Colocalizations of noncommutative spectra and bootstrap categories
aa r X i v : . [ m a t h . K T ] A ug COLOCALIZATIONS OF NONCOMMUTATIVE SPECTRA ANDBOOTSTRAP CATEGORIES
SNIGDHAYAN MAHANTA
Abstract.
We construct a compactly generated and closed symmetric monoidal stable ∞ -category NSp ′ and show that hNSp ′ op contains the suspension stable homotopy categoryof separable C ∗ -algebras ΣHo C ∗ constructed by Cuntz–Meyer–Rosenberg as a fully faithfultriangulated subcategory. Then we construct two colocalizations of NSp ′ , namely, NSp ′ [ K − ]and NSp ′ [ Z − ], both of which are shown to be compactly generated and closed symmetricmonoidal. We prove that Kasparov KK-category of separable C ∗ -algebras sits inside the ho-motopy category of KK ∞ := NSp ′ [ K − ] op as a fully faithful triangulated subcategory. Hence KK ∞ should be viewed as the stable ∞ -categorical incarnation of Kasparov KK-categoryfor arbitrary pointed noncommutative spaces (including nonseparable C ∗ -algebras). As anapplication we find that the bootstrap category in hNSp ′ [ K − ] admits a completely algebraicdescription. We also construct a K-theoretic bootstrap category in hKK ∞ that extends theconstruction of the UCT class by Rosenberg–Schochet. Motivated by the algebraizationproblem we finally analyse a couple of equivalence relations on separable C ∗ -algebras thatare introduced via the bootstrap categories in various colocalizations of NSp ′ . Contents
1. A generalization of the suspension-stable homotopy category of SC ∗ NSp ′ K -colocalization of noncommutative spectra 122.1. Coproducts and products in hKK ∞ hKK ∞ and its dual hKK ∞ op KK ∞ Z -colocalization of noncommutative spectra 194.1. Bootstrap category in ZZ ∞ Mathematics Subject Classification.
Key words and phrases. noncommutative spectra, stable ∞ -categories, triangulated categories, bootstrapcategories, (co)localizations, C ∗ -algebras, bivariant K-theory.This research was supported by the Deutsche Forschungsgemeinschaft (SFB 878 and SFB 1085), ERCthrough AdG 267079, and the Humboldt Professorship of M. Weiss. ntroduction In [26] we constructed a compactly generated stable ∞ -category of noncommutative spec-tra NSp primarily with the intention of proving that the noncommutative stable homotopycategory is topological according to the definition in [39]. The stable ∞ -category NSp af-fords an ideal framework for the stable homotopy theory of noncommutative spaces. In[24] we demonstrated that
NSp is closed symmetric monoidal, which enabled us to producesmashing colocalizations of
NSp that generalize bivariant (connective) E-theory category andsome variants thereof. One aim of our project is to construct similar interesting stable ∞ -subcategories of noncommutative spectra (possibly by colocalizations) and ideally give purelyalgebraic descriptions of their homotopy categories. This is the algebraization problem thatpertains to computational aspects. Concurrently this project also settles the long-standingproblem of constructing generalized (co)homology theories on the category of C ∗ -algebras.In fact, thanks to Brown representability in this setup (see Theorem 2.23 of [26] and Re-mark 1.7), noncommutative spectra parametrize all generalized (co)homology theories. Thecrucial property is the carefully designed compact generation of noncommutative spectra.Amongst the bivariant homology theories present in the literature KK-theory plays adistinguished role as it has proved to be remarkably effective in tackling various problems intopology and geometry (see, for instance, [19, 35, 5, 15, 44]). The assumption of separabilityis inherent in Kasparov’s original definition of KK-theory [18, 17]. However, for certainapplications to index theory and problems in non-metrizable topology this assumption istoo restrictive. Moreover, the construction of the Kasparov (composition) product is a verydelicate issue in this setup. Extending the Cuntz picture of KK-theory it is possible toconstruct a KK-theory kk C ∗ for nonseparable C ∗ -algebras (see, for instance, [8]), where thecomposition product can be established quite easily. In Remark 8.29 of [8] the authors stateAlthough kk C ∗ is defined for inseparable C ∗ -algebras, it does not seem theright generalisation of Kasparov theory to this realm .....One motivation of this article is to address this point. After the appearance of [41]and [28] bivariant homology theories of separable C ∗ -algebras have been treated via tensortriangulated categories. Triangulated categories do not offer the full strength of homotopytheoretic techniques. The stable ∞ -category of noncommutative spectra NSp resolves thisissue satisfactorily. Hence in this article we construct a stable ∞ -categorical incarnationof the KK-category that is also able to treat nonseparable C ∗ -algebras. Along the way weprove that the KK-category is topological and construct a generalization of the Rosenberg–Schochet bootstrap category in this setting. The existence of the Kasparov (composition)product also follows effortlessly in our setup. The article is organised as follows:In Section 1 we construct a variant of noncommutative spectra (denoted by NSp ′ ). Theexcisive behaviour of hNSp op and the KK-category are not compatible, although the differencecan often be ignored. The triangulated category hNSp ′ op eliminates this difference entirely.Then we show that NSp ′ is a compactly generated stable ∞ -category that is also closedsymmetric monoidal (see Proposition 1.6 and Theorem 1.4). Theorem 1.9 explains howpointed noncommutative spaces and spectra generalize their commutative counterparts. TheKK-category kk C ∗ is constructed as a localization of a suspension-stable homotopy categoryof C ∗ -algebras ΣHo C ∗ . We show that the two triangulated categories ΣHo C ∗ and hNSp ′ op agreewhen restricted to separable C ∗ -algebras (see Proposition 1.8 for a precise formulation);however, hNSp ′ is a compactly generated triangulated category in contrast with ( ΣHo C ∗ ) op hat does not even admit arbitrary products. Since NSp ′ is symmetric monoidal, one canconstruct smashing colocalizations thereof with respect to coidempotent objects. We mostlyfocus our attention on two smashing colocalizations, namely, NSp ′ [ K − ] and NSp ′ [ Z − ]. Thefirst colocalization is designed to construct a stable ∞ -categorical KK-category with goodhomotopy theoretic properties, whereas the second one is chosen to address the algebraizationproblem in a tractable setting. In subsection 1.2 we discuss the basic construction of thebootstrap category generated by a set of compact objects in a closed symmetric monoidaland compactly generated stable ∞ -category. The intuitive picture is that the objects inthe bootstrap category are precisely the ones that can be constructed by simple operationsstarting from the chosen set of compact objects as the basic building blocks.If a noncommutative space X is a coidempotent object in N S ∗ , then so is its suspensionspectrum Σ ∞ S ′ ( X ) in NSp ′ . The coidempotent objects in N S ∗ include the C ∗ -algebra of com-pact operators K as well as any strongly self-absorbing C ∗ -algebra (like the Jiang–Su algebra Z ). In Section 2 we show that the smashing colocalization NSp ′ [ K − ] is a compactly gener-ated and closed symmetric monoidal stable ∞ -category (see Theorem 2.3). Then we showthat there is a fully faithful exact functor from Kasparov KK-category into hNSp ′ [ K − ] op (see Theorem 2.4), which also shows that the KK-category is topological (see Proposition2.10). Thus we denote NSp ′ [ K − ] op by KK ∞ and it is our proposed bivariant K-theory ∞ -category for arbitrary pointed noncommutative spaces, whose construction was alluded toin Remark 2.29 of [26]. The category of pointed noncommutative spaces also accommodatesnonseparable C ∗ -algebras. Hence the stable ∞ -category KK ∞ produces a bivariant K-theoryfor nonseparable C ∗ -algebras (see Remark 2.15) with (arguably) better formal propertiesthan the counterpart in [8]. The main advantage of our approach is the compact generationof hNSp ′ [ K − ]. Our method is flexible enough to have a much wider scope of applicability;for instance, it can also be used to construct a bivariant K-theory purely in the algebraicsetting (see Remark 2.8). A bivariant K-theory for C ∗ -spaces was constructed using ideasfrom motivic homotopy theory and model categories by Østvær [31]; the precise relationshipbetween our construction and that of [31] has yet to be clearly understood.In Section 3 we study the bootstrap category KK bt ∞ in KK ∞ and show that there is apurely algebraic description of the bootstrap category, i.e., there is an additive equivalence( hKK bt ∞ ) op ≃ D ( Z [ u, u − ]) (see Theorem 3.4 and Remark 3.5). The bivariant K-theory groupsof the noncommutative spaces belonging to KK bt ∞ satisfy a universal coefficient theorem (UCT)(see Proposition 3.6). They are computable in terms of K-homology groups. The classi-cal UCT in KK-theory [36] expresses the bivariant K-theory groups in terms of K-theory.The category of separable C ∗ -algebras satisfying this form of UCT can be described as a(co)homological localization of Kasparov KK-category [8]. In subsection 3.1 we generalizethis idea to construct a K-theoretic bootstrap category in KK ∞ that truly generalizes theRosenberg–Schochet UCT category to the setting of pointed noncommutative spaces.The global structure of the stable ∞ -category of noncommutative spectra NSp ′ appears tobe quite difficult, since it contains the stable ∞ -category of spectra Sp as a full subcategory.Thus it seems prudent to concentrate on certain subcategories that lie away from Sp . Thefollowing diamond diagram of colocalizations of NSp ′ is illustrative. Note that we havecarefully selected the colocalizations to arrive at the diamond shape; there are numerousother colocalizations arising from coidempotent objects of NSp ′ that have been omitted. Sp ′ [ O − ] x x qqqqqqqqqqq ( ( PPPPPPPPPPPP
NSp ′ [ Q − ] (cid:15) (cid:15) NSp ′ [( Q ˆ ⊗ O ∞ ) − ] (cid:15) (cid:15) NSp ′ [ M − ∞ ] & & ◆◆◆◆◆◆◆◆◆◆◆ NSp ′ [ O − ∞ ] v v ♥♥♥♥♥♥♥♥♥♥♥♥ NSp ′ [ Z − ](1)Actually we also have a sequence of colocalizations between NSp ′ [( Q ˆ ⊗ O ∞ ) − ] and NSp ′ [ O − ∞ ]: NSp ′ [( Q ˆ ⊗ O ∞ ) − ] ֒ → · · · ֒ → NSp ′ [( M ∞ ˆ ⊗ M ∞ ˆ ⊗ O ∞ ) − ] ֒ → NSp ′ [( M ∞ ˆ ⊗ O ∞ ) − ] ֒ → NSp ′ [ O − ∞ ] . Here M ∞ , M ∞ , M ∞ , etc. are UHF algebras of infinite type and so are their C ∗ -tensorproducts. They are also examples of strongly self-absorbing C ∗ -algebras [42]. In Section 4we analyse the colocalization of NSp ′ with respect to the Jiang–Su algebra Z , denoted by NSp ′ [ Z − ]. In this case we fall short of an algebraic description of the bootstrap category. Thehindrance is our lack of understanding of the graded endomorphism ring of the unit object in hNSp ′ [ Z − ] (see Remark 4.4). In stable homotopy theory one tries to understand the stablehomotopy category via Bousfield localizations with respect to various spectra, since the lo-calized categories are often more tractable and occasionally admit algebraic approximations.It is also important to understand the interrelationship between these localizations. Guidedby such considerations we introduce two equivalence relations on C ∗ -algebras (see Definition4.5 and Definition 4.11) and analyse some examples (see Theorem 4.10). Notations and conventions:
Throughout this article ˆ ⊗ will denote the maximal C ∗ -tensor product. All C ∗ -algebras are assumed to be separable unless otherwise stated. Forany ∞ -category C we denote by h C its homotopy category. In the context of ∞ -categoriesa functor (resp. limit or colimit) will implicitly mean an ∞ -functor (resp. ∞ -limit or ∞ -colimit). There is a Yoneda embedding j : SC ∗∞ op → N S ∗ and a separable C ∗ -algebra A isviewed as a noncommutative space via j ( A ). In the sequel for brevity we suppress j fromthe notation. By compact (resp. compactly generated) we shall tacitly mean ω -compact(resp. ω -compactly generated). The triangulated category ΣHo C ∗ stands for the suspension-stable homotopy category of C ∗ -algebras. In the sequel we often denote the full triangulatedsubcategory spanned by the (de)suspensions of separable C ∗ -algebras also by ΣHo C ∗ andthe difference will be clear from the context. We freely use the notation from the articles[23, 22, 26, 24]. For the benefit of the reader we enlist some important ones below:(1) S fin ∗ = ∞ -category of finite pointed spaces [Notation 1.4.2.5 of [22]].(2) S ∗ = ∞ -category of pointed spaces [Notation 1.4.2.5 of [22]](3) SC ∗∞ = ∞ -category of separable C ∗ -algebras [Definition 2.2 of [26]](4) N S ∗ = ∞ -category of pointed noncommutative spaces [Definition 2.13 of [26]](5) Sp = stable ∞ -category of spectra [Definition 1.4.3.1 of [22]](6) NSp = stable ∞ -category of noncommutative spectra [Definition 2.19 of [26]] cknowledgements: The author would like to thank T. Nikolaus for helpful discussions.The author has benefited from the hospitality of Max Planck Institute for Mathematics, Bonnand Hausdorff Research Institute for Mathematics, Bonn under various stages of developmentof this project. The author is also extremely grateful to the anonymous referee for theconstructive report suggesting several improvements.1.
A generalization of the suspension-stable homotopy category of SC ∗ In Section 8.5 of [8] the authors constructed a suspension-stable homotopy category ofall C ∗ -algebras denoted by ΣHo C ∗ . Although the construction in [8] works for all (possiblynonseparable) C ∗ -algebras, we may (and later on we shall) restrict our attention to separable C ∗ -algebras. The aim in this section is to construct a variant of the stable ∞ -category ofnoncommutative spectra, denoted by NSp ′ , such that hNSp ′ op and ΣHo C ∗ agree when restrictedto separable C ∗ -algebras. The triangulated category hNSp ′ is actually quite large (it iscompactly generated) so that it is able to accommodate nonseparable C ∗ -algebras. Ourconstruction will differ from that of [8] both in methodology and end result for genuinelynonseparable C ∗ -algebras. The triangulated category hNSp ′ has better formal properties (seeRemark 3.1 of [26]). Recall that a presentable symmetric monoidal ∞ -category is closed ifthe tensor product preserves colimits separately in each variable. For the benefit of thereader we record a couple of results from [24]. Proposition 1.1.
The maximal C ∗ -tensor product on SC ∗ leads to the following:(1) The ∞ -categories SC ∗∞ and N S ∗ are symmetric monoidal. Moreover, the presentable ∞ -category N S ∗ is closed symmetric monoidal and the Yoneda functor j : SC ∗∞ op → N S ∗ is symmetric monoidal.(2) There is a closed symmetric monoidal stable ∞ -category Sp ( N S ∗ ), such that thestabilization functor Σ ∞ : N S ∗ → Sp ( N S ∗ ) is symmetric monoidal.Hence there is a symmetric monoidal functor Stab : SC ∗∞ op → Sp ( N S ∗ ) that arises as acomposition of two symmetric monoidal functors SC ∗∞ op j ֒ → N S ∗ Σ ∞ → Sp ( N S ∗ ). Recall thatan extension of C ∗ -algebras 0 → A → B → C → semisplit if the surjective ∗ -homomorphism B → C admits a completely positive contractive section. Let C be anycompactly generated stable ∞ -category and let V be a set of compact objects of C . Then h V i denotes the smallest full stable ∞ -subcategory of C generated the translations (in bothdirections) and cofibers of the objects of V . Consider the collection of morphisms in SC ∗∞ op that can be chosen to be a small set T ′ = { C( f ) → ker( f ) | → ker( f ) → B f → C → } (see Remark 2.4 of [26]). We set S ′ = { Stab ( θ ) | θ ∈ T ′ } , which is also a small set of morphisms in Sp ( N S ∗ ). This defines an exact localization L S ′ : Sp ( N S ∗ ) → S ′− Sp ( N S ∗ ) as follows: Set V = { cone( θ ) | θ ∈ S ′ } and let A = h V i denotethe stable ∞ -subcategory of Sp ( N S ∗ ) generated by V . Note that A is a subcategory of thecompact objects of Sp ( N S ∗ ). Hence Ind ω ( A ) is a compactly generated stable ∞ -subcategoryof Sp ( N S ∗ ) and we let S ′ denote the collection of maps in Sp ( N S ∗ ), whose cones lie in Ind ω ( A ).The collection of maps S ′ defines an accessible localization L S ′ : Sp ( N S ∗ ) → S ′− Sp ( N S ∗ ), hich is the desired one (for the details see Section 2.4 of [26]). By construction there is ashort exact sequence of stable presentable ∞ -categories:Ind ω ( A ) → Sp ( N S ∗ ) → S ′− Sp ( N S ∗ ) . Definition 1.2.
Due to the obvious analogy with the ∞ -category of noncommutative spectra NSp = S − Sp ( N S ∗ ), we denote the ∞ -category S ′− Sp ( NSp ) by
NSp ′ . The stable ∞ -category NSp ′ is yet another candidate for noncommutative spectra as we are shortly going to demon-strate (see Theorem 1.9 below). Remark 1.3.
Evidently, S ′ ⊂ S and S ′ ⊂ S from which we obtain a commutative diagramof stable presentable ∞ -categories (up to equivalence) Sp ( N S ∗ ) L S ′ / / L S * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ S ′− Sp ( N S ∗ ) = NSp ′ (cid:15) (cid:15) S − Sp ( N S ∗ ) = NSp . Theorem 1.4.
There is a colimit preserving symmetric monoidal functor Σ ∞ S ′ = L S ′ ◦ Σ ∞ : N S ∗ → NSp ′ between presentable closed symmetric monoidal ∞ -categories. Proof.
Thanks to Proposition 1.1 part (2) the functor Σ ∞ : N S ∗ → Sp ( N S ∗ ) is a colimitpreserving symmetric monoidal functor between presentable closed symmetric monoidal ∞ -categories. Being an accessible localization the functor L S ′ : Sp ( N S ∗ ) → S ′− Sp ( N S ∗ ) = NSp ′ is a colimit preserving functor between presentable ∞ -categories. It remains to show that thefunctor L S ′ is also symmetric monoidal between closed symmetric monoidal ∞ -categories.Let f : X → Y be a local equivalence, i.e., L S ′ ( f ) is an equivalence in S ′− Sp ( N S ∗ ). Byconstruction this means that the cofiber of f lies in Ind ω ( A ), in other words, there is acofiber sequence X f → Y g → Z in Sp ( N S ∗ ) with Z ∈ Ind ω ( A ). For any Q ∈ Sp ( N S ∗ ) theinduced diagram X ⊗ Q f ⊗ id → Y ⊗ Q g ⊗ id → Z ⊗ Q is again a cofiber sequence. If we can showthat Z ⊗ Q ∈ Ind ω ( A ) then it would imply that f ⊗ id : X ⊗ Q → Y ⊗ Q is also a localequivalence. Hence by Proposition 2.2.1.9 and Example 2.2.1.7 of [22] (see also Lemma 3.4of [14]) this would prove that the localization is symmetric monoidal (or compatible withit). It would follow that NSp ′ is closed symmetric monoidal (see Remark 3.5 of [14]) and weshall have completed the proof.To this end write Z = colim α Z α with Z α ∈ A and set Y α = Y × Z Z α and consider themap of cofiber sequences X α f α / / (cid:15) (cid:15) Y α g α / / (cid:15) (cid:15) Z α (cid:15) (cid:15) X f / / Y g / / Z, where f α : X α → Y α is the fiber of g α . If V is a set of compact objects in a compactly gener-ated stable ∞ -category C , then h V i denotes the smallest stable ∞ -subcatgory of C generatedby V . Observe that Sp ( N S ∗ ) is compactly generated by the objects of h Stab ( SC ∗∞ op ) i . Now α need not be compact but we may write Y α = colim β Y αβ with each Y αβ ∈ h Stab ( SC ∗∞ op ) i .Set X αβ = X α × Y α Y αβ and we obtain a ladder diagram of cofiber sequences X αβ f αβ / / (cid:15) (cid:15) Y αβ (cid:15) (cid:15) g αβ / / Z αβ (cid:15) (cid:15) X α f α / / (cid:15) (cid:15) Y α g α / / (cid:15) (cid:15) Z α (cid:15) (cid:15) X f / / Y g / / Z, The top left square is by construction a pullback square and since we are in a stable ∞ -category it is also a pushout square. Thus we have equivalences Z αβ ∼ → Z α and X α ∼ → X .Let Sp ( N S ∗ ) c denote the stable ∞ -category of compact object in Sp ( N S ∗ ). We thus have acofiber sequence X αβ f αβ → Y αβ g αβ → Z αβ (2)in Sp ( N S ∗ ) c with Z αβ ∈ A . Moreover,colim αβ X αβ → colim αβ Y αβ → colim αβ Z αβ is equivalent to the cofiber sequence X f → Y g → Z . Now we write Q = colim γ Q γ with each Q γ compact in Sp ( N S ∗ ). Using the fact that ⊗ commutes with colimits we find that thecofiber sequence X ⊗ Q f ⊗ id → Y ⊗ Q g ⊗ id → Z ⊗ Q is equivalent tocolim αβγ ( X αβ ⊗ Q γ ) → colim αβγ ( Y αβ ⊗ Q γ ) → colim αβγ ( Z αβ ⊗ Q γ ) . (3)Our aim is to show that ( Z αβ ⊗ Q γ ) ∈ A . From Proposition 2.17 of [26] we know that there isa fully faithful exact functor Stab = Π op : HoSC ∗ [Σ − ] op → hSp ( N S ∗ ), whose image lies inside hSp ( N S ∗ ) c . The functor Stab is also symmetric monoidal with respect to ˆ ⊗ on HoSC ∗ [Σ − ].The cofiber sequence (2) gives rise to an exact triangle in the triangulated category hSp ( N S ∗ ) c ;by construction we may assume that it is of the form X αβ → Stab ( B, m ) → Stab ( C, n ) → Σ X αβ with ( B, m ) , ( C, n ) ∈ HoSC ∗ [Σ − ] op . Using the fully faithful exact functor Stab we mayview the exact triangle to be (
B, m ) h ← ( C, n ) ← C( h ) ← Σ( B, m ) with
Stab (C( h )) ≃ X αβ in hSp ( N S ∗ ). There is also an exact triangle associated with the cofiber sequence X αβ ⊗ Q γ → Y αβ ⊗ Q γ → Z αβ ⊗ Q γ . If Q γ = Stab ( A, k ), then using the exactness of ˆ ⊗ in the triangulatedcategory HoSC ∗ [Σ − ] we may write this exact triangle as( B, m ) ˆ ⊗ ( A, k ) h ˆ ⊗ id ← ( C, n ) ˆ ⊗ ( A, k ) ← C( h ) ˆ ⊗ ( A, k ) ← Σ( B, m ) ˆ ⊗ ( A, k )with
Stab ( C, n ) ˆ ⊗ ( A, k )) ≃ Stab ( C, n )) ⊗ Stab ( A, k ) ≃ Z αβ ⊗ Q γ . We know that ( C, n )belongs to the triangulated subcategory T of HoSC ∗ [Σ − ] generated by { cone( θ ) | θ ∈ T } sothat Stab ( T ) ≃ h A . Using the fact that − ˆ ⊗ ( A, k ) is an exact functor on
HoSC ∗ [Σ − ] it is clear( C, n ) ˆ ⊗ ( A, k ) belongs to T as well. Thus Stab ( C, n ) ˆ ⊗ ( A, k )) ≃ Z αβ ⊗ Q γ belongs to A andhence colim αβγ Z αβ ⊗ Q γ ∈ Ind ω ( A ), i.e., f ⊗ id : X ⊗ Q → Y ⊗ Q is a local equivalence. (cid:3) emark 1.5. Let T ′ (resp. T ) denote the strongly saturated collections of morphisms in N S ∗ generated by j ( T ′ ) (resp. j ( T )), where T = { C( f ) → ker( f ) | → ker( f ) → B f → C → } . One can also construct an ∞ -category of noncommutative spaces, which comes equippedwith a canonical functor T ′− N S ∗ → T − N S ∗ . Moreover, the suspension spectrum functorΣ ∞ S ′ factors as N S ∗ → T ′− N S ∗ → NSp ′ . Proposition 1.6.
The stable ∞ -categories NSp and
NSp ′ are compactly generated. Proof.
It is shown in Lemma 2.22 of [26] that
NSp is compactly generated. The proof of thecompact generation of
NSp ′ is similar. Indeed, by construction N S ∗ is compactly generatedand hence so is its stabilization Sp ( N S ∗ ) (see Propostion 1.4.3.7 of [22]). Therefore, theaccessible localization L S ′ : Sp ( N S ∗ ) → NSp ′ is also compactly generated. Note that S ′ is astrongly saturated collection of morphisms generated by a small set, such that the domainand the codomain of every morphism in this small set is compact. One can also argue asfollows: by construction there is a short exact sequence of stable presentable ∞ -categoriesInd ω ( A ) → Sp ( N S ∗ ) → NSp ′ , which induces a short exact sequence of triangulated categories h Ind ω ( A ) → hSp ( N S ∗ ) → hNSp ′ . We know that hSp ( N S ∗ ) is compactly generated and sincethe objects of A are compact in Sp ( N S ∗ ) so is h Ind ω ( A ). From Theorem 7.2.1 (2) of [21]we deduce that hNSp ′ is a compactly generated triangulated category. Since NSp ′ is a stable ∞ -category, it must itself be compactly generated (see Remark 1.4.4.3 of [22]). (cid:3) Remark 1.7.
Using arguments similar to Theorem 2.23 and Remark 2.25 of [26] one canshow that both hNSp ′ and ( hNSp ′ ) op satisfy Brown representability.Now we compare our construction with ΣHo C ∗ (restricting our attention to separable C ∗ -algebras). Recall that the objects in ΣHo C ∗ are pairs ( A, n ) with A ∈ SC ∗ and n ∈ Z . Itsmorphisms are defined as ΣHo C ∗ (( A, n ) , ( B, m )) := lim −→ k [ J n + kcpc A, Σ m + k B ] , where the functor J cpc A is defined by the short exact sequence 0 → J cpc A → T cpc A → A → SC ∗∞ op j → N S ∗ Σ ∞ → Sp ( N S ∗ ) , whose opposite functor is denoted by Π : SC ∗∞ → Sp ( N S ∗ ) op and another composite functor SC ∗∞ op j → N S ∗ Σ ∞ S ′ → NSp ′ , whose opposite functor is denoted by π : SC ∗∞ → NSp ′ op . Proposition 1.8.
There is a fully faithful functor Θ :
ΣHo C ∗ → hNSp ′ op induced by thefunctor π : SC ∗∞ → NSp ′ op ; in particular, for A, B ∈ SC ∗ there is a natural isomorphism ΣHo C ∗ ( A, B ) ∼ = hNSp ′ op ( π ( A ) , π ( B )) . Proof.
The homotopy category hSp ( N S ∗ ) op is triangulated. The canonical composite functorΠ : hSC ∗∞ = HoSC ∗ → hN S ∗ op → hSp ( N S ∗ ) op inverts the suspension functor Σ. Thus it factorsthrough HoSC ∗ [Σ − ], i.e., we have the following commutative diagram: hSC ∗∞ Π / / ι & & ▲▲▲▲▲▲▲▲▲▲ hSp ( N S ∗ ) op HoSC ∗ [Σ − ] . Π ♦♦♦♦♦♦ he dashed functor Π is fully faithful (see Proposition 2.17 of [26]), i.e., one has HoSC ∗ [Σ − ]( ι ( A ) , ι ( B )) ∼ = lim −→ k [Σ k A, Σ k B ] ∼ = hSp ( N S ∗ ) op (Π( A ) , Π( B )) . Thanks to the universal characterization of
ΣHo C ∗ (see Section 8.5 of [8]) one can obtain itas a Verdier quotient HoSC ∗ [Σ − ] V → ΣHo C ∗ of triangulated categories with respect to the set ofmaps ι ( T ′ ). We have an exact colocalization L op S ′ : hSp ( N S ∗ ) op → hSp ( N S ∗ ) op , whose essentialimage hNSp ′ op is spanned by the S ′ -colocal objects. Now ker( V ) is the thick subcategory of HoSC ∗ [Σ − ] generated by { cone( f ) | f ∈ ι ( T ′ ) } and ker( L op S ′ ) is the colocalizing subcategoryof hSp ( N S ∗ ) op generated by Π(ker( V )). Thus we obtain a unique functor Θ : ΣHo C ∗ → hNSp ′ op making the following diagram commute: ker( V ) (cid:15) (cid:15) Π / / ker( L op S ′ ) (cid:15) (cid:15) hSC ∗∞ ι / / ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ HoSC ∗ [Σ − ] V (cid:15) (cid:15) Π / / hSp ( N S ∗ ) op L op S ′ (cid:15) (cid:15) ΣHo C ∗ Θ / / ❴❴❴❴❴ hNSp ′ op . Observe that the composite functor L op S ′ ◦ Π ◦ ι is π and V acts as identity on objects whence Θacts as π on objects. Taking the opposite of the above diagram one can argue as in Theorem2.26 of [26] to show that Θ is fully faithful. The argument relies on Neeman’s generalizationof Thomason localization theorem [30] (see also [21] for the formulation used in [26]). (cid:3) Theorem 1.9.
Let S ∗ denote the ∞ -category of pointed spaces and Sp denote the stable ∞ -category of spectra. We have the following:(1) There is a fully faithful ω -continuous functor S ∗ ֒ → N S ∗ .(2) If C denotes either of the two compactly generated stable ∞ -categories NSp , NSp ′ ,then there is a fully faithful colimit preserving exact functor Sp ֒ → C . Proof.
For the first assertion notice that the Gel’fand–Na˘ımark correspondence gives a fullyfaithful functor S fin ∗ ֒ → SC ∗∞ op . Hence there is a fully faithful ω -continuous functor S ∗ =Ind ω ( S fin ∗ ) ֒ → Ind ω ( SC ∗∞ op ) = N S ∗ (see Proposition 5.3.5.11 (1) of [23]).Let us prove that there is a fully faithful colimit preserving exact functor Sp ֒ → NSp . Let S denote the sphere spectrum that generates the stable ∞ -category of finite spectra Sp fin undertranslations (in both directions) and cofibers in Sp . Sending S to π op ( C ) ∈ NSp sets up anexact functor Sp fin → NSp , whose image lies inside the compact objects of
NSp . By Theorem2.26 of [26] this functor is fully faithful at the level of homotopy categories and hence it isfully faithful. Once again by Proposition 5.3.5.11 (1) of [23] it extends to a fully faithful ω -continuous exact functor Sp = Ind ω ( Sp fin ) → NSp . Thus it preserves small coproducts andhence by Proposition 1.4.4.1 (2) of [22] all colimits. The proof of the corresponding assertionfor
NSp ′ is similar using Proposition 1.8 and hence it is left to the reader.The referee has kindly pointed out that in conjunction with Theorem 2.26 of [26] abovewe are using that the Gel’fand–Na˘ımark correspondence induces a fully faithful functor hSp fin ֒ → NSH op . This can be seen as follows: for finite pointed CW complexes ( X, x ) and Y, y ) one has [C(
X, x ) = the C ∗ -algebra of continuous functions on X that vanish at x ] hSp fin (( X, x ) , ( Y, y )) ∼ = lim −→ n [Σ n ( X, x ) , Σ n ( Y, y )] ∼ = lim −→ n [C(Σ n ( Y, y )) , C(Σ n ( X, x ))]and
NSH (C(
Y, y ) , C( X, x )) = lim −→ n [[Σ n C( Y, y ) , Σ n C( X, x )]] ∼ = lim −→ n [[C(Σ n ( Y, y )) , C(Σ n ( X, x ))]],where [[ − , ?]] denotes asymptotic homotopy classes of asymptotic homomorphisms. Finally,it follows from Corollary 17 of [9] that the canonical homomorphism[C(Σ n ( Y, y )) , C(Σ n ( X, x ))] → [[C(Σ n ( Y, y )) , C(Σ n ( X, x ))]]is an isomorphism. (cid:3)
Remark 1.10.
Let us consider the category
ΣHo C ∗ for separable C ∗ -algebras. From Theorem1.4 and Proposition 1.8 we get a symmetric monoidal functor π : HoSC ∗ = hSC ∗∞ → hNSp ′ op .The functor π : hSC ∗∞ → hNSp ′ op uniquely determines Θ below hSC ∗∞ π / / ι (cid:15) (cid:15) hNSp ′ op . ΣHo C ∗ Θ ttttt The category
ΣHo C ∗ is actually a tensor triangulated category by setting ( A, m ) ⊗ ( B, n ) =( A ˆ ⊗ B, m + n ), i.e., the tensor structure is compatible with that on hSC ∗∞ induced by themaximal C ∗ -tensor product so that ι is symmetric monoidal. The explicit nature of thetensor structure on ΣHo C ∗ can now be used to verify that Θ must be symmetric monoidal.1.1. Colocalizations of
NSp ′ . Computations in
NSp ′ are presumably as hard as those inthe stable ∞ -category of spectra Sp . Therefore, we try to understand colocalizations of NSp ′ with respect to certain coidempotent objects (see Definition 3.1 of [24]) that lie away from Sp . The following Lemma follows easily from Theorem 1.4. Lemma 1.11.
Let A be a coidempotent object in N S ∗ . Then Σ ∞ S ′ ( A ) is a coidempotentobject in NSp ′ .If A = K or A = D , where D is any strongly self-absorbing C ∗ -algebra, then j ( A ) is acoidempotent object in N S ∗ (see Lemma 3.2 of [24]). In this case Σ ∞ S ′ ( A ) is a coidempotentobject in NSp ′ and using the dual of Proposition 4.8.2.4 of [22] one concludes that the functor R Σ ∞ S ′ ( A ) : NSp ′ → NSp ′ sending X X ⊗ Σ ∞ S ′ ( A ) is a colocalization. We simplify the notationby setting R A = R Σ ∞ S ′ ( A ) and denote the essential image R A ( NSp ′ ) by NSp ′ [ A − ]. For any A ∈ SC ∗ we set A K = A ˆ ⊗ K , i.e., the K -stabilization of A . Remark 1.12.
Let C be any symmetric monoidal ∞ -category and let CAlg ( C ) denote the ∞ -category of commutative algebra objects in C (see Definition 2.1.3.1 of [22]). Let E be anidempotent object of C . Then the localization L E = − ⊗ E : C → C with L E C ≃ E endows L E C with the structure of a symmetric monoidal ∞ -category. Moreover, the right adjoint L E C ֒ → C induces a fully faithful functor CAlg ( L E C ) ֒ → CAlg ( C ) (see Proposition 4.8.2.9 of[22]). Hence E itself is a commutative algebra object in C . This observation in the specialcase C = NSp ′ op will play an important role in the sequel. Proposition 1.13.
Let A ∈ SC ∗ be such that Σ ∞ S ′ ( A ) is a coidempotent object in NSp ′ . Thenthe stable ∞ -category NSp ′ [ A − ] is compactly generated and closed symmetric monoidal. roof. It follows from Lemma 3.4 (and Remark 3.5) of [14] that
NSp ′ [ A − ] is a closed sym-metric monoidal ∞ -category. It is clear that the ∞ -category NSp ′ [ A − ] is stable. Thus itsuffices to show that its triangulated homotopy category hNSp ′ [ A − ] is compactly generated(see Remark 1.4.4.3 of [22]). Proposition 1.6 above shows that NSp ′ is compactly generatedwhence so is its triangulated homotopy category. We observe that R A : hNSp ′ → hNSp ′ isa coproduct preserving colocalization of triangulated categories, whose essential image is hNSp ′ [ A − ]. From the functorial triangle R A ( X ) → X → L A ( X ) → Σ R A ( X ) we deducethat the corresponding localization L A : hNSp ′ → hNSp ′ also preserves coproducts. HenceIm( L A ) ≃ ker( R A ) is a compactly generated triangulated category (see Remark 5.5.2 of[21]). Observe that hNSp ′ / ker( R A ) ≃ Im( R A ) = hNSp ′ [ A − ] whence hNSp ′ [ A − ] is compactlygenerated (see Theorem 5.6.1 of [21]). (cid:3) Proposition 1.14.
Let D , D ′ be separable C ∗ -algebras such that Σ ∞ S ′ ( D ) and Σ ∞ S ′ ( D ′ ) areboth coidempotent objects in NSp ′ . Moreover, let ι : D → D ′ be a ∗ -homomorphism, suchthat D ˆ ⊗ D ′ ι ⊗ id D ′ → D ′ ˆ ⊗ D ′ is homotopic to an isomorphism. Then there is a colocalization θ : NSp ′ [ D − ] → NSp ′ [ D ′− ] given by θ ( − ) = − ⊗ Σ ∞ S ′ ( D ⊗ D ′ ), such that R D ′ ≃ θ ◦ R D . Proof.
Since Σ ∞ S ′ ( D ′ ) ∼ = Σ ∞ S ′ ( D ′ ) ⊗ Σ ∞ S ′ ( D ′ ) it follows thatΣ ∞ S ′ ( D ′ ) ∼ = Σ ∞ S ′ ( D ′ ) ⊗ Σ ∞ S ′ ( D ′ ) → Σ ∞ S ′ ( D ) ⊗ Σ ∞ S ′ ( D ′ ) ≃ Σ ∞ S ′ ( D ⊗ D ′ )is homotopic to an equivalence. Since R D (resp. R D ′ ) is − ⊗ Σ ∞ S ′ ( D ) (resp. − ⊗ Σ ∞ S ′ ( D ′ ))and Σ ∞ S ′ ( D ) ⊗ Σ ∞ S ′ ( D ′ ) ≃ Σ ∞ S ′ ( D ⊗ D ′ ), the assertion follows. (cid:3) Example 1.15.
We present two pertinent examples of the above scenario.(1) If D → D ′ is a unital embedding between strongly self-absorbing C ∗ -algebras, wededuce from the Proposition on page 4027 of [42] that D ˆ ⊗ D ′ → D ′ ˆ ⊗ D ′ is homotopicto an isomorphism.(2) For any strongly self-absorbing C ∗ -algebra D the corner embedding D ι → D ˆ ⊗ K hasthe property that ι ⊗ id D ˆ ⊗ K is homotopic to an isomorphism (see Proposition 2.9 of[24]). Note that K itself is not a strongly self-absorbing C ∗ -algebra. Remark 1.16.
Observe that the functor
NSp ′ [ D ′− ] → NSp ′ [ D − ], which is the left adjointto θ in the above Proposition 1.14, is a colimit preserving fully faithful functor betweencompactly generated stable ∞ -categories.The Jiang–Su algebra Z was introduced in [16] and it plays a crucial role in Elliott’sClassification Program. It is itself strongly self-absorbing and for any other strongly self-absorbing C ∗ -algebra D there is a unique (up to homotopy) unital embedding Z → D [43].There is also a canonical unital ∗ -homomorphism O ∞ → O ∞ ˆ ⊗ Q , where Q is the universalUHF algebra. Hence we obtain the following sequence of colocalizations of NSp ′ (see Equation1 and the comment thereafter): NSp ′ [( O ∞ ˆ ⊗ Q ) − ] ֒ → · · · ֒ → NSp ′ [( M ∞ ˆ ⊗ O ∞ ) − ] ֒ → NSp ′ [ O − ∞ ] ֒ → NSp ′ [ Z − ] ֒ → NSp ′ . (4)There is yet another sequence of colimit preserving fully faithful functors between stable andcompactly generated ∞ -categories: · · · ֒ → NSp ′ [( K ˆ ⊗ O ∞ ) − ] ֒ → NSp ′ [( K ˆ ⊗ Z ) − ] ֒ → NSp ′ [ Z − ] ֒ → NSp ′ . (5) emark 1.17. In sequence (5) all the stable ∞ -categories to left of NSp ′ [ Z − ] can be viewedas fully faithful subcategories of KK ∞ op that we are shortly going to introduce in Section 2.1.2. Bootstrap categories.
Let us explain the construction of the bootstrap category ina general setting. Let C be a compactly generated and closed symmetric monoidal stable ∞ -category with unit object C , which is a compact object. Let V denote a set of compactobjects of C and let h V i denote the full stable ∞ -subcategory of C generated by the transla-tions (in both directions) and cofibers of the objects in V . Observe that all objects of h V i areagain compact in C . Therefore, Ind ω ( h V i ) is a compactly generated full stable ∞ -subcategoryof C (see Proposition 5.3.5.11 (1) of [23]), which is the bootstrap category in C generated by V . Two different sets V, V ′ may generate equivalent bootstrap categories. If V = { C } issingleton, then Ind ω ( h C i ) is called the bootstrap category in C ; we also refer to Ind ω ( h C i ) op as the bootstrap category in C op . The bootstrap category Ind ω ( h V i ) generated by a set V ofcompact noncommutative spectra is intuitively the subcategory of noncommutative spectrathat can be constructed by using the objects of V as basic building blocks. Example 1.18.
Typical choices for V constitute a small set of simple separable C ∗ -algebras.(1) The bootstrap category in NSp ′ generated by { C } is the stable ∞ -category of spectra Sp (see Theorem 1.9).(2) If V = { M n ( C ) | n ∈ N } then the bootstrap category in NSp ′ generated by V is thecategory of noncommutative stable cell complexes [12, 41, 27].2. K -colocalization of noncommutative spectra It was remarked by the author in [26] that the opposite of a stable ∞ -categorical modelfor KK-theory can be constructed as an accessible localization of the stable presentable ∞ -category Sp ( N S ∗ ) (see Remark 2.29 of [26]). Since NSp ′ is symmetric monoidal we are ableto show that a smashing colocalization of noncommutative spectra furnishes us with an ∞ -categorical incarnation of (the opposite of) KK-category. An argument similar to Lemma 3.3of [24] shows that Σ ∞ S ′ ( K ) is a coidempotent object in NSp ′ . It follows from Proposition 3.4 of[24] that the functor R = R K : NSp ′ → NSp ′ sending X X ⊗ Σ ∞ S ′ ( K ) is a colocalization. Wedenote the essential image of the colocalization by NSp ′ [ K − ] := R ( NSp ′ ), which is a closedsymmetric monoidal ∞ -category (see Proposition 1.13). Definition 2.1.
We set KK ∞ := NSp ′ [ K − ] op . The symmetric monoidal stable ∞ -category KK ∞ is our ∞ -categorical model for the bivariant K-theory category.There is a composite functor N S ∗ Σ ∞ S ′ → NSp ′ R → NSp ′ [ K − ] , whose opposite functor is denoted by k : N S ∗ op → KK ∞ . Definition 2.2.
For any pointed noncommutative space X ∈ N S ∗ we defineK ∗ ( X ) = NSp ′ [ K − ](k( C ) , Σ ∗ k( X )) = hKK ∞ (Σ ∗ k( X ) , k( C ))K ∗ ( X ) = NSp ′ [ K − ](k( X ) , Σ ∗ k( C )) = hKK ∞ (Σ ∗ k( C ) , k( X ))where K ∗ ( X ) (resp K ∗ ( X )) is the K-homology (resp. K-theory) of X . Theorem 2.3.
The functor k : N S ∗ op → KK ∞ is symmetric monoidal and it preserves limits. roof. Since R is a smashing colocalization the functor k : N S ∗ op → KK ∞ is symmetricmonoidal (see Example 2.2.1.7 and Proposition 2.2.1.9 of [22]). The functor k preserveslimits because its opposite functor R ◦ Σ ∞ S ′ preserves colimits. (cid:3) Now we justify that the above definitions are good ones. We continue to denote byk : hN S ∗ op → hKK ∞ the induced functor at the level of homotopy categories. By abuse ofnotation we also denote the composite functor SC ∗∞ j op → N S ∗ op k → KK ∞ as well as the functorthat it induces at the level of homotopy categories by k. Let KK denote the Kasparov bivariantK-theory category for separable C ∗ -algebras, whose morphisms are given by KK ( A, B ) =KK ( A, B ) and the composition of morphisms is induced by Kasparov product. There is acanonical functor SC ∗ → KK , which is identity on objects. Theorem 2.4.
For any two separable C ∗ -algebras A, B there is a natural isomorphism KK ( A, B ) ∼ = hKK ∞ (k( A ) , k( B )) = hNSp ′ [ K − ] op (k( A ) , k( B )) . In other words, the functor k induces a fully faithful functor KK → hKK ∞ . Proof.
There is a fully faithful exact functor Θ :
ΣHo C ∗ → hNSp ′ op (see Proposition 1.8).There is another natural identification ΣHo C ∗ ( A ˆ ⊗ K , B ˆ ⊗ K ) ∼ = KK ( A, B ) (see Theorem 8.28of [8]). Using Theorem 13.7 of [8] and the comment thereafter one deduces that there isa localization of triangulated categories
ΣHo C ∗ → KK → ΣHo C ∗ , where the first functor actson objects as A A ˆ ⊗ K and the second functor is fully faithful. Since the localization R op : hNSp ′ op → KK ∞ is smashing we get a commutative diagram: hSC ∗∞ ι / / $ $ ❍❍❍❍❍❍❍❍❍ ΣHo C ∗ Θ / / (cid:15) (cid:15) hNSp ′ op R op (cid:15) (cid:15) KK θ / / (cid:15) (cid:15) hKK ∞ (cid:15) (cid:15) ΣHo C ∗ Θ / / hNSp ′ op , where the composite left vertical functor ΣHo C ∗ → KK → ΣHo C ∗ is the localization describedabove. The composition R op ◦ Θ ◦ ι is the functor k and the middle horizontal functor θ : KK → hKK ∞ continues to be fully faithful. Hence we get the desired natural isomorphism KK ( A, B ) ∼ = hKK ∞ (k( A ) , k( B )) . (cid:3) Let ι : C → K denote the ∗ -homomorphism which sends 1 to e . This induces a mapk( ι ) : k( C ) → k( K ) in KK ∞ . Corollary 2.5.
The map k( ι ) is an equivalence in KK ∞ . Proof.
It is well-known that the map ι : C → K is a KK-equivalence. By the above Theorem2.4 k( ι ) must descend to an isomorphism in hKK ∞ . (cid:3) Remark 2.6.
The bivariant K-theory category for separable C ∗ -algebras KK is also a ten-sor triangulated category, where the tensor structure is induced by the maximal C ∗ -tensorproduct. The minimal C ∗ -tensor product would have worked equally well over here; see, for nstance, [11] for some interesting features of this tensor triangulated category. Our projectinitiated with the noncommutative stable homotopy category NSH that was shown to be atensor triangulated category with respect to the maximal C ∗ -tensor product (see Theorem3.3.7 of [41]). Hence for the sake of consistency we have used the maximal C ∗ -tensor productthroughout. As before one can view KK as a tensor triangulated subcategory of hKK ∞ .2.1. Coproducts and products in hKK ∞ . For a countable family of separable C ∗ -algebras { A n } n ∈ N the infinite sum C ∗ -algebra in SC ∗ is defined as ⊕ n ∈ N A n := lim −→ F ⊂ N ⊕ i ∈ F A i , where F runs through all finite subsets of N . The infinite sum C ∗ -algebra is neither the coproduct northe product in SC ∗ . Nevertheless, when viewed inside the Kasparov category KK for separable C ∗ -algebras it acts as a countable coproduct, i.e., KK ( ⊕ n ∈ N A n , B ) ∼ = Q n ∈ N KK ( A n , B ) for anyseparable C ∗ -algebra B viewed as an object of KK (see Theorem 1.12 of [36]). This result isoptimal, i.e., KK does not admit arbitrary coproducts. In sharp contrast we have Proposition 2.7.
The triangulated category hKK ∞ admits all small coproducts. Proof.
Being an accessible localization of the stable presentable ∞ -category Sp ( N S ∗ ), thestable ∞ -category NSp ′ is itself presentable. It also follows from Corollary 5.5.2.4 of [23]that it admits all small limits whence the triangulated homotopy category hNSp ′ admitsall small products. Therefore, hNSp ′ op admits all small coproducts. Now by construction R : NSp ′ → NSp ′ is a smashing colocalization, whose essential image is KK ∞ op . It followsthat R op : hNSp ′ op → hNSp ′ op is a smashing localization of triangulated categories, whichpreserves small coproducts. Hence its image hKK ∞ is closed under taking small coproducts(see Remark 5.5.2 of [21]). (cid:3) Remark 2.8.
The above construction of bivariant K-theory is quite flexible and can becarried out purely in the algebraic setting. For instance, let k be a commutative ring withunit. Consider a small full subcategory of the category of k -algebras with algebraic homotopyequivalences as a category with weak equivalences. One technical point is to ensure that thechosen subcategory admits finite homotopy limits. Applying the Dwyer–Kan localizationone obtains a simplicial category. Taking the fibrant replacement of this simplicial categoryin the model structure on simplicial categories constructed in [1] and applying the homotopycoherent nerve to it produces an ∞ -category. The (algebraic) homotopy equivalences between k -algebras become equivalences in this ∞ -category. Now one can follow the steps as abovereplacing K -stability by matrix stability that needs to be enforced by (co)localization.2.2. Brown representability in hKK ∞ and its dual hKK ∞ op . Let T be a triangulatedcategory with arbitrary coproducts. A localizing subcategory of T is a thick subcategorythat is closed under taking small coproducts. Following [20] one says that T is perfectlygenerated by a small set X of objects of T provided the following holds:(1) There is no proper localizing subcategory of T containing all the objects in X ,(2) given a countable family of morphisms { X i → Y i } i ∈ I in T , such that the map T ( C, X i ) → T ( C, Y i ) is surjective for all C ∈ X and i ∈ I , the induced map T ( C, a i X i ) → T ( C, a i Y i )is surjective.Predictably a triangulated category T with coproducts is called perfectly cogenerated if T op is perfectly generated by some small set of objects. Finally, a triangulated category is called ompactly generated if it is perfectly generated by a small set of compact objects. Recall thatan object C of T is compact if ` j ∈ J T ( C, D j ) ∼ = T ( C, ` j ∈ J D j ) for every set indexed familyof objects { D j } j ∈ J of T . A very intuitive and equivalent definition of compact generation isthe following: a triangulated category T admitting small coproducts is compactly generated if there is a small set T of compact objects that generate T , i.e., each X ∈ T is compact and T ( X, Y ) = 0 for every X ∈ T implies that Y is itself 0 (see Definition 1.7 of [29]). For thecorresponding notion in the setting of ∞ -categories see Section 5.5.7 of [23]. Lemma 2.9.
The triangulated category hNSp ′ [ K − ] = hKK ∞ op is compactly generated. Proof.
The assertion is a consequence of Proposition 1.13. (cid:3)
Recall that a triangulated category is said to be topological if it is triangle equivalent tothe homotopy category of a stable cofibration category (see Definition 1.4 of [39]).
Proposition 2.10.
Kasparov category KK is topological. Proof.
The above Theorem 2.4 shows that KK is equivalent to a full triangulated subcategoryof hKK ∞ = hNSp ′ [ K − ] op via the functor k. Since NSp ′ [ K − ] is a stable presentable ∞ -category(see Lemma 2.9) one can establish the result following the proof of Theorem 2.27 of [26]. (cid:3) Remark 2.11.
Note that our methods actually show that both KK and KK op are topological.Indeed, our methods exhibit KK op naturally as a full triangulated subcategory of a stablemodel category whence it is topological. Since the notion of a stable model category isself-dual, one also deduces that KK is topological. Proposition 2.12.
The triangulated category hKK ∞ is perfectly generated. Proof.
The above Lemma shows that hKK ∞ op is compactly generated. Thus it follows that hKK ∞ op is perfectly cogenerated (see Section 5.3 of [21]) whence hKK ∞ is perfectly generated. (cid:3) Theorem 2.13.
Both hKK ∞ and hKK ∞ op satisfy Brown representability, i.e., a functor F : T op → Ab is cohomological and sends all coproducts in T to products in Ab if and only if F ( − ) ∼ = T ( − , X ) for some object X ∈ T , where T = hKK ∞ or hKK ∞ op . Proof.
We already observed that hKK ∞ op satisfies Brown representability due to its compactgeneration [30]. Thanks to its perfect generation Brown representability for hKK ∞ followsfrom Theorem A of [20]. (cid:3) Corollary 2.14.
The triangulated category hKK ∞ admits all small products. Proof.
We refer the readers to Remark 5.1.2 (2) of [21]. (cid:3)
Remark 2.15.
As we mentioned before there is a bivariant K-theory category specificallydesigned for nonseparable C ∗ -algebras kk C ∗ that was constructed in [8]. Our formalismalso covers the bivariant K-theory of nonseparable C ∗ -algebras (or pointed noncommutativecompact Hausdorff spaces). Indeed, any nonseparable C ∗ -algebra gives rise to a filtereddiagram of its separable C ∗ -subalgebras. Now one can take its filtered colimit in N S ∗ op after applying j op : SC ∗∞ → N S ∗ op . Finally one can apply the functor k : N S ∗ op → KK ∞ to land inside KK ∞ . Our bivariant K-theory for genuinely nonseparable C ∗ -algebras will ingeneral not agree with that of [8], although both kk C ∗ and hKK ∞ contain Kasparov category or separable C ∗ -algebras KK as a fully faithful triangulated subcategory. Clearly hKK ∞ hasbetter formal properties (see Remark 8.29 of [8]). Moreover, in the stable ∞ -category KK ∞ one can compute limits and colimits, which carry more refined information than the weak(co)limits and sequential homotopy (co)limits in the triangulated category hKK ∞ or kk C ∗ . Remark 2.16.
It would be interesting to understand the relationship between hKK ∞ andthe bivariant K-theory for σ - C ∗ -algebras [7] (see also [25]). Note that the notion of a σ - C ∗ -algebra is more restrictive than that of an arbitrary noncommutative (pointed) space.3. Bootstrap category in KK ∞ The Universal Coefficient Theorem (UCT) is a milestone in the development of bivariantK-theory and it is very natural to seek a generalization of this result beyond the categoryof separable C ∗ -algebras (or that of pointed noncommutative compact metrizable spaces).The original construction of the UCT class in Kasparov bivariant K-theory category is dueto Rosenberg–Schochet [36]. A separable C ∗ -algebra A belongs to the UCT class if for every B ∈ SC ∗ there is a natural short exact sequence of Z / → Ext ∗ (K ∗ +1 ( A ) , K ∗ ( B )) → KK ∗ ( A, B ) → Hom ∗ (K ∗ ( A ) , K ∗ ( B )) → . The UCT class can also be characterized as consisting of those C ∗ -algebras, which lie in thereplete triangulated subcategory of KK generated by C that is also closed under countablecoproducts. Henceforth we set Σ C ∗ = C ((0 , Σ C ∗ ˆ ⊗ ( − ) ∼ = Σ N S ∗ ( − ). The following result generalizes the argumentsof the proof of Bott periodicity in [6]. A nice observation made by the anonymous refereeenabled us to formulate the result in its current form and streamline the proof. Proposition 3.1.
There is an isomorphism of endofunctors Σ − KK ∞ ( − ) ∼ = Id( − ) of hKK ∞ . Proof.
Since k( C ) is the tensor unit of hKK ∞ , we have Σ − KK ∞ ( − ) ∼ = Σ − KK ∞ k( C ) ⊗ ( − ) andk( C ) ⊗ ( − ) ∼ = Id( − ). Note that Σ − KK ∞ k( C ) = Ω KK ∞ k( C ) ≃ k(Σ N S ∗ C ) ≃ k( Σ C ∗ ˆ ⊗ C ) ≃ k( Σ C ∗ ).Here we have used that fact that Σ C ∗ ˆ ⊗ C ≃ Σ N S ∗ C in N S ∗ (see Lemma 2.9 of [26]). Hencethere is an isomorphism of endofunctors Σ − KK ∞ ( − ) ∼ = k( Σ C ∗ ) ⊗ k( Σ C ∗ ) ⊗ ( − ). Thus it sufficesto show that k( Σ C ∗ ) ⊗ k( Σ C ∗ ) ∼ = k( C ) in hKK ∞ .Consider the reduced Toeplitz extension 0 → K → T → Σ C ∗ →
0. Applying the functork we obtain a diagram k( K ) → k( T ) → k( Σ C ∗ ) in KK ∞ . It is a (co)fiber sequence in KK ∞ thanks to excision with respect to semisplit extenstions and it gives rise to an exact trianglein hKK ∞ . For any X ∈ hKK ∞ applying the functor hKK ∞ ( X, − ) we get a long exact sequence · · · → hKK ∞ ( X, k( K )) → hKK ∞ ( X, k( T )) → hKK ∞ ( X, k( Σ C ∗ )) → · · · . Hence we get a boundary map hKK ∞ ( X, k( Σ C ∗ ) ⊗ k( Σ C ∗ )) → hKK ∞ ( X, k( K )) . Now we claim that(1) hKK ∞ ( X, k( T )) = 0 and(2) hKK ∞ ( X, k( K )) ∼ = hKK ∞ ( X, k( C )).Since the reduced Toeplitz algebra T is KK-equivalent to 0, we get (1) from Theorem 2.4.For (2) we simply invoke Corollary 2.5. Thus we have shown that hKK ∞ ( X, k( Σ C ∗ ) ⊗ k( Σ C ∗ )) ∼ = hKK ∞ ( X, k( C )) or every X ∈ hKK ∞ . Using the Yoneda Lemma the assertion follows. (cid:3) Lemma 3.2.
For any A ∈ SC ∗ there are natural isomorphisms: hKK ∞ (k( C ) , k( A )) ∼ = K ( A ) and hKK ∞ (k( Σ C ∗ ) , k( A )) ∼ = K ( A ). Proof.
The assertion follows from Theorem 2.4 and the above Proposition. (cid:3)
We spell out the construction of the bootstrap category following subsection 1.2. Recall thatthere is a composite functor N S ∗ Σ ∞ S ′ → NSp ′ R → NSp ′ [ K − ] , whose opposite functor gives us k : N S ∗ op → KK ∞ . Let C denote the stable ∞ -subcategoryof KK ∞ op generated by k( C ). It is the closure of k( C ) under translations (in both directions)and cofibers. Now we set KK bt ∞ := Ind ω ( C ) op , which is a stable ∞ -subcategory of KK ∞ . Hencethe homotopy category of Ind ω ( C ) is a localizing subcategory of hKK ∞ op compactly generatedby k( C ), since k( C ) is compact in KK ∞ op . Definition 3.3.
We define KK bt ∞ (resp. hKK bt ∞ ) to be the bootstrap category in KK ∞ (resp. in hKK ∞ ) (see also Remark 3.7 and Definition 3.9 below).Let Z [ u, u − ] be a differential graded algebra with trivial differentials and deg( u ) = 2 andlet D ( Z [ u, u − ]) denote its unbounded derived category of differential graded modules. Thederived category D ( Z [ u, u − ]) is also the homotopy category of a stable model category. Ourbivariant K-theory category possesses the correct formal properties from the viewpoint ofhomotopy theory. Thus we are able to use a result of Bousfield [4] and Franke [13] (writtenup carefully in [32]; see also [33]) to arrive at an algebraic description of the triangulatedcategory ( hKK bt ∞ ) op . Theorem 3.4.
There is an additive equivalence of categories ( hKK bt ∞ ) op ≃ D ( Z [ u, u − ]). Proof.
Since ( KK bt ∞ ) op := Ind ω ( C ) is a presentable stable ∞ -category the triangulated category( hKK bt ∞ ) op admits infinite coproducts. It follows from Theorem 2.4 that the graded Homobject Hom ( hKK bt ∞ ) op (Σ ∗ k( C ) , k( C )) ∼ = Hom ( hKK ∞ ) op (Σ ∗ k( C ) , k( C )) is isomorphic to Z [ u, u − ]with deg( u ) = 2. Since the graded global dimension of Z [ u, u − ] is 1 and it is concentratedin even dimensions, one deduces the assertion from Proposition 5.2.3 of [32]. (cid:3) Remark 3.5.
It is not clear whether the above additive equivalence of categories is actuallyan exact equivalence of triangulated categories (see Remark 5. 2.4 of [32]). Hence we arenot able to conclude that ( hKK bt ∞ ) op is algebraic as a triangulated category according to thedefinition in [38].Now we are going to justify the notation KK bt ∞ with the help of two simple propositions. Proposition 3.6.
Let T denote the triangulated category ( hKK bt ∞ ) op . Then for any X, Y ∈ T there is a natural short exact sequence0 → Ext ∗ (k( C )) (K ∗ (Σ X ) , K ∗ ( Y )) → T ( X, Y ) → Hom K ∗ (k( C )) (K ∗ ( X ) , K ∗ ( Y )) → . In particular, X is isomorphic to Y in T if and only if K ∗ ( X ) and K ∗ ( Y ) are isomorphic asgraded K ∗ (k( C )) ≃ Z [ u, u − ]-modules. Proof.
The assertion is a consequence of Proposition 5.1.1 of [32]. (cid:3) emark 3.7. The above result is a universal coefficient theorem in hKK ∞ via K-homology. Proposition 3.8.
Let A be a nuclear separable C ∗ -algebra satisfying UCT with finitelygenerated K-theory. Then k( A ) belongs to hKK bt ∞ . Proof.
Under the assumptions A is KK-equivalent to C( X, x ), where (
X, x ) is a finite pointedCW complex (see Corollary 7.5 of [36]). It is clear that k(C(
X, x )) belongs to hKK bt ∞ . Theassertion now follows from Theorem 2.4. (cid:3) The K-theoretic bootstrap category.
It follows from Theorem 2.4 that there is acanonical fully faithful functor k : KK ֒ → hKK ∞ . Let B C ∗ denote the triangulated subcategoryof KK consisting of those separable C ∗ -algebras that satisfy UCT. Let K ∗ ( − ) denote the Z / KK . One interpretation of B C ∗ is that it is the the Verdierquotient KK / ker(K ∗ ) (see Theorem 13.11 of [8]).We know from Proposition 2.7 that the triangulated category hKK ∞ admits arbitrary coprod-ucts. We denote the coproduct in hKK ∞ by ` . Let Ab Z / denote the category of Z / C ) ` k( Σ C ∗ ) corepresents a functor hKK ∞ → Ab Z / that gener-alizes the functor K ∗ : KK → Ab Z / , i.e., for any separable C ∗ -algebra A one has hKK ∞ (k( C ) a k( Σ C ∗ ) , k( A )) ∼ = hKK ∞ (k( C ) , k( A )) ⊕ hKK ∞ (k( Σ C ∗ ) , k( A )) ∼ = K ( A ) ⊕ K ( A ) = K ∗ ( A ) . Let us denote the corepresented functor hKK ∞ (k( C ) ` k( Σ C ∗ ) , − ) : hKK ∞ → Ab Z / by K . Wehave the following commutative diagram: KK k / / K ∗ (cid:15) (cid:15) hKK ∞ K (cid:15) (cid:15) Ab Z / / / Ab Z / . Let N denote the triangulated subcategory of hKK ∞ spanned by the objects in the image ofk(ker(K ∗ )). Since N op is contained in the compact objects of ( hKK ∞ ) op , one concludes thatthe localizing subcategory hh N op ii of ( hKK ∞ ) op generated by N op is compactly generated. Itfollows that there is a coproduct preserving (Bousfield) localization of triangulated categories( hKK ∞ ) op → ( hKK ∞ ) op / hh N op ii and hence the product preserving (Bousfield) colocalization hKK ∞ → B , where B is the opposite of the triangulated category ( hKK ∞ ) op / hh N op ii . Definition 3.9.
We define the triangulated category B to be the K -theoretic bootstrap cat-egory . A justification for this nomenclature will be provided below (see Theorem 3.12). Remark 3.10.
It follows from Theorem 7.2.1 (2) of [21] that the triangulated category B op is compactly generated. Proposition 3.11.
There is a fully faithful exact functor B C ∗ → B . Proof.
The proof follows from standard arguments along the lines of Theorem 2.26 of [26]. (cid:3)
Theorem 3.12.
Set π ∗ ( − ) = hKK ∞ (Σ ∗ k( C ) , − ) and let X ∈ B . Then for any Y ∈ hKK ∞ there is a natural short exact sequence0 → Ext ( π ∗ +1 ( X ) , π ∗ ( Y )) → hKK ∞ ( X, Y ) → Hom( π ∗ ( X ) , π ∗ ( Y )) → . roof. Since the triangulated category hKK ∞ admits coproducts (see Proposition 2.7) theargument in the proof of Theorem 13.11 (and Exercise 13.13) of [8] goes through. (cid:3) Remark 3.13.
Observe that π ∗ (k( X )) = K ∗ ( X ), where K ∗ ( X ) is the K-theory of X ∈ N S ∗ (see Definition 2.2). Thus the objects of B satisfy a K-theoretic universal coefficient theorem.The ad hoc notation π ∗ ( − ) in Theorem 3.12 is deployed to make the short exact sequenceresemble the usual UCT sequence in KK-theory. Remark 3.14.
Let ker( K ) denote the full triangulated subcategory of hKK ∞ consisting ofthose objects X with K ( X ) ≃
0. The triangulated category hKK ∞ admits arbitrary products(see Corollary 2.14) and the subcategory ker( K ) ⊆ hKK ∞ is colocalizing. However, it is notclear whether ker( K ) op is compactly generated (this is related to the smashing conjecture in hNSp ′ [ K − ] that is an interesting problem in its own right). Hence the predictable analogueof Proposition 3.11 may not hold in this case.4. Z -colocalization of noncommutative spectra Here we use the basic terminology of rings and modules in the context of ∞ -categories fromSections 3 and 4 of [22]. Recall from subsection 1.1 that there is a smashing colocalization R Z : NSp ′ → NSp ′ [ Z − ]. We set ZZ ∞ = NSp ′ [ Z − ] op so that there is a composite functor N S ∗ Σ ∞ S ′ → NSp ′ R Z → NSp ′ [ Z − ] , whose opposite functor is denoted by z : N S ∗ op → ZZ ∞ . Recall from Proposition 1.8 thatthere is a fully faithful functor Θ : ΣHo C ∗ → hNSp ′ op , which maps A viewed as ( A, ∈ ΣHo C ∗ to Σ ∞ S ′ ( A ). From Remark 1.12 we deduce that z( C ) ≃ Θ( Z ) is a commutative algebra objectin NSp ′ op . Observe that z( C ) = Σ ∞ S ′ ( Z ) as an object in NSp ′ op .4.1. Bootstrap category in ZZ ∞ . Let A be a separable C ∗ -algebra, such that Σ ∞ S ′ ( A )is a coidempotent object in NSp ′ . Then we have seen that C = NSp ′ [ A − ] is a compactlygenerated and closed symmetric monoidal stable ∞ -category, whose unit object C = Σ ∞ S ′ ( A )is compact. Applying the construction from subsection 1.2 we obtain the bootstrap categoryin NSp ′ [ A − ]. Let us now investigate the bootstrap category of ZZ ∞ . Definition 4.1.
Specializing to the case A = Z produces a compactly generated stable ∞ -subcategory Ind ω ( h Σ ∞ S ′ ( Z ) i ) of NSp ′ [ Z − ]. The subcategory ZZ bt ∞ := Ind ω ( h Σ ∞ S ′ ( Z ) i ) op of ZZ ∞ = NSp ′ [ Z − ] op is said to be the bootstrap category inside ZZ ∞ . Remark 4.2.
By construction hZZ bt ∞ is the colocalizing subcategory of hZZ ∞ generatedby Σ ∞ S ′ ( Z ), since h Ind ω ( h Σ ∞ S ′ ( Z ) i ) is the localizing subcategory of hNSp ′ [ Z − ] generated byΣ ∞ S ′ ( Z ), i.e., the closure in hNSp ′ [ Z − ] under translations, cofibers, and arbitrary coproducts.The first step towards understanding ZZ ∞ is to investigate the bootstrap category ZZ bt ∞ .We get a description of its opposite category as a module category using the classification ofstable model categories by Schwede–Shipley [40]. More precisely, using the enhanced versionin the symmetric monoidal setup (see Proposition 7.1.2.7 of [22]) we get Proposition 4.3.
Let C = ( ZZ bt ∞ ) op so that C = Σ ∞ S ′ ( Z ). Then there is an equivalence C ≃ Mod R , where R = End C ( C ) is an E ∞ -ring. roof. By construction C is generated by the tensor unit C , which is a compact object. Thetensor product preserves finite colimits separately in each variable in the stable ∞ -category h C i . Hence it preserves small colimits separately in each variable in Ind ω ( h C i ) = C (seeCorollary 4.8.1.13 of [22]). The assertion now follows from Proposition 7.1.2.7 of [22]. (cid:3) Remark 4.4.
Our construction of the bootstrap category ZZ bt ∞ (or ZZ bt ∞ op ) enables us togive an algebraic recipe to compute its morphism groups. More precisely, let C = ( ZZ bt ∞ ) op and set Z ( X ) = C ( C , X ) (the mapping spectrum) for every X ∈ C . Note that Z ( X ) is amodule spectrum over R = End C ( C ). Then for any X , X ∈ C there is an equivalence ofspectra C ( X , X ) ∼ → Mod R ( Z ( X ) , Z ( X )) and there is a convergent spectral sequence (see,for instance, Corollary 4.15 of [2])E p,q = Ext p,qπ −∗ ( R ) ( π −∗ ( Z ( X )) , π −∗ ( Z ( X ))) ⇒ π − p − q Mod R ( Z ( X ) , Z ( X )) . We now introduce an equivalence relation on C ∗ -algebras. Let A ∈ SC ∗ , such that Σ ∞ S ′ ( A ) isa coidempotent object in NSp ′ . As explained before one can consider the bootstrap categorygenerated by a set of compact objects V in NSp ′ [ A − ]. Definition 4.5.
Let
B, C be two separable C ∗ -algebras. Then(1) we say that B, C are A -equivalent (denoted B ∼ A C ) if { R A ◦ Σ ∞ S ′ ( B ) } and { R A ◦ Σ ∞ S ′ ( C ) } generate equivalent bootstrap categories in NSp ′ [ A − ] and(2) we say B A C if the bootstrap category generated by { R A ◦ Σ ∞ S ′ ( B ) } is containedin the bootstrap category generated by { R A ◦ Σ ∞ S ′ ( C ) } . Remark 4.6.
It is clear that ∼ A is reflexive, symmetric, and transitive, i.e., an equivalencerelation and A is reflexive, anti-symmetric, and transitive, i.e., a partial order. For A, B ∈ SC ∗ it is an interesting problem to determine whether A ∼ C B , i.e., whether { Σ ∞ S ′ ( A ) } and { Σ ∞ S ′ ( B ) } generate equivalent bootstrap categories in NSp ′ . Remark 4.7.
It is conceivable that a finer equivalence relation than the one in Definition4.5 is more interesting. One can define
B, C to be thick A -equivalent (denoted B ∼ tA C ) if { R A ◦ Σ ∞ S ′ ( B ) } and { R A ◦ Σ ∞ S ′ ( C ) } generate equivalent thick subcategories in hNSp ′ [ A − ].For an E -ring (or an A ∞ -ring spectrum) R one denotes the stable ∞ -category of right R -modules by RMod R (see Chapter 7 of [22]). Much like classical Morita theory in algebra,modulo technicalities, Morita theory in stable homotopy theory tries to ascertain whentwo E -rings have equivalent module categories. One key result in this direction is theclassification of stable model categories (under some hypotheses) by Schwede–Shipley [40]. Proposition 4.8. If A ∼ C B , then RMod
End(Σ ∞ S ′ ( A )) ≃ RMod
End(Σ ∞ S ′ ( A )) . Proof.
Observe first that Ind ω ( h Σ ∞ S ′ ( A ) i ) and Ind ω ( h Σ ∞ S ′ ( B ) i ) are both stable and presentable ∞ -categories that are compactly (graded) generated by Σ ∞ S ′ ( A ) and Σ ∞ S ′ ( B ) respectively.Hence by the Schwede–Shipley classification there are equivalencesInd ω ( h Σ ∞ S ′ ( A ) i ) ≃ RMod
End(Σ ∞ S ′ ( A )) and Ind ω ( h Σ ∞ S ′ ( B ) i ) ≃ RMod
End(Σ ∞ S ′ ( B )) (see also Theorem 7.1.2.1 of [22] for the version needed here). Now A ∼ C B implies bydefinition that Ind ω ( h Σ ∞ S ′ ( A ) i ) ≃ Ind ω ( h Σ ∞ S ′ ( B ) i ). (cid:3) Remark 4.9.
Algebraic K-theory and topological Hochschild homology are both Moritainvariant functors. For an E -ring R the algebraic K-theory functor takes as input the stable ∞ -category of compact R -modules. Such functors can be deployed to test whether A ∼ C B . et p, q be relatively prime infinite supernatural numbers and Z p,q be a prime dimensiondrop C ∗ -algebra. Let Q be the universal UHF algebra of infinite type. Theorem 4.10. In Z -colocalized noncommutative spectra NSp ′ [ Z − ] the following hold:(1) C ∼ Z Z (2) C Z Z p,q (3) K ≁ Z K ˆ ⊗ Q Proof.
Observe that R A ◦ Σ ∞ S ′ ( B ) ≃ Σ ∞ S ′ ( B ˆ ⊗ A ). Thus for (1) we need to show that thebootstrap categories generated by { Σ ∞ S ′ ( Z ) } and { Σ ∞ S ′ ( Z ˆ ⊗ Z ) } are equivalent. Since Z isstrongly self-absorbing the unital embedding id ⊗ Z : Z → Z ˆ ⊗ Z is a unital ∗ -homomorphismbetween strongly self-absorbing C ∗ -algebras that is homotopic to an isomorphism. HenceΣ ∞ S ′ ( Z ) ∼ ← Σ ∞ S ′ ( Z ˆ ⊗ Z ) in NSp ′ [ Z − ], from which the assertion follows.Thanks to (1) for (2) we simply need to show Z Z Z p,q , i.e., we need to show thatthe bootstrap category generated by { Σ ∞ S ′ ( Z ˆ ⊗ Z ) } is contained in that of { Σ ∞ S ′ (Z p,q ˆ ⊗ Z ) } .Recall that it is shown in Proposition 3.5 of [34] that there are unital embeddings Z → Z p,q and Z p,q → Z , so that the composition is a unital endomorphism Z → Z . Since thespace of unital endomorphisms of Z is contractible (see Theorem 2.3 of [10]) the abovecomposition is homotopic to the identity. By tensoring with Z and applying Σ ∞ S ′ ( − ) we findthat the composition Σ ∞ S ′ ( Z ˆ ⊗ Z ) → Σ ∞ S ′ (Z p,q ˆ ⊗ Z ) → Σ ∞ S ′ ( Z ˆ ⊗ Z ) is homotopic to the identity in NSp ′ [ Z − ]. Hence we conclude that Σ ∞ S ′ ( Z ˆ ⊗ Z ) is a retract of Σ ∞ S ′ (Z p,q ˆ ⊗ Z ) in NSp ′ [ Z − ]. Sincethe bootstrap category generated by { Σ ∞ S ′ (Z p,q ˆ ⊗ Z ) } is a localizing subcategory of NSp ′ [ Z − ],it must be closed under retracts, i.e., Σ ∞ S ′ ( Z ˆ ⊗ Z ) belongs to this bootstrap category. Now theassertion follows from Proposition 5.3.5.11 (1) of [23].We prove (3) by contradiction. Set E = End(Σ ∞ S ′ ( K )) and E = End(Σ ∞ S ′ ( K ˆ ⊗ Q )) andassume K ∼ Z K ˆ ⊗ Q . This implies that the topological Hochschild homology E and E areequivalent. However, this is false. (cid:3) Bousfield equivalence.
An important concept in the global structure of the stablehomotopy category is
Bousfield equivalence [3]. It is possible to consider a variant of it fornoncommutative spectra. We present a definition that is slightly different from the routinegeneralization of Bousfield equivalence for spectra. Indeed, unlike localizations with respectto arbitrary spectra we focus on smashing colocalizations with respect to certain C ∗ -algebras(their stabilizations are compact objects in NSp ′ ). Definition 4.11.
Let
A, B ∈ SC ∗∞ op , such that Σ ∞ S ′ ( A ) , Σ ∞ S ′ ( B ) are coidempotent objects in NSp ′ (see subsection 1.1). Then(1) we say that A and B are Bousfield equivalent if NSp ′ [ A − ] ≃ NSp ′ [ B − ] (as stable ∞ -categories) and(2) we say that A (cid:22) B if there is a fully faithful functor NSp ′ [ A − ] ֒ → NSp ′ [ B − ]. Example 4.12.
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