ -homomorphisms. Following the common convention inthe field, the term C ∗ -algebra or ∗ -homomorphism will always mean non-unital .The Gelfand correspondence implies that the functor X C ( X ) : CM ∗ −→ CSC ∗ op that assigns to every pointed compact metrizable space X the commutativeseparable C ∗ -algebra of continuous complex valued functions on X that vanishat the basepoint, is an equivalence of categories. It is thus natural to regard SC ∗ op as the category of noncommutative pointed compact metrizable spaces.Consider CM ∗ as a topologically enriched category, where for every X, Y ∈ CM ∗ we endow the set of pointed continuous maps CM ∗ ( X, Y ) with the compactopen topology . Now we take the topological nerve [Lur1, Section 1.1.5] of thistopological category and obtain the ∞ -category ( CM ∗ ) ∞ . It is well-known that( CM ∗ ) ∞ admits finite ∞ -colimits and that ∞ -pushouts can be calculated usingthe standard cylinder object.Let us construct the ∞ -category of pointed spaces in a way that admitsa natural generalization to the noncommutative case. We denote by CW f ∗ thesmallest full subcategory of CM ∗ that contains S and is closed under finitehomotopy-colimits using the standard cylinder object. Thus CW f ∗ is the topo-logical category of finite pointed CW-complexes. We will also consider CW f ∗ asan ∞ -category, by applying the coherent nerve functor to it. We will use thesame notation CW f ∗ to indicate both the ordinary (topologically enriched) andthe ∞ -categorical incarnation of the category, trusting that it is clear from thecontext which is meant. The ∞ -category of pointed spaces can be defined asthe ∞ -categorical Ind construction of CW f ∗ . Note that under the Gelfand corre-spondence, S corresponds to C , which is the only nonzero finite dimensionalsimple commutative C ∗ -algebra.We now turn to the noncommutative analogue. We first recall from [Mah1,Section 2.1] the construction of the ∞ -category SC ∗∞ . Consider SC ∗ as a topo-logically enriched category, where for every A, B ∈ SC ∗ we endow the set of ∗ -homomorphisms SC ∗ ( A, B ) with the topology of pointwise norm convergence.Now we take the topological nerve of this topological category and obtain the ∞ -category SC ∗∞ . It is shown in [Mah1, Section 2.1] that SC ∗∞ is (essentially)small, pointed, and finitely complete. Remark . Recall that any relative category, that is a pair ( C , W ) consisting ofa category C an a subcategory W ⊆ C , has a canonically associated ∞ -category C ∞ , obtained by formally inverting the morphisms in W , in the infinity cate-gorical sense. There is also a canonical localization functor C → C ∞ satisfyinga universal property. We refer the reader to [Hin1] for a thorough account,and also to the discussion in [BHH, Section 2.2]. We refer to C ∞ as the ∞ -8ocalization of C (with respect to W ). If C is a model category or a (co)fibrationcategory, we always take W to be the set of weak equivalences in C .Using ∞ -localization, there is another natural way of considering separa-ble C ∗ -algebras as an ∞ -category. There is a well known notion of homotopyequivalence between C ∗ -algebras. We can consider SC ∗ as a relative category,with the weak equivalences given by the homotopy equivalences, and take its ∞ -localization. This is the point of view taken, for instance, in [AG, Uuy]. Itfollows from [BJM, Proposition 3.17] that we obtain an ∞ -category equivalentto SC ∗∞ . Remark . It is well-known that SC ∗ is cotensored over the category of pointedfinite CW-complexes [AG]. If K is a finite pointed CW-complex and A ∈ SC ∗ then the cotensoring of A and K is given by the C ∗ -algebra of pointed continuousfunctions from K to A . One can define finite homotopy limits in SC ∗ usingthis cotensoring. Consequently, the ∞ -pullbacks in SC ∗∞ can be calculated ashomotopy pullbacks using the standard path object [Mah1, Proposition 2.7]. Definition 2.3.
We denote by
NCW f the opposite of the smallest full subcategoryof SC ∗ that contains the nonzero finite dimensional simple algebras in SC ∗ (whichare just the matrix algebras over C ) and is closed under finite homotopy-limitsusing the standard path object. We call NCW f the category of finite pointed non-commutative CW-complexes . The category NCW f is an “ordinary” topologicallyenriched category. We will also consider NCW f as an ∞ -category, by applyingthe coherent nerve functor to it. Like in the commutative case, will use thesame notation NCW f to indicate both the ordinary (topologically enriched) andthe ∞ -categorical incarnation of the category, trusting that it is clear from thecontext which is meant.The topological category NCW f is the category of noncommutative CW-complexes as defined in [ELP, Section 2.4]. Using [Ped, Theorem 11.14], the same proofas in [Mah2, Proposition 1.1] gives that the (maximal) tensor product of C ∗ -algebras induces a symmetric monoidal structure on the ∞ -category NCW f , thatpreserves finite colimits in each variable separately. Note also that since thetopological category SC ∗ is cotensored over pointed finite CW-complexes, thetopological category NCW f is tensored over pointed finite CW-complexes. We willdenote the tensoring of a finite CW-complex K and a noncommutative finitecomplex X by K ∧ X .We now define the ∞ -category of noncommutative pointed CW-complexes tobe the ∞ -categorical ind-completion of NCW f , NCW := Ind(
NCW f ) . The ∞ -category NCW f is (essentially) small, pointed and finitely cocomplete so NCW is a compactly generated pointed ∞ -category. By [Lur2, Corollary 4.8.1.14]there is an induced closed symmetric monoidal structure on the ∞ -category NCW ≃ Ind(
NCW f ) such that the natural embedding j : NCW f −→ NCW is symmetricmonoidal. 9n [Mah1], Mahanta defined the ∞ -category N S ∗ as the ind-completion of( SC ∗∞ ) op , and called it the ∞ -category of pointed noncommutative spaces . Bydefinition our category NCW f is a full subcategory of ( SC ∗∞ ) op . Since ∞ -pushoutsin both NCW f and ( SC ∗∞ ) op can be calculated as homotopy pushouts using thestandard cylinder object, we see that the inclusion NCW f ֒ → ( SC ∗∞ ) op commuteswith finite colimits. Passing to ind-completions we get that the induced inclusion NCW ֒ → N S ∗ admits a right adjoint (or in other words, NCW is a coreflective fullsubcategory of N S ∗ ) i : NCW ⇄ N S ∗ : R. We call a morphism g : X → Y in N S ∗ a weak homotopy equivalence if R ( g ) isan equivalence in NCW , or equivalently, if for every object W in NCW the inducedmap g ∗ : Map N S ∗ ( i ( W ) , X ) → Map N S ∗ ( i ( W ) , Y )is an equivalence in S . Since every object in NCW is a small colimit of matrixalgebras and i commutes with small colimits, we see that g is a weak equivalenceiff for every n ≥ g ∗ : Map N S ∗ ( M n , X ) → Map N S ∗ ( M n , Y )is an equivalence in S . This is analogous to weak homotopy equivalences betweentopological spaces. The counit of the adjunction above i ◦ R → Id N S ∗ is alevelwise weak equivalence, and thus can be thought of as a CW approximationto elements in N S ∗ . If X and Y are noncommutative CW-complexes then g is aweak equivalence iff it is an equivalence in N S ∗ . Since the weak equivalences in( SC ∗∞ ) op are the homotopy equivalences, the category N S ∗ is somewhat analogousto the infinity category modeled by the Strøm model category on topologicalspaces [Str]. In Theorem 1.1 we make use of enriched infinity categories. There are a fewapproaches to this theory (see, for example, [GH, Lur2]) but so far only in [Hin2]the Yoneda embedding is defined and its basic properties are shown. Since weneed these results, we chose to follow Hinich’s approach in this paper. In thissection we give an overview of the basic definitions and constructions needed forlater on. We also present some new material in subsection 3.0.1, concerning theconnection between model categories and ∞ -categories in the enriched setting.Let Cat denote the ∞ -category of ∞ -categories, and let Cat L denote the ∞ -subcategory of Cat whose objects are ∞ -categories having small colimitsand whose morphisms preserve these colimits. The category Cat is symmet-ric monoidal under the cartesian product, while
Cat L has a symmetric monoidalstructure induced from the cartesian structure on Cat (see [Lur2, 4.8.1.3, 4.8.1.4]).With this structure on
Cat L , Map Cat L ( P ⊗ L, M ) is the subspace of Map
Cat ( P × L, M ) consisting of functors preserving small colimits along each argument.10ote that a monoidal ∞ -category is equivalent to an associative algebra ob-ject in Cat , while an associative algebra in
Cat L is equivalent to a monoidal ∞ -category with colimits, whose monoidal product commutes with colimits ineach variable. We define a closed monoidal ∞ -category to be an associativealgebra in Cat L .If M is a closed monoidal ∞ -category, then a category left-tensored over M is by definition a left module over M in Cat L . More generally, if O is an ∞ -operad, an O -monoidal category is an algebra over O in Cat L . If M is an O -monoidal category one can define an O -algebra in M .Let Ass be the associative operad and LM be the two colored operad of leftmodules. Algebras over Ass are associative algebras and algebras over LM consistof an associative algebra and a left module over it. Thus, an Ass -monoidalcategory is just a closed monoidal ∞ -category and an LM -monoidal category isa pair consisting of a closed monoidal ∞ -category and a category left-tensoredover it.Let M be a closed monoidal ∞ -category. For every space (i.e., an ∞ -groupoid) X , Hinich constructs (see [Hin2, Sections 3 and 4]) a closed monoidalstructure on the ∞ -category of QuiversQuiv X ( M ) := Fun( X op × X, M ) . Hinich’s monoidal structure is an ∞ -categorical version of the usual convolutionproduct that one uses to define ordinary enriched categories. For B a categoryleft-tensored over M , Hinich constructs a left action of the closed monoidal ∞ -category Quiv X ( M ) on the ∞ -category Fun( X, B ). In his notation we obtainan LM -monoidal categoryQuiv LM X ( M , B ) := (Quiv X ( M ) , Fun( X, B )) . Definition 3.1. A M -enriched category, with space of objects X is an associa-tive algebra in Quiv X ( M ). Remark . Hinich uses the term M -enriched precategory for an associativealgebra in Quiv X ( M ). He reserves the term M -enriched category for precate-gories satisfying a version of Segal completeness condition (see [Hin2, Definition7.1.1]). In this paper we are not concerned with Segal completeness, so we willnot distinguish between enriched categories and precategories. We will just say“enriched category” where Hinich might have said “enriched precategory”. Remark . If M is the ∞ -category of spaces, then the category of M -enrichedcategories with space of objects X is equivalent to the category of simplicialspaces satisfying the Segal condition and equalling X in simplicial degree zero.See [Hin2, Corollary 5.6.1], where a more general statement is proved. Inother words, a category enriched in spaces is the same thing as an ordinary ∞ -category. For a general closed monoidal ∞ -category M , there is a monoidal“forgetful” functor from M to spaces, given by Map M ( , − ). In this way we ob-tain a forgetful functor from the category of M -enriched categories to ordinaryinfinity categories (compare with [Hin2, Definition 7.1.1]).11 emark . As we show in next subsection 3.0.1, the theory of monoidal modelcategories and categories enriched or tensored over them extends nicely to thetheory presented above upon application of ∞ -localization.In [Hin2, Section 6] Hinich defines the notion of an M -functor from an M -enriched category to a category left-tensored over M . Let A be an M -enrichedcategory with space of objects X and B a category left-tensored over M . Then A is an associative algebra in Quiv X ( M ) and Fun( X, B ) is left tensored overQuiv X ( M ). An M -functor A → B is defined to be a left module over A inFun( X, B ), and the ∞ -category of M -functors A → B is defined to beFun M ( A , B ) := LMod A (Fun( X, B )) . For x, y objects of B , we define the presheaf Hom B ( x, y ) ∈ P ( M ) byHom B ( x, y )( K ) := Map B ( K ⊗ x, y ) . Clearly Hom B ( x, y ) : M op → S preserves limits, but it is not necessarily repre-sentable. If it happens to be representable, then the representing object servesas an internal mapping object from b to c . Every M -functor F : A → B inducesmaps in P ( M ) h A ( x,y ) → Hom B ( F ( x ) , F ( y ))for x, y ∈ X . The M -functor F is called M -fully faithful if all these maps areequivalences.If Hom B ( b, c ) : M op → S is representable for all objects b, c of B , then B is enriched as well as left tensored. More generally and more precisely, Hinichproves the following proposition (see [Hin2, Proposition 6.3.1 and Corollary6.3.4]). We note that if M is presentable, then any functor M op → S thatpreserves limits is representable (see [Lur1, Proposition 5.5.2.2]). Proposition 3.5.
Let M be a closed monoidal ∞ -category and B left tensoredover M . Let C be a class of objects of B . If for all x, y ∈ C , the presheaf Hom B ( x, y ) is representable then there exists an M -enriched category C whoseclass of objects is C , such that for any two objects x, y of C , the morphismobject C ( x, y ) is a representing object for the functor Hom B ( x, y ) . There is afully-faithful M -enriched functor C → B , extending the inclusion of C into B . Using [Hin2, Lemma 6.3.3] it is not hard to see that given the conditions ofTheorem 3.5 all the categories C that can be obtained are canonically equivalent(via the choice of X as the full subspace of B spanned by C in [Hin2, Corollary6.3.4]). We can thus call C the full enriched subcategory of B spanned by C . Taking C to be the class of all objects in B we see that if Hom B ( x, y ) isrepresentable for all x, y , then B is enriched as well as tensored over M . Let
Cat denote the category of small categories and functors between them andlet S M denote the category of simplicial sets. We have the usual nerve functor N : Cat → S M . N is limit preserving and in particular it is a (cartesian) monoidalfunctor.In [Hor, Section 2.1], Horel constructs a (large, coloured) Cat -operad de-noted ModCat . The colours in
ModCat are model categories, while the category ofmultilinear operations Map
ModCat ( M , · · · , M n , N ) is the category of left Quillen n -functors M × · · · × M n → N and natural weak equivalences (on cofibrant objects) between them. Since N isa monoidal functor, we obtain a simplicial operad from ModCat by composingwith N . We denote this simplicial operad also by ModCat .Let S ∆ op M denote the category of simplicial objects in S M with Rezk’s modelstructure. This is a combinatorial simplicial cartesian closed symmetric monoidalmodel category with all objects cofibrant. Let CSS denote the full simplicialsubcategory of S ∆ op M spanned by the fibrant objects. Then CSS is a monoidalsimplicial category (under the cartesian product) whose simplicial nerve is nat-urally equivalent to the monoidal ∞ -category Cat . Horel also constructs in [loc.cit.] another full simplicial subcategory
Cat ∞ ⊆ S ∆ op M containing CSS and closedunder the cartesian product. He shows that the inclusion
CSS → Cat ∞ inducesan equivalence of ∞ -categories after application of simplicial nerve.Let ModCat c be the full sub simplicial operad of ModCat on model cat-egories that are Quillen equivalent to a combinatorial model category. For
M ∈
ModCat c , let N R ( M ) denote the Resk nerve construction on the cofibrantobjects in M and weak equivalences between them. By [Hor, Theorem 2.16], N R extends to a map of simplicial operads ModCat c → Cat ∞ . Applying simplicialnerve, we obtain a map of ∞ -operads N S ( ModCat c ) → Cat . By [Hor, Remark2.17], this map factors through
Cat L . Since Resk’s nerve is one of the modelsfor ∞ -localization (see, for example, [BHH, Section 2.2]), we obtain a map of ∞ -operads ( − ) ∞ : N S ( ModCat c ) → Cat L which acts as ∞ -localization on objects.Let M be the nonsymmetric operad (in Set) freely generated by an operationin degree 0 and 2. An algebra over M in Set is a set with a binary multiplicationand a base point. Let P be the operad in Cat which is given in degree n bythe groupoid whose objects are points of M ( n ) and a with a unique morphismbetween any two objects. Then an algebra over P in Cat is a monoidal category.The nerve of P is a simplicial operad which we also denote by P . Clearly, wehave an equivalence of ∞ -operads N S P ≃ Ass .Let M be a monoidal model category, Quillen equivalent to a combinatorialmodel category. Then M is an algebra over P in ModCat c (as operads in Cat and thus in S M ). Applying the simplicial nerve we get that M is an algebraover N S P ≃ Ass in N S ( ModCat c ). It follows that M ∞ is an algebra over Ass in Cat L , so M ∞ is a presentable closed monoidal ∞ -category. Furthermore, thelocalization functor M → M ∞
13s lax monoidal.Now let C be a model category, Quillen equivalent to a combinatorial one.Suppose that C is an M -model category, in the sense that C is tensored closedover M and satisfies the Quillen SM7 axiom. As above, we can constructa simplicial operad Q whose simplicial nerve is equivalent to LM and such that( M , C ) is an algebra over Q in ModCat c . Applying the simplicial nerve we get that( M , C ) is an algebra over N S Q ≃ LM in N S ( ModCat c ). It follows that ( M ∞ , C ∞ )is an algebra over LM in Cat L , so C ∞ is a presentable ∞ -category left tensoredover M ∞ . Furthermore, the localization functor( M , C ) → ( M ∞ , C ∞ )is LM -lax monoidal.Let ( − ) f and ( − ) c denote fibrant and cofibrant replacement functors in amodel category. The model category C is an M -model category so for every A ∈ C we have a Quillen pair( − ) ⊗ A c : M ⇄ C : Hom C ( A c , − ) . By [Maz] we have an induced adjunction of ∞ -categories L ( − ) ⊗ A c : M ∞ ⇄ C ∞ : R Hom C ( A c , − ) . Thus we have equivalences natural in
A, B ∈ C :Map M ∞ ( K, R Hom C ( A c , B )) ≃ Map C ∞ ( L K ⊗ A c , B )Clearly L ( − ) ⊗ A c : M ∞ → C ∞ represents the tensor product ( − ) ⊗ A of C ∞ as tensored over M ∞ so we have natural equivalencesMap M ∞ ( K, Hom C ( A c , B f )) ≃ Map C ∞ ( K ⊗ A, B ) . We see that Hom C ( A c , B f ) ∈ M ∞ is a representing object for Hom C ∞ ( A, B ) ∈ P ( M ∞ ) and thus we haveHom C ( A c , B f ) ≃ Hom C ∞ ( A, B ) . Definition 3.6.
Let A be a strictly enriched category over M . Let X denotethe discrete space on the set of objects of A . Then A is an algebra over Ass inthe monoidal category Quiv X ( M ) (see [Hin4]). The localization functor M → M ∞ is lax monoidal, so the induced functorQuiv X ( M ) → Quiv X ( M ∞ )is also lax monoidal. We define the enriched ∞ -localization functor to be com-position with this last functor:( − ) ∞ : Alg Ass (Quiv X ( M )) → Alg
Ass (Quiv X ( M ∞ )) . A ∞ is an enriched ∞ -category over M ∞ .Let F : A → C be a strict M -functor. We have an LM -monoidal categoryQuiv LM X ( M , C ) := (Quiv X ( M ) , Fun( X, C ))and F is just a module in Fun( X, C ) over A (see [Hin4]). In this case ( A , F ) isan algebra over LM in the LM -monoidal category Quiv LM X ( M , C ). The localizationfunctor ( M , C ) → ( M ∞ , C ∞ )is LM -lax monoidal, so the induced functorQuiv LM X ( M , C ) → Quiv LM X ( M ∞ , C ∞ )is also LM -lax monoidal. It follows that we obtain a functor that we denote( − ) ∞ : Alg LM (Quiv LM X ( M , C )) → Alg LM (Quiv LM X ( M ∞ , C ∞ )) . Clearly this functor lifts the functor above so we have( A , F ) ∞ = ( A ∞ , F ∞ ) . Thus, F ∞ is an M ∞ -functor from A ∞ to C ∞ and we call it the enriched ∞ -localization of F .We call F homotopy fully faithful if for every x, y ∈ X the composition A ( x, y ) F −→ Hom C ( F ( x ) , F ( y )) → Hom C ( F ( x ) c , F ( y ) f )is an equivalences in the model category M . Theorem 3.7.
Let A be a strictly enriched category over M and let F : A → C be a strict M -functor which is homotopy fully faithful. Then the M ∞ -functor F ∞ : A ∞ → C ∞ is M ∞ -fully faithful (in the sense described after Remark 3.4).Proof. Lat L : M → M ∞ denote the localization functor. Then for every x, y ∈ X we have a commutative square in M ∞ L ( A ( x, y )) * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ LF / / (cid:15) (cid:15) L (Hom C ( F ( x ) , F ( y ))) (cid:15) (cid:15) A ∞ ( x, y ) F ∞ / / Hom C ∞ ( F ∞ ( x ) , F ∞ ( y )) . The left map is an equivalence by definition of A ∞ and the right map is equiva-lent to L applied to the map Hom C ( F ( x ) , F ( y )) → Hom C ( F ( x ) c , F ( y ) f ). Since F is homotopy fully faithful, the diagonal map is an equivalence, and thus alsothe bottom map. Corollary 3.8.
Let A be a strictly enriched category over M and let F : A → C be a strict fully faithful M -functor that lands in the fibrant cofibrant objects.Then the M ∞ -functor F ∞ : A ∞ → C ∞ is M ∞ -fully faithful. nriched Yoneda Lemma Hinich formulates and proves a version of enriched Yoneda lemma, which is ofkey importance to us. We will review this part of Hinich’s work next.Let M be a closed monoidal ∞ -category and let A be an M -enriched cate-gory with space of objects X . Hinich defines the opposite category A op , whichis an M rev -enriched category with space of objects X op , and constructs a struc-ture of a category left-tensored over M on the ∞ -category of M -presheaves P M ( A ) := Fun M rev ( A op , M ) . Here, M is considered as a right M -module which is the same as a left M rev -module. Remark . In the case of interest to us, M is the category of spectra, which is a symmetric monoidal category. This means that there is a canonical equivalenceof monoidal categories M ≃ M rev .Hinich also constructs an M -fully faithful functor called the enriched Yonedaembedding Y : A → P M ( A ) . In [Hin3] it is shown that this construction has the following universal property:If B is any category left-tensored over M then precomposition with Y inducesan equivalence Map LMod M ( P M ( A ) , B ) ≃ Map M ( A , B ) . In [Hin3], all the above is done more generally relative to an ∞ -operad O .Taking O = Com to be the terminal ∞ -operad and noting that Com ⊗ Ass ≃ Com ,we obtain the following. Suppose M is a closed symmetric monoidal ∞ -category.Then the category of M -left-tensored categories is symmetric monoidal and wedefine a symmetric monoidal M -left-tensored category to be a commutativealgebra in the category of M -left-tensored categories. Similarly, the categoryof M -enriched categories is symmetric monoidal and we define a symmetricmonoidal M -enriched category to be to be a commutative algebra in the cate-gory of M -enriched categories. Moreover, one can define the notion of a sym-metric monoidal M -functor from a symmetric monoidal M -enriched categoryto a symmetric monoidal M -left-tensored category.If A is a symmetric monoidal M -enriched category, the category of presheaves P M ( A ) acquires a canonical symmetric monoidal M -left-tensored structure(Day convolution), and the Yoneda embedding Y : A → P M ( A ) acquires astructure of a symmetric monoidal M -functor. Moreover, this constructionhas the following universal property: If B is any symmetric monoidal M -left-tensored category then precomposition with Y induces an equivalenceMap ComLMod M ( P M ( A ) , B ) ≃ Map
Com M ( A , B ) . Stable ∞ -categories and spectral presheaves In this section we consider the notion of stable ∞ -categories. We show that acompactly generated stable ∞ -category is equivalent to P Sp ( A ) for some small Sp -enriched category A .Let D be a pointed finitely cocomplete ∞ -category. We define the suspensionfunctor on D Σ D : D → D by the formula Σ D ( X ) := ∗ a X ∗ . Alternatively, the suspension functor can be defined as the smash product S ∧ X , using the fact that a pointed finitely cocomplete ∞ -category is tensored overpointed spaces.If the suspension functor is an equivalence of categories, then D is called sta-ble . A stable presentable ∞ -category is naturally left-tensored over the closedmonoidal ∞ -category of spectra Sp (see [Lur2, Proposition 4.8.2.18]). Moreover, Sp is presentable, so for every b, c ∈ D the presheaf Hom D ( b, c ) ∈ P ( Sp ) is rep-resentable (we will denote the representing object also by Hom D ( b, c ) ∈ Sp ). ByProposition 3.5 it follows that a stable presentable ∞ -category is canonically en-riched over Sp (this was observed by Gepner-Haugseng in [GH, Example 7.4.14],where they also pointed out that the presentability assumption is unnecessary). Theorem 4.1.
Let D be a cocomplete stable ∞ -category. Suppose that thereis a small set C of compact objects in D , that generates D under colimits anddesuspensions. Thinking of D as left-tensored over Sp , we let C be the full Sp -enriched subcategory of D spanned by C . Then we have a natural functor ofcategories left-tensored over Sp P Sp ( C ) ∼ −→ D , which is an equivalence of the underlying ∞ -categories and sends each repre-sentable presheaf Y ( c ) ∈ P Sp ( C ) to c ∈ C .Remark . Theorem 4.1 appears in [Lur2, Theorem 7.1.2.1] for the case that | C | = 1. The general case, formulated in the language of model categories, canbe found in [ScSh1, Theorem 3.3.3]. In [GM] the last result can be found undermore general hypotheses Proof.
By defintion of full enriched subcategory, there is a fully-faithful Sp -functor i : C → D . By the universal property of the Yoneda embedding, wehave an induced functor of categories left-tensored over Sp I : P Sp ( C ) → D , such that I ◦ Y ≃ i . We thus get an equivalence I ( Y ( c )) ≃ i ( c ) ≃ c , for every c ∈ C , and it remains to show that I is an equivalence.17he functor I is a morphism of left modules over Sp in Cat L , so in particular I commutes with colimits. The ∞ -category D is presentable by [Lur1, Theorem5.5.1.1]. Let X denote the full subspace of D generated by C and recall that the ∞ -category P Sp ( C ) is defined as the category of left C op -modules with values inFun( X op , Sp ). Since Fun( X op , Sp ) is stable and presentable, so is P Sp ( C ) (see[Lur2, 1.1.3.1 and 4.2.3.5]). Thus, by the adjoint functor theorem I has a rightadjoint J : I : P Sp ( C ) ⇄ D : J. We first show that the unit Y ( c ) → J ( I ( Y ( c ))) of the adjunction I ⊣ J isan equivalence for every c ∈ C . It is not hard to show that J preserves Sp -enrichment, so both Y and J ◦ I ◦ Y are Sp -functors C → P Sp ( C ), and that theunit induces a map Y → J ◦ I ◦ Y of Sp -functors. Note thatFun Sp ( C , P Sp ( C )) = LMod C (Fun( X, P Sp ( C ))) = LMod C (Fun( X, Fun Sp rev ( C op , Sp ))) = LMod C (Fun( X, LMod C op (Fun( X op , Sp )))) = LMod C ( LMod C op (Fun( X, Fun( X op , Sp )))) = LMod C ( RMod C (Fun( X op × X, Sp ))) , so an Sp -functor C → P Sp ( C ) is the same as a C - C -bimodule in the categoryFun( X op × X, Sp ). Thus we can view Y → J ◦ I ◦ Y as a map of C - C -bimodulesin Fun( X op × X, Sp ) and we need to show that it is an equivalence. The for-getful functor to Fun( X op × X, Sp ) reflects equivalences, and an equivalence inFun( X op × X, Sp ) can be verified objectwise, so we can fix two objects c, d in C and show that the induced map of spectra Y ( c, d ) → ( J ◦ I ◦ Y )( c, d )is an equivalence. But since Y and i are Sp -fully faithful, we have( J ◦ I ◦ Y )( c, d ) = J ( I ( Y ( d )))( c ) ≃ Hom P Sp ( C ) ( Y ( c ) , J ( I ( Y ( d )))) ≃ Hom D ( I ( Y ( c )) , I ( Y ( d ))) ≃ Hom D ( i ( c ) , i ( d )) ≃ C ( c, d ) ≃ Hom P Sp ( C ) ( Y ( c ) , Y ( d )) ≃ Y ( d )( c ) ≃ Y ( c, d ) . Since I ( Y ( c )) ≃ c and J ( c ) ≃ Y ( c ), the counit I ( J ( c )) → c of I ⊣ J isalso an equivalences, for every c ∈ C . Note that C generates D under colimits, { Y ( c ) | c ∈ C } generates P Sp ( C ) under colimits and the functor I commutes withcolimits. Thus, if we can show that J also commutes with colimits, it wouldfollow that the unit and counit of I ⊣ J are equivalences, and we are done.Let us show first that J commutes with filtered colimits. So let d = colim i ∈ I d i be a filtered colimit diagram in D . We need to verify that the induced mapcolim i ∈ I J ( d i ) → J ( d ) is an equivalence. Recall that P Sp ( C ) = Fun Sp rev ( C op , Sp ) = LMod C op (Fun( X op , Sp )) . The forgetful functor U from P Sp ( C ) to Fun( X op , Sp ) commutes with colimits(see [Lur2, 4.2.3.5]) and reflects equivalences, so it is enough to verify thatcolim i ∈ I U ( J ( d i )) → U ( J ( d ))18s an equivalence. Now colimits in Fun( X op , Sp ) are pointwise, so we can fix c ∈ C and show that colim i ∈ I U ( J ( d i ))( c ) → U ( J ( d ))( c )in an equivalence. We have an equivalence natural in e ∈ D U ( J ( e ))( c ) = J ( e )( c ) = Hom P Sp ( C ) ( Y ( c ) , J ( e )) ≃ Hom D ( I ( Y ( c )) , e ) ≃ Hom D ( i ( c ) , e ) ≃ Hom D ( c, e ) , so it is enough to show thatcolim i ∈ I Hom D ( c, d i ) → Hom D ( c, d )is an equivalence, which is true by the compactness of c in D .Since both range and domain of J are stable, and in a stable ∞ -categoryevery pullback square is a pushout square and vice versa, it follows that J sends pushout squares to pushout squares. Thus J commutes with all smallcolimits.Using the results in [Hin3] one can prove an extension to Theorem 4.1: Theorem 4.3.
In the situation of Theorem 4.1, suppose D is symmetric monoidaland the set C is closed under the monoidal product in D and contains the unitof D . Then C acquires a canonical symmetric monoidal Sp -enriched structure,the category of presheaves P Sp ( C ) acquires a canonical symmetric monoidal left Sp -tensored structure and the equivalence P Sp ( C ) ∼ −→ D acquires a canonicalsymmetric monoidal left Sp -tensored structure. In this section we review the notion of stabilization of an ∞ -category. We willuse the framework established by Lurie in [Lur2, Section 1.4]. We present insubsection 5.0.1 a similar procedure that can be applied to an ordinary topo-logically enriched category. We will compare the strict and the ∞ -categoricalversions of stabilization.Let C be a pointed ∞ -category. To ensure that C has all the good propertieswe may want, we will assume that C ≃
Ind( C ) where C is a small pointed ∞ -category that is closed under finite colimits. This includes NCW ≃ Ind(
NCW f ).By [Lur1, Theorem 5.5.1.1] C is presentable, and therefore has small limits andcolimits [op. cit., Corollary 5.5.2.4]. Furthermore, filtered colimits commutewith finite limits in C by the remark immediately following [Lur1, Definition5.5.7.1]. It follows, in particular, that C is differentiable in the sense of [Lur2,Definition 6.1.1.6] and therefore the results of [op. cit., Chapter 6] apply to C .Recall that CW f ∗ is the ∞ -category of pointed finite CW-complexes. Let F : CW f ∗ → C be a functor. Recall that F is called reduced if F ( ∗ ) is a final object19f C , and F is called 1 -excisive if F takes pushout squares to pullback squares.Let linear functors be functors that are both reduced and 1-excisive. Linearfunctors provide a good framework for defining spectra in the context of general ∞ -categories. Definition 5.1 ([Lur2], Definition 1.4.2.8) . A spectrum object in C is a linearfunctor CW f ∗ → C . Let Sp ( C ) be the ∞ -category of linear functors CW f ∗ → C . Sp ( C ) is called the category of spectra of C , or the stabilization of C . By re-sults in [Lur2], Sp ( C ) is a stable and presentable ∞ -category (Corollary 1.4.2.17and Proposition 1.4.4.4 respectively).Let Fun ∗ ( CW f ∗ , C ) be the category of all pointed functors from CW f ∗ to C . Then Sp ( C ) is by definition a full subcategory of Fun ∗ ( CW f ∗ , C ). The fully faithfulfunctor Sp ( C ) ֒ → Fun ∗ ( CW f ∗ , C ) has a left adjoint L : Fun ∗ ( CW f ∗ , C ) → Sp ( C )called linearization. Explicitly, if F : CW f ∗ → C is a pointed functor, then thelinearization of F is given by the following formula L F ( X ) = colim n →∞ Ω n C F (Σ n X ) . See [Lur2, Example 6.1.1.28] for a discussion of this formula in the contextof ∞ -categories (of course this formula is older than [Lur2] and goes back atleast to [Goo]). The stabilization Sp ( C ) is the left Bousfield localization ofFun ∗ ( CW f ∗ , C ) at the stable equivalences and L is the localization functor (a mapbetween functors is a stable equivalence if it induces an equivalence betweenlinearizations).There is an adjoint pair of functorsΣ ∞C : C ⇆ Sp ( C ) : Ω ∞C , where Σ ∞C x ( K ) = colim n →∞ Ω n C Σ n C ( K ∧ x ) and Ω ∞C G = G ( S ). This formulafor Ω ∞C agrees with the one in [Lur2, Notation 1.4.2.20], and therefore our Σ ∞C ,being left adjoint to Ω ∞C is also equivalent to Lurie’s. The functor Σ ∞C satisfiesthe following universal property: For every stable presentable ∞ -category D ,pre-composition with Σ ∞C induces an equivalence of ∞ -categories Fun L ( Sp ( C ) , D ) ≃ −→ Fun L ( C , D ) , where Fun L denotes left functors (that is, colimit preserving functors).An important special case is when C = S ∗ is the ∞ -category of pointedspaces. In this case Sp := Sp ( S ∗ ) is the classical ∞ -category of spectra, pre-sented as the category of linear functors from CW f ∗ to S ∗ . Whenever C = S ∗ wewrite Sp , Σ ∞ or Ω ∞ , omitting the subscript C .There is another useful way to construct Sp ( C ) when C = Ind( C ), with C closed under finite colimits. We will now describe it.20 efinition 5.2. Let C be an ∞ -category closed under finite colimits. Definethe Spanier-Whitehead category of C , which we denote by SW ( C ), to be thecolimit of the sequence C C −−→ C C −−→ · · · in Cat .Thus, the objects of SW ( C ) are pairs ( X, n ) where X ∈ C and n ∈ N . Thepair ( X, n ) represents the n -fold desuspension of X . The mapping spaces in SW ( C ) are given byMap SW ( C ) (( X, n ) , ( Y, m )) = colim k ∈ N Map C (Σ k − n C X, Σ k − m C Y ) , where the colimit is taken in the ∞ -category of spaces. Clearly, SW ( C ) is astable ∞ -category. It is closed under finite colimits, but not under arbitrarycolimits. It plays the role of the category of finite spectra over C . There is afinite suspension spectrum functorΣ ∞C f : C → SW ( C )given by X ( X, ∞ -category D , pre-composition with Σ ∞C f induces an equivalence of ∞ -categories Fun fc ∗ ( SW ( C ) , D ) ≃ −→ Fun fc ∗ ( C , D ) , where Fun fc ∗ denotes pointed finite colimit preserving functors.One has the following description of Sp (Ind( C )): Proposition 5.3.
Let C be a small pointed finitely cocomplete ∞ -category, andlet C := Ind( C ) . Then there is a natural equivalence Sp ( C ) ≃ Ind( SW ( C )) . and under this equivalence, Σ ∞C : Ind( C ) ≃ C → Sp ( C ) ≃ Ind( SW ( C )) , is just the prolongation of Σ ∞C f : C → SW ( C ) . Proof.
This is proved in [Lur2, Chapter 1.4] in the case when C = CW f ∗ and C = Ind( CW f ∗ ) ≃ S ∗ . The proof in the general case is similar. In brief, one cancheck that Ind( SW ( C )) satisfies the same universal property as Sp ( C ).This proposition has the following rather important corollary. Corollary 5.4.
Let C , C be as before. Suppose A is a set of objects of C thatgenerates C under finite colimits, in the sense that C is the only subcategoryof C that contains A and is closed under finite colimits and equivalences. Thenthe set of suspension spectra { Σ ∞C x | x ∈ A } generates SW ( C ) under finitecolimits and desuspensions, and generates Sp ( C ) under arbitrary colimits anddesuspensions. Definition 5.5.
Suppose C is a pointed ∞ -category with finite colimits. Let x and y be object of C . Then the functor G C x,y : CW f ∗ → S ∗ is defined by G C x,y ( K ) =Map C ( x, K ∧ y ). Sometimes we will omit the superscript C and write simply G x,y . Lemma 5.6. If C is a stable ∞ -category then G x,y is linear for any two objects x, y .Proof. We need to prove that G x,y ( ∗ ) ≃ ∗ and that G x,y is 1-excisive. The firstcondition holds because ∗ ∧ y is equivalent to a final object of C . Now let usprove that G x,y is 1-excisive. We have equivalencesMap C ( x, K ∧ y ) ≃ −→ Map C ( S ∧ x, S ∧ K ∧ y ) ≃ −→ Map C ( x, Ω( S ∧ K ∧ y )) . Here the first map is an equivalence because C is stable, and the second equiva-lence is a standard adjunction. The composite equivalence can be reinterpretedas saying that the canonical map G x,y → Ω G x,y Σ is an equivalence. It followsthat the map G x,y → L G x,y is an equivalence, so G x,y is linear.If C is a stable ∞ -category then G C x,y is in fact equivalent to the canonicalenrichment of C over spectra. The next lemma and remark say that moregenerally the linearization of G C gives, in favorable circumstances, the spectralenrichment of the stabilization of C . Lemma 5.7.
Let C be a small finitely cocomplete ∞ -category. To simplifynotation, let S : C → SW ( C ) be the finite suspension spectrum functor. Thenthe natural map G C x,y → G SW ( C ) S ( x ) ,S ( y ) induced by the finite suspension spectrum functor is equivalent to the lineariza-tion map G C x,y → L G C x,y . Proof.
By definition, we have equivalences G SW ( C ) S ( x ) ,S ( y ) ( K ) = Map SW ( C ) ( S ( x ) , K ∧ S ( y )) = colim n →∞ Map C ( S n ∧ x, S n ∧ K ∧ y ) == colim n →∞ Ω n Map C ( x, S n ∧ K ∧ y ) = colim n →∞ Ω n G C x,y ( S n ∧ K ) = L G C x,y ( K )and the map from G C x,y is precisely the linearization map. Remark . If C = Ind C , with C as in the previous lemma, and x, y areobjects of C , it is not true that G Sp ( C )Σ ∞C ( x ) , Σ ∞C ( y ) is equivalent to the linearizationof G C x,y . Rather, there is an equivalence G Sp ( C )Σ ∞C ( x ) , Σ ∞C ( y ) ( K ) = Map C ( x, colim n →∞ Ω n ( S n ∧ y )) . This is equivalent to L G C x,y ( K ) if x is a compact object, but not in general. Inparticular, it is true when x ∈ C , which is the case considered in the previouslemma. 22 .0.1 Spectral enrichment of pointed topological categories Suppose C is an ∞ -category and x, y are objects of C . We saw that functors ofthe form G C x,y can be used to define the spectral enrichment of the stabilizationof C . If C is an ordinary topological category, one can use similar functors G C x,y to define a spectral enrichment of C , using the more traditional view ofspectra as modeled by the Quillen model category of continuous functors. Inthis subsection we define a strict spectral enrichment of pointed topologicalcategories and compare it with the ∞ -categorical enrichment.Let Top denote the category of pointed compactly generated weak Hausdorffspaces with the standard model structure of Quillen [Qui]. Every object in Topis fibrant, and every CW-complex is cofibrant. The model category Top isa model for the ∞ -category of pointed spaces S ∗ . This means that the ∞ -localization of Top with respect to the weak equivalences (see Remark 2.1) iscanonically equivalent to S ∗ . Note that any pointed topological category isnaturally enriched in Top. Definition 5.9.
A pointed topological category C is called tensored closed over CW f ∗ if we are given a bi-continuous left action ∧ : CW f ∗ × C → C such that thefollowing hold:1. The ∞ -category C ∞ is finitely cocomplete, where C ∞ is the topologicalnerve of C .2. After application of the topological nerve the functor ∧ ∞ : CW f ∗ × C ∞ → C ∞ commutes with finite colimits in each variable. Definition 5.10.
Let C be a pointed topological category, tensored closed over CW f ∗ . Let x and y be objects of C . In keeping with notation we introduced inDefinition 5.5, we define the pointed topological functor G C x,y : CW f ∗ → Top by G C x,y ( K ) = Map C ( x, K ∧ y ). Again, we may omit the superscript C and writesimply G x,y . Remark . We saw earlier that pointed ∞ -functors from CW f ∗ to S ∗ providea way of defining the ∞ -category of spectra. This is known also in the moretraditional approach to spectra via model categories. There is a Quillen modelstructure on the category Fun ∗ ( CW f ∗ , Top) of continuous pointed functors, calledthe stable model structure, and it provides one of the models for the category ofspectra. We refer the reader to [Lyd, MMSS] for more details about this modelstructure.We denote the category Fun ∗ ( CW f ∗ , Top) with the stable model structure by Sp M . We denote the category Fun ∗ ( CW f ∗ , Top) with the projective model struc-ture simply by Fun ∗ ( CW f ∗ , Top). The model category Sp M is a left Bousfieldlocalization of Fun ∗ ( CW f ∗ , Top), so we have a Quillen pairId : Fun ∗ ( CW f ∗ , Top) ⇄ Sp M : Id . ∞ -localization, this Quillen pair becomes the localization adjunction L : Fun ∗ ( CW f ∗ , S ∗ ) ⇄ Sp : ı . Let C be a pointed topological category, tensored closed over CW f ∗ and let x, y ∈ C . By the previous definition we have a functor G C x,y : CW f ∗ → Top. Underthe identification Fun ∗ ( CW f ∗ , Top) ∞ ≃ Fun ∗ ( CW f ∗ , S ∗ ), G C x,y can be thought ofas an object in Fun ∗ ( CW f ∗ , S ∗ ). We will now compare this with the functor G C ∞ x,y : CW f ∗ → S ∗ from Definition 5.5.Clearly, the bi-functor ∧ ∞ : CW f ∗ × C ∞ → C ∞ is a left action of the monoidal ∞ -category CW f ∗ on C ∞ . It follows that we havean induced action on the ind-categories ∧ ∞ : S ∗ × Ind( C ∞ ) → Ind( C ∞ )and this action commutes with small colimits in each variable. Since S ∗ is amode in the sense of [CSY, Section 5], this action coincides with the canonicalaction of S ∗ on Ind( C ∞ ) as a presentable pointed ∞ -category. In particular weget that the restriction ∧ ∞ : CW f ∗ × C ∞ → C ∞ coincides with the canonical action of CW f ∗ on C ∞ as an ∞ -category with finitecolimits.It follows that under the identification Top ∞ ≃ S ∗ , for any K ∈ CW f ∗ and z ∈ C we have natural equivalencesMap C ( x, z ) ≃ Map C ∞ ( x, z ) K ∧ y ≃ K ∧ ∞ y. Thus we have G C ∞ x,y ( K ) = Map C ∞ ( x, K ∧ ∞ y ) ≃ Map C ( x, K ∧ y ) = G C x,y ( K ) (1)or G C ∞ x,y ≃ G C x,y . Definition 5.12.
Let C be a pointed topological category, tensored closed over CW f ∗ . We define a strict enrichment of C over Sp M as follows: If x and y areobjects of C we define Hom C ( x, y ) := G C x,y ∈ Sp M . Let x, y, z be objects of C and let K and L be finite CW-complexes. Notethat there is a natural map G x,y ( K ) ∧ G y,z ( L ) → G x,z ( K ∧ L ), defined as thecomposite:Map C ( x, K ∧ y ) ∧ Map C ( y, L ∧ z ) →→ Map C ( x, K ∧ y ) ∧ Map C ( K ∧ y, K ∧ L ∧ z ) →→ Map C ( x, K ∧ L ∧ z )24here the first map is induced by the topological functor K ∧ ( − ) : C → C and the second map is given by composition. This map induces a natural map G x,y ⊗ G y,z → G x,z , where ⊗ denotes Day convolution which is the tensorproduct in Sp M . Thus we have defined composition and it can be checked thatthe above indeed defines a strict enrichment of C over Sp M . Theorem 5.13.
Let C be a small pointed topological category, tensored closedover CW f ∗ . Then under the identification Sp M ∞ ≃ Sp , for any two objects x and y of C we have a natural equivalence Hom C ( x, y ) ≃ Hom Sp (Ind( C ∞ )) (Σ ∞ ( x ) , Σ ∞ ( y )) . Proof.
Let x and y be objects in C . Recall that Hom C ( x, y ) = G C x,y and consider G C x,y as an object in Fun ∗ ( CW f ∗ , Top). By (1), we have G C x,y ≃ G C ∞ x,y . We have a commutative squareFun ∗ ( CW f ∗ , Top) ∞ L Id (cid:15) (cid:15) ∼ / / Fun ∗ ( CW f ∗ , S ∗ ) L (cid:15) (cid:15) Sp M ∼ / / Sp Let ( G C x,y ) f be a fibrant replacement to G C x,y in Sp M . Then the map G C x,y → ( G C x,y ) f , considered in Fun ∗ ( CW f ∗ , Top), translates to G C ∞ x,y → L G C ∞ x,y under thetop horizontal map. By Lemma 5.7 the last map is equivalent to G C ∞ x,y → G SW ( C ∞ ) S ( x ) ,S ( y ) . After applying L , this map becomes an equivalence, so we have anatural equivalence in Sp G C x,y ≃ G SW ( C ∞ ) S ( x ) ,S ( y ) ≃ Hom SW ( C ∞ ) ( S ( x ) , S ( y )) . By Proposition 5.3 we are done. ∞ -category of noncommutative CW-spectra In this section we define the ∞ -category of noncommutative CW-spectra NSp .The suspension spectra of matrix algebras form a set of compact generators of
NSp . We denote by M the full spectral subcategory of NSp spanned by this setof generators (see Proposition 3.5). Thus M is an Sp -enriched ∞ -category. Weprove our main theorem: there is an equivalence of ∞ -categories between NSp and the category of spectral presheaves on M . We give two versions and twoindependent proofs of this result. One version is formulated fully in the languageof enriched ∞ -categories, using Hinich’s theory. In the second approach, we firstdefine a strict version of M , denoted M s , which is a category strictly enrichedin Sp M . We then prove that NSp is modelled by a Quillen model category of Sp M -valued presheaves on M s . Finally we prove that our two models of M are25quivalent, in the sense that M is equivalent to the enriched ∞ -localization of M s (see Definition 3.6).Let us proceed with the definition of NSp . Recall that in Section 2 we de-fined the ∞ -category of finite noncommutative CW-complexes and denoted it by NCW f . We then defined the ∞ -category of all noncommutative CW-complexesby the formula NCW := Ind(
NCW f ) . We now define the ∞ -category of noncommu-tative CW-spectra to be NSp := Sp ( NCW ) . By the results in Section 5 we know that
NSp is a presentable stable ∞ -category.In particular, NSp is naturally left-tensored over spectra. By [Lur2, Corollary4.8.2.19] the monoidal structure on
NCW induces a closed symmetric monoidalstructure on
NSp , such that Σ ∞ NC : NCW −→ NSp is symmetric monoidal.Recall that M n is the algebra of n × n matrices over C . Since the set of objects { M i | i ∈ N } generates NCW f under finite colimits, it follows by Corollary 5.4 that M := { Σ ∞ NC M i | i ∈ N } generates NSp under small colimits and desuspensions.The following is one of the main definitions of the paper:
Definition 6.1.
Let M be the full Sp -enriched subcategory of NSp spanned bythe spectra { Σ ∞ NC M i | i ∈ N } .Since M is closed under the monoidal product in NSp , the following theoremis a special case of Theorem 4.3:
Theorem 6.2.
The Sp -enriched category M acquires a canonical symmetricmonoidal structure, the category of presheaves P Sp ( M ) acquires a canonicalsymmetric monoidal left Sp -tensored structure and we have a natural symmetricmonoidal left Sp -tensored functor P Sp ( M ) ∼ −→ NSp , which is an equivalence of the underlying ∞ -categories and sends each repre-sentable presheaf Y (Σ ∞ NC M n ) ∈ P Sp ( C ) to Σ ∞ NC M n . M In this subsection we give a strict model for the category M as a monodialspectrally enriched category, as well as a strict version of Theorem 6.2.In the context of Section 5.0.1, let us consider the example C = NCW f , con-sidered as a topological category. Then NCW f is a pointed topological category,tensored closed over CW f ∗ (see Definition 5.9). As explained in Definition 5.12,we have a strict enrichment of NCW f over the model category of spectra Sp M usingthe functors G NCW f x,y .The topological category NCW f has a continuous symmetric monoidal struc-ture induced by tensor product in SC ∗ . The spectral enrichment respects themonoidal structure, in the sense that given objects x, x , y, y , there is a naturaltransformation G NCW f x,y ( K ) ∧ G NCW f x ,y ( L ) → G NCW f x ⊗ x ,y ⊗ y ( K ∧ L ) . NCW f is symmetric monoidal. Definition 6.3.
Let M s be the full (strict) Sp M -enriched subcategory of NCW f spanned by { M n | n ∈ N } . That is, the objects of M s are { M n | n ∈ N } andfor any m, n ∈ N we haveHom M s ( M m , M n ) = G NCW f M m ,M n ∈ Sp M . Since M s is a category enriched over Sp M , we can define the strict categoryof spectral presheaves on M s , which we denote by P Sp M ( M s ), to be the categoryof enriched functors M op s → Sp M . We endow P Sp M ( M s ) with the projectivemodel structure (see, for example, [GM] on the projective model structure inthe enriched setting). Since both M op s and Sp M have a symmetric monoidalstructure, the category P Sp M ( M s ) has a symmetric monoidal structure given byenriched Day convolution turning it into a symmetric monoidal model category.Consider NCW f as a category enriched in Sp M . There is a canonical strictspectral functor RY : NCW f → P Sp M ( M s ) , (2)which is the composition of the enriched Yoneda embedding NCW f → P Sp M ( NCW f )followed by restriction P Sp M ( NCW f ) → P Sp M ( M s ). It is well-known, and easy tocheck that RY is lax symmetric monoidal.We call a map A → B in NCW f a weak equivalence if it is a homotopyequivalence in NCW f considered as a topological category. Lemma 6.4.
The functor RY sends weak equivalences to weak equivalences.Proof. Let A → B be a weak equivalence in NCW f . We need to show that RY ( A ) → RY ( B ) is a levelwise weak equivalence in P Sp M ( M s ). Let n ≥ NCW f ( M n , A ) → Hom
NCW f ( M n , B ) isa weak equivalence in Sp M . Since Sp M is a localization of the projective modelstructure, is enough to show that Hom NCW f ( M n , A ) → Hom
NCW f ( M n , B ) is alevelwise weak equivalence in Fun ∗ ( CW f ∗ , Top). That is, it is enough to showthat for every finite pointed CW-complex K ,Map NCW f ( M n , K ∧ A ) → Map
NCW f ( M n , K ∧ B )is a weak equivalence. Since NCW f is a topological category, and a weak equiva-lence in NCW f is just a homotopy equivalence, we are done.By the lemma above, we can apply ∞ -localization with respect to weakequivalences (see Remark 2.1) to RY and obtain a functor of ∞ -categories RY ∞ : NCW f ∞ → P Sp M ( M s ) ∞ . Lemma 6.5.
The ∞ -category NCW f ∞ is naturally equivalent to NCW f definedabove as the topological nerve of the topological category NCW f . roof. The category SC ∗ op (defined in the beginning of Section 2) has the struc-ture of a category of cofibrant objects with the weak equivalences given by thehomotopy equivalences and the cofibrations by Schochet cofibrations (see, forinstance, in [AG, Uuy]). We say that a map in NCW f is a weak equivalence (resp.cofibration) if it is a weak equivalence (resp. cofibration) when regarded as amap in SC ∗ op . Since SC ∗ op is a category of cofibrant objects and NCW f ⊆ SC ∗ op isa full subcategory which is closed under weak equivalences and pushouts alongcofibrations it follows that NCW f inherits a structure of a category of cofibrantobjects. In exactly the same way as in [BHH, Lemma 7.1.1] one can show thatthe natural map between ∞ -localizations with respect to weak equivalences:( NCW f ) ∞ −→ ( SC ∗ op ) ∞ is fully faithful.By [BJM, Proposition 3.17] we have that the ∞ -localization of SC ∗ op isequivalent to the topological nerve of the topological category structure on SC ∗ op described in Section 2 (see Remark 2.1 and the paragraph before). Since NCW f is a full topological subcategory of SC ∗ op , we are done. Lemma 6.6.
The functor RY ∞ preserves finite colimits.Proof. By [Lur1, Corollary 4.4.2.5], it is enough to prove that the functor pre-serves initial objects and pushout squares. Since
NCW f is a pointed category, theinital object of NCW f is also the final object, and the first condition obviouslyholds.In both NCW f ∞ and P Sp M ( M s ) ∞ pushouts can be calculated as homotopypushouts in an appropriate structure. Suppose we have a homotopy pushoutdiagram in NCW f y → y ↓ ↓ y → y . (3)We want to prove that for any x ∈ M s the induced diagram of functors is ahomotopy pushout in the stable model structureMap NCW f ( x, − ∧ y ) → Map
NCW f ( x, − ∧ y ) ↓ ↓ Map
NCW f ( x, − ∧ y ) → Map
NCW f ( x, − ∧ y ) . Since we are working in the stable model structure, a square is a homotopypushout if and only if it is a homotopy pullback. A square of functors is ahomotopy pullback in the stable model structure if the induced square of lin-earizations is a homotopy pullback. But the linearization of the functorMap
NCW f ( x, − ∧ y ) : CW f ∗ → Topevaluated at K is the same as the linearization of the functorMap NCW f ( x, K ∧ − ) : NCW f → Top28valuated at y . Indeed, the two linearizations are given by the equivalent for-mulashocolim n →∞ Ω n Map
NCW f ( x, Σ n K ∧ y ) = hocolim n →∞ Ω n Map
NCW f ( x, K ∧ Σ n NCW f y ) . We have been thinking of the functor Map
NCW f ( x, K ∧ − ) : NCW f → Top as astrict functor, but now let us think of it as a functor between ∞ -categories byapplying the ∞ -localization. The ∞ -category NCW f has finite colimits and a finalobject. The conditions of [Lur1, Lemma 6.1.1.33] are satisfied, and thereforethe linearization of this functor really is linear, i.e., takes homotopy pushoutsquares to homotopy pullback squares. Therefore applying the linearization tothe square (3) yields a homotopy pullback square, which is what we wanted toprove.Since P Sp M ( M s ) ∞ is a stable ∞ -category, we have, by the lemma above, that RY ∞ extends canonically to a finite-colimit preserving functor RY ∞ : SW ( NCW f ) → P Sp M ( M s ) ∞ . This functor extends, in turn, to an all-small-colimit-preserving functor RY ∞ : NSp = Ind( SW ( NCW f )) → P Sp M ( M s ) ∞ . (4)This functor takes an object Σ ∞ NC M n to the presheaf represented by M n . Theorem 6.7.
The functor RY ∞ : NSp → P Sp M ( M s ) ∞ is an equivalence of ∞ -categories.Proof. First let us prove that RY ∞ is fully faithful, that is, that for all objects x, y of NSp the map of spectral mapping functors G NSp x,y → G P SpM ( M s ) ∞ RY ∞ ( x ) ,RY ∞ ( y ) (5)is an equivalence. First, consider the case x, y ∈ Σ ∞ M s , i.e., x = Σ ∞ NC M k , y =Σ ∞ NC M l for some k, l . In this case x, y are in the image of the finite suspensionfunctor NCW f → SW ( NCW f ). The functor SW ( NCW f ) → NSp is fully faithful, so itinduces an equivalence G SW ( NCW f ) x,y ≃ −→ G NSp x,y . By Lemma 5.7, the map G NCW f M k ,M l → G SW ( NCW f ) x,y is stabilization.The functor RY : NCW f → P Sp M ( M s ) when restricted to M s is just theenriched Yoneda embedding of M s Y : M s → P Sp M ( M s ) . Since the unit in Sp M is cofibrant, RY ∞ (Σ ∞ M k ) = Y ( M k ) is cofibrant in theprojective model structure on P Sp M ( M s ) (see [GM, Theorem 4.32]). The fibrant29eplacement in P Sp M ( M s ) is levelwise, so using the (strict) enriched Yonedalemma we get G P SpM ( M s ) ∞ RY ∞ ( x ) ,RY ∞ ( y ) = Hom P SpM ( M s ) ∞ ( Y ( M k ) , Y ( M l )) ≃ Hom P SpM ( M s ) ( Y ( M k ) , Y ( M l ) f ) ∼ = ( Y ( M l ) f )( M k ) ≃ Y ( M l )( M k ) f = ( G NCW f M k ,M l ) f . But the map G NCW f M k ,M l → ( G NCW f M k ,M l ) f translates to the stabilization G NCW f M k ,M l →L G NCW f M k ,M l under ∞ -localization (see the proof of Theorem 5.13). Thus the map G NCW f M k ,M l → G P SpM ( M s ) ∞ RY ∞ ( x ) ,RY ∞ ( y ) is also the stabilization. By the uniqueness of the stabilization map, we get thatthe map (5) is an equivalence in the case when x, y are suspension spectra ofmatrix algebras.Next, let x be a fixed suspension spectrum of a matrix algebra, but let y vary. We may consider the functor y G NSp x,y as a functor
NSp → Sp . Thisfunctor preserves all small colimits, because x is compact in NSp and both
NSp and Sp are stable. Similarly, the functor y G P SpM ( M s ) ∞ RY ∞ ( x ) ,RY ∞ ( y ) is also a functor NSp → Sp that preserves small colimits. It follows that the category of objects y for which the map (5) is an equivalence is closed under colimits and alsodesuspensions. Since this category contains M s , it is all of NSp .Now fix y , and consider the functors x G NSp x,y , G P SpM ( M s ) ∞ RY ∞ ( x ) ,RY ∞ ( y ) as contravari-ant functors from NSp to spectra. Since both functors take small colimits tolimits, a similar argument shows that this map is an equivalence for all x ∈ NSp .We have shown that the functor RY ∞ : NSp → P Sp M ( M s ) ∞ is fully faithfuland also preserves all small colimits, so it is a left adjoint. It follows that theimage of RY ∞ is closed under small colimits. Since the image contains therepresentable presheaves, RY ∞ is essentially surjective. It follows that RY ∞ isan equivalence of categories.Thus RY ∞ is an explicit monoidal model for the inverse of the equivalencegiven by Theorem 6.2. This also allows us to show that M s is indeed a stricti-fication of M from Definition 6.1. Theorem 6.8.
We have a natural equivalence ( M s ) ∞ ≃ M between the en-riched ∞ -localization of M s and M .Proof. By definition, M s is an Sp M -enriched category, whose set of objects isthe set of natural numbers N . Let Sp M N - Cat be the category of all Sp M -enrichedcategories, whose set of objects is N , and whose morphisms are functors thatare the identity on objects. This category has a Quillen model structure, wherefibrations and weak equivalences are defined levelwise, and where cofibrant ob-jects are levelwise cofibrant [ScSh2, Proposition 6.3]. Thus, M s is an object in30 p M N - Cat . Let M s → M fs be a fibrant replacement of M s in Sp M N - Cat . Weget a Quillen adjunctionLKan i : P Sp M ( M s ) ⇄ P Sp M ( M fs ) : i ∗ . It follows from [GM, Proposition 2.4] that this adjunction is a Quillen equiva-lence. To give a little more detail, it follows from the general result of Guillouand May that it is enough to show that for every cofibrant object M of Sp M andevery two objects x, y of M s , the following induced map is an equivalence M ∧ Hom M s ( x, y ) → M ∧ Hom M fs ( x, y ) , where Hom( − , − ) denotes the spectral mapping object. The mapHom M s ( x, y ) → Map M fs ( x, y )is an equivalence by definition of M fs . It follows by [MMSS, Proposition 12.3]that the induced map is an equivalence for all cofibrant M .Applying ∞ -localization we obtain an equivalence P Sp M ( M s ) ∞ ∼ −→ P Sp M ( M fs ) ∞ . Now, P Sp M ( M fs ) is an Sp M -model category, and it is Quillen equivalent to acombinatorial model category. The enriched Yoneda embedding Y : M fs → P Sp M ( M fs )is a fully faithful Sp M -enriched functor and it clearly lands in the fibrant cofibrantobjects. Thus, by Corollary 3.8, the Sp -functor Y ∞ : ( M fs ) ∞ → P Sp M ( M fs ) ∞ is Sp -fully faithful. Note that this claim is made with respect to the action of Sp on P Sp M ( M fs ) ∞ induced by the structure of P Sp M ( M fs ) as an Sp M -model category.Since Sp is a mode in the sense of [CSY, Section 5], this action coincides withthe canonical action of Sp on P Sp M ( M fs ) ∞ as a presentable stable ∞ -category.Now, using Theorem 6.7, we have the following composition( M fs ) ∞ Y ∞ −−→ P Sp M ( M fs ) ∞ ∼ −→ P Sp M ( M s ) ∞ ∼ −→ NSp . We get a fully faithful Sp -enriched functor ( M fs ) ∞ → NSp , with essential image M , so that ( M fs ) ∞ ≃ M . References [AG] Andersen K. K. S., Grodal J.
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