Algebraic K-theory of stable ∞ -categories via binary complexes
aa r X i v : . [ m a t h . K T ] J a n ALGEBRAIC K –THEORY OF STABLE ∞ –CATEGORIES VIABINARY COMPLEXES DANIEL KASPROWSKI AND CHRISTOPH WINGES
Abstract.
We adapt Grayson’s model of higher algebraic K –theory usingbinary acyclic complexes to the setting of stable ∞ –categories. As an appli-cation, we prove that the K –theory of stable ∞ –categories preserves infiniteproducts. Introduction
Using binary acyclic complexes, Grayson [Gra12] gave a description of higheralgebraic K –theory for exact categories in terms of generators and relations. Thepresent article shows that Grayson’s picture of higher algebraic K –theory admitsa concise description in the context of stable ∞ –categories, using the languageintroduced by Blumberg–Gepner–Tabuada in [BGT13]; see Section 2 for a quickrecollection of the notions of additive and localizing invariants as well as the asso-ciated categories of motives. Let U add and U loc denote the universal additive andlocalizing invariants from [BGT13].In Section 3, we define analogs F q C and B q C of acyclic complexes and Grayson’sbinary acyclic complexes in the setting of stable ∞ –categories. Following Grayson’sargument, we obtain the following analog of [Gra12, Corollary 6.5].1.1. Theorem.
The cofiber Γ C of the ‘diagonal’ U loc (∆) : U loc ( F q C ) → U loc ( B q C ) is naturally equivalent to Ω U loc ( C ) . Each binary acyclic complex has a length. One may ask to which extent theabelian group π K (Γ r C ) defined in terms of binary acyclic complexes of a fixedlength r differs from the abelian group π K (Γ C ). Following the construction from[KW], we show in Section 4 that binary acyclic complexes of length 5 generatethe whole group π K (Γ C ), and we construct a split to binary acyclic complexes oflength 7. More explicitly, we show the following.1.2. Theorem.
Let C be an idempotent complete stable ∞ –category. The canonicalmap π K (Γ C ) → π K (Γ C ) is a surjection and the canonical map π K (Γ C ) → π K (Γ C ) ∼ = K ( C ) admits a natural section. The K –theory functor preserves filtered colimits and finite products for essen-tially trivial reasons. By work of Carlsson [Car95], it is known that the alge-braic K –theory functor commutes with infinite products of exact categories andof Waldhausen categories with a cylinder functor. This result plays an essentialrole in proofs of the integral Novikov conjecture, e.g. for virtually polycyclic groups[CP95, Ros04] and linear groups with a finite dimensional model for the classifyingspace for proper actions [RTY14, Kas15, Kas16]. As an application of Theorem 1.2,we will imitate the proof of Carlsson’s theorem given in [KW] and show the followingresult in Section 5. Mathematics Subject Classification.
Primary 19D99; Secondary 18F25.
Key words and phrases. binary acyclic complexes, higher algebraic K –theory, stable ∞ –categories. Theorem.
For every family {C i } i ∈ I of small stable ∞ -categories, the naturalmap K Y i ∈ I C i ! → Y i ∈ I K ( C i ) is an equivalence. The same result holds for connective algebraic K –theory K in-stead of non-connective K –theory K . The case of connective algebraic K –theory admits the following generalizationto the universal additive invariant.1.4. Theorem.
The universal additive invariant U add : Cat ex ∞ → M add commuteswith arbitrary products. In particular, any additive invariant that becomes corepre-sentable on the category of non-commutative motives M add commutes with arbitraryproducts. Acknowledgements.
We thank Akhil Mathew for helpful discussions and thereferee for useful comments on a previous version of the article. The authors aremembers of the Hausdorff Center for Mathematics at the University of Bonn. Thesecond author furthermore acknowledges support by the Max Planck Society andby Wolfgang L¨uck’s ERC Advanced Grant “KL2MG-interactions” (no. 662400).2.
Algebraic K –theory of stable ∞ –categories We understand algebraic K –theory as an invariant of stable ∞ –categories inthe sense of Lurie [Lur]. In this context, the work of Blumberg–Gepner–Tabuada[BGT13] provides a conceptual approach to algebraic K –theory which, in additionto some elementary facts about the class group K , is sufficient for the purposesof this article. The current section serves to recall the necessary statements andterminology.2.1. Algebraic K –theory via universal invariants. We work in the settingof quasicategories, the term ∞ –category being used synonymously throughout. Aquasicategory is stable if it admits all finite limits and colimits, and pushout squaresand pullback squares coincide [Lur, Proposition 1.1.3.4]. A functor between sta-ble ∞ –categories is exact if it preserves finite limits and colimits [Lur, Proposi-tion 1.1.4.1]. We denote by Cat ex ∞ the ∞ –category of small stable ∞ –categoriesand exact functors. Inside Cat ex ∞ we have the full subcategory of idempotent com-plete stable ∞ –categories Cat perf ∞ . The idempotent completion of any small stable ∞ –category is stable [Lur, Corollary 1.1.3.7], and the idempotent completion func-tor Idem : Cat ex ∞ → Cat perf ∞ provides a left adjoint to the inclusion functor.We are mainly interested in localizing and additive invariants in the sense ofBlumberg–Gepner–Tabuada [BGT13]. In order to give the definitions of these no-tions, we recall some additional terminology.2.1. Definition.
Let A and B be stable ∞ –categories, and let A ⊆ B be a fullsubcategory. We define the Verdier quotient B / A as the localization of B at thecollection of arrows in B whose cofiber is equivalent to an object in A .2.2. Definition (cf. [BGT13, Definition 5.12 and Proposition 5.13]) . An exactsequence in Cat ex ∞ is a sequence of exact functors A → B → C satisfying the following properties:(1) The functor
A → B is fully faithful.(2) The composed functor
A → C is trivial.(3) The induced exact functor Idem( B / A ) → Idem( C ) is an equivalence. LGEBRAIC K –THEORY OF STABLE ∞ –CATEGORIES VIA BINARY COMPLEXES 3 Our definition of an exact sequence of stable ∞ –categories does not matchexactly the definition and characterization in [BGT13]. However, [NS, Proposi-tion I.3.5] shows that Definition 2.2 is equivalent to the one in [BGT13]. We pre-fer the present formulation because it applies more directly in the arguments ofSection 3.2.3. Definition ([BGT13, Definition 8.1]) . Let D be a presentable stable ∞ –category. A functor F : Cat ex ∞ → D is a localizing invariant if it satisfies thefollowing properties:(1) F commutes with filtered colimits.(2) F sends exact sequences to cofiber sequences.2.4. Remark.
Note that [BGT13, Definition 8.1] also requires that F sends thecanonical functor C →
Idem( C ) to an equivalence for every C ∈
Cat ex ∞ . But this isa consequence of 2 by considering 0 → C → Idem( C ).2.5. Theorem ([BGT13, Theorem 8.7]) . There exists a presentable stable ∞ –category M loc and a localizing invariant U loc : Cat ex ∞ → M loc such that ( U loc ) ∗ : Fun Lex ( M loc , D ) → Fun loc (Cat ex ∞ , D ) is an equivalence for every presentable stable ∞ –category D , where Fun
Lex denotesthe ∞ –category of colimit-preserving functors, and Fun loc denotes the ∞ –categoryof localizing invariants. For the moment, we contend ourselves with the characterization of non-connectivealgebraic K –theory K : Cat ex ∞ → Sp as the localizing invariant corepresented by U loc (Sp ω ), where Sp ω denotes the stable ∞ –category of compact spectra [BGT13,Theorem 9.8]. That is,(2.6) K ( A ) ≃ Map( U loc (Sp ω ) , U loc ( A )) . The notion of an additive invariant is closely related to that of a localizing invariant.The difference lies in the fact that we require fewer exact sequences to be turnedinto cofiber sequences.2.7.
Definition ([BGT13, Definition 5.18]) . An exact sequence A f −→ B g −→ C ofstable ∞ –categories is split-exact if f and g admit right adjoints i : B → A and j : C → B , respectively, such that the unit id → if and counit gj → id are equiva-lences.2.8. Remark.
Note that in the above definition the unit id → if is automaticallyan equivalence since f is fully faithful. One can also prove that the counit gj → idis an equivalence by showing that j is fully faithful.2.9. Example.
The following is the canonical non-trivial example of a split-exactsequence of stable ∞ –categories. Let C be a stable ∞ –category, and consider thestable ∞ –category E ( C ) of cofiber sequences in C . Then E ( C ) fits into a split-exactsequence C k −→ E ( C ) q −→ C , in which the functor q projects to the cofiber, while k sends an object X to thecanonical cofiber sequence X id −→ X → Definition ([BGT13, Definition 6.1]) . Let D be a presentable stable ∞ –category. A functor F : Cat ex ∞ → D is an additive invariant if it satisfies the follow-ing properties:(1) F commutes with filtered colimits. DANIEL KASPROWSKI AND CHRISTOPH WINGES (2) F sends the canonical functor C →
Idem( C ) to an equivalence for every C ∈
Cat ex ∞ .(3) For every split-exact sequence A B C fi gj in Cat ex ∞ , the morphism F ( f ) + F ( j ) : F ( A ) ⊕ F ( C ) → F ( B ) is an equiva-lence.2.11. Theorem ([BGT13, Theorem 6.10]) . There exists a presentable stable ∞ –category M add and an additive invariant U add : Cat ex ∞ → M add such that ( U add ) ∗ : Fun Lex ( M add , D ) → Fun add (Cat ex ∞ , D ) is an equivalence for every presentable stable ∞ –category D , where Fun
Lex denotesthe ∞ –category of colimit-preserving functors, and Fun add denotes the ∞ –categoryof additive invariants. Assuming the non-connective algebraic K –theory functor K to be known, con-nective algebraic K –theory K : Cat ex ∞ → Sp can be characterized as the functor τ ≥ ◦ K , where τ ≥ : Sp → Sp denotes the functor taking connective covers. By[BGT13, Theorem 7.13], connective algebraic K –theory, considered as a colimit-preserving functor on M add , is corepresented by U add (Sp ω ). Note that with theabove definition, the connective algebraic K –theory of C only depends on Idem( C ).It is possible to define connective algebraic K -theory such that the values at C andIdem( C ) may differ, but here we are following [BGT13].2.2. Lower algebraic K –theory. Our arguments do require some additional in-formation about the lower algebraic K –groups of a small stable ∞ –category. For C a small stable ∞ –category, we denote by K ( C ) the abelian group generated byequivalence classes of objects in C , subject to the relation [ C ] + [ C ] = [ C ] forevery cofiber sequence C → C → C in C . In general, we have an isomorphism K (Idem( C )) ∼ = π K ( C ) ∼ = π K ( C ) . The next lemma gives a convenient criterion for equality of elements in K ( C ).2.12. Lemma (Heller’s criterion) . Let X and Y be objects in a stable ∞ –category C . Then [ X ] = [ Y ] ∈ K ( C ) if and only if there are cofiber sequences A → X ⊕ S → B and A → Y ⊕ S → B for some objects A , B , S ∈ C .Proof. The proof of [KW, Lemma 3.2] applies almost verbatim. (cid:3)
Fix an uncountable regular cardinal κ . The suspension of a stable ∞ –category C is defined to be Σ C := Ind ℵ ( C ) κ / C , where Ind ℵ ( C ) κ denotes the full subcategory of κ -compact objects in the Ind-completion of C , see also [BGT13, Section 2.4 and Section 9.1]. The stable ∞ –category Ind ℵ ( C ) κ admits a canonical Eilenberg swindle since it admits countablecoproducts, hence has contractible K –theory. Non-connective K –theory can beconstructed from connective K –theory by the formula K ( C ) ≃ colim n ∈ N Ω n K (Σ n C ) , LGEBRAIC K –THEORY OF STABLE ∞ –CATEGORIES VIA BINARY COMPLEXES 5 see [BGT13, Definition 9.6]. It follows that K ( C ) ≃ Ω n K (Σ n C ) for every n ∈ N ,and hence the negative K –groups of any stable ∞ –category can be described bythe formula π − n K ( C ) ∼ = K (Idem(Σ n C )) . Grayson’s model for stable ∞ –categories Let C be a stable ∞ –category.3.1. Definition (cf. [Lur, Definition 1.2.2.2]) . For a linearly ordered set I , denoteby I [1] the subposet of I × I containing all pairs ( i, j ) satisfying i ≤ j . An I –complexin C is a functor X : N ( I [1] ) → C such that X i,i is a zero object for all i , and suchthat for i ≤ j ≤ k the square X i,j X i,k X j,j X j,k is a pushout. Denote by S I C the full subcategory of Fun( I [1] , C ) spanned by the I –complexes. For I = [ n ], we also write S n C instead of S [ n ] C .An N –complex X is called bounded if there exists a natural number r ∈ N suchthat X ,i → X ,i +1 is an equivalence for i ≥ r . In this case, we say that X issupported on [0 , r ]. Let F C denote the full subcategory of S N C spanned by thebounded N –complexes. We call F C the stable ∞ –category of bounded complexes .Let F q C ⊆ F C be the full stable subcategory spanned by those bounded com-plexes X such that X ,i ≃ i ≫ Definition.
Let N δ denote the set of natural numbers considered as a discreteposet. The map N δ → N [1] , i ( i, i + 1) induces the functorgr : F C → M N C which associates to a bounded complex its (underlying) graded object .3.3. Definition.
The category of binary complexes B C is defined by the followingpullback square of ∞ –categories: B C F C F C L N C ⊤⊥ grgr The functor ⊤ is called the top projection , while ⊥ is the bottom projection . Bysubstituting F q C at appropriate places, we obtain the following variations of B C : B t C F q C F C L N C ⊤⊥ grgr B b C F C F q C L N C ⊤⊥ grgr B q C F q C F q C L N C ⊤⊥ grgr The objects of B t C , B b C and B q C are called binary ⊤ –acyclic complexes , binary ⊥ –acyclic complexes and binary acyclic complexes , respectively.3.4. Remark.
Occasionally, we will require an explicit description of maps K → B C , where K is some simplicial set. Since limits of stable ∞ –categories may becomputed in the category of all ∞ –categories [Lur, Theorem 1.1.4.4], it follows DANIEL KASPROWSKI AND CHRISTOPH WINGES from [Rie14, Example 17.7.3 and Remark 17.7.4] that maps p : K → B C may bedescribed by a pair ( p ⊤ , p ⊥ ) of maps K → F C together with a natural equivalencegr ◦ p ⊤ ∼ −→ gr ◦ p ⊥ in L N C . To specify a map K → B t C , K → B b C or K → B q C , weadditionally have to require that p ⊤ , p ⊥ or both of them map to the full subcategory F q C .Note that B q C , B t C , B b C and B C are all stable. Moreover, these categories fitinto a commutative square B q C B t C B b C B C in which all functors are fully faithful and exact. The identity functor on F C inducesthe diagonal functor ∆ : F C → B C . Similarly, the identity functor on F q C inducesa functor ∆ : F q C → B q C , and these fit into a commutative square of exact functors F q C B q C F C B C ∆∆ By definition, the diagonal functor ∆ is split by both ⊤ and ⊥ .3.5. Definition.
Let C be a small stable ∞ –category. Define the Grayson construc-tion Γ C on C to be the motiveΓ C := cofib( U loc ( F q C ) U loc (∆) −−−−−→ U loc ( B q C )) . Our goal is to show that Γ C represents Ω U loc ( C ). The proof follows closelyGrayson’s arguments in [Gra12]. Note that the functor ∆ : F q C → B q C is not fullyfaithful and hence we cannot directly form the Verdier quotient in Cat ex ∞ .Objects in the ∞ –category F C are filtered objects in C with a choice of allpossible filtration quotients. To make some of the upcoming proofs easier to read,we also define versions of these categories which do not include choices of filtrationquotients, which will make it easier to map into these categories.Let f C ⊆
Fun( N ( N ) , C ) be the full subcategory of bounded filtrations , i.e. the fullsubcategory spanned by those functors X which are essentially constant in all butfinitely many degrees and satisfy X ≃
0. By [Lur, Lemma 1.2.2.4], the forgetfulfunctor u : F C → f C induced by the map N → N [1] , i (0 , i ) is an equivalence.Denote by f q C ⊆ f C the full stable subcategory spanned by those X satisfyingcolim X ≃ u : F C ∼ −→ f C , we also obtain a functorgr : f C → L N C . This allows us to define stable ∞ –categories b C , b t C , b b C and b q C in analogy to Definition 3.3.The next lemma, which is reminiscent of the Gillet–Waldhausen theorem, butmuch easier to prove, tells us that we may concentrate on describing the K –theory of F C /F q C . Let ι : C → f C denote the functor induced by the projection N ( N ) → ∆ ,and let π : f C → C denote the colimit functor (which exists since filtrations in f C become essentially constant). The structure maps of the colimit provide a naturaltransformation τ : id → ι ◦ π .3.6. Lemma.
The functors ι : C → f C /f q C and π : f C /f q C → C induced by ι and π are equivalences.Proof. There is an evident equivalence π ◦ ι ≃ id C . Since π vanishes on f q C , thereis an induced exact functor π : f C /f q C → C as well as a natural transformation τ : id → ι ◦ π . We still have π ◦ ι ≃ id. Moreover, τ is a natural equivalence by LGEBRAIC K –THEORY OF STABLE ∞ –CATEGORIES VIA BINARY COMPLEXES 7 Definition 2.1 of the Verdier quotient since cofib( τ X ) lies in f q C for every X ∈ f C .The claim follows. (cid:3) In what follows, we will omit the overline decoration on ι and π ; it should alwaysbe clear from context whether we consider these as functors to/from the quotientcategory or the original category.3.7. Lemma.
There is a natural equivalence Γ C ≃ fib( U loc ( B q C ) U loc ( ⊤ ) −−−−−→ U loc ( F q C )) . Proof.
Since ⊤ ◦ ∆ ≃ id, we have the following commutative diagram, natural in C :0 U loc ( F q C ) U loc ( F q C )fib( U loc ( B q C ) U loc ( ⊤ ) −−−−−→ U loc ( F q C )) U loc ( B q C ) U loc ( F q C )fib( U loc ( B q C ) U loc ( ⊤ ) −−−−−→ U loc ( F q C )) Γ C id U loc (∆) idid U loc ( ⊤ ) ≃ (cid:3) Definition.
Let f r C ⊆ f C denote the full subcategory of bounded filtrationssupported on [0 , r ], and define b r C as the pullback b r C f r C f r C L N C ⊤⊥ grgr Definition.
Let p r : N ( N ) → ∆ denote the map characterized by sending i to 0 for i ≤ r and i to 1 for i ≥ r + 1. Furthermore, choose a zero object 0 in C anda section s of the trivial fibration C / → C . Define the functor ι r +1 : C → f r +1 C asthe composition C s −→ C / ⊆ Fun(∆ , C ) p ∗ r −→ f r +1 C . Lemma.
The map U loc ( ⊤ ) : U loc ( B C ) → U loc ( F C ) induced by the top projec-tion is an equivalence.Proof. Since we have a commutative square B C F C b C f C ⊤≃ ≃⊤ it suffices to prove the claim for the lower horizontal arrow.Since f C ≃ colim r f r C and finite limits commute with directed colimits (combine[Lur, Theorem 1.1.4.4 and Proposition 1.1.4.6] with [Lur09, Lemma 5.4.5.6]), wealso have b C ≃ colim r b r C . As U loc commutes with filtered colimits, it is enough toprove that U loc ( ⊤ ) : U loc ( b r C ) → U loc ( f r C ) is an equivalence for all r . We do thisby induction on r , the case r = 0 being trivial.Let gr r +1 : f r +1 C → C be the composition of gr : f r +1 C → L C with the projec-tion onto the ( r + 1)–th component. Then gr r +1 ◦ ι r +1 ≃ id C , and there exists anatural transformation id → ι r +1 ◦ gr r +1 . Since gr r +1 vanishes on f r C , there is an DANIEL KASPROWSKI AND CHRISTOPH WINGES induced functor gr r +1 : f r +1 C /f r C → C . On the quotient, the natural transforma-tion id → ι r +1 ◦ gr r +1 becomes an equivalence since its fiber is contained in f r C .Hence, gr r +1 : f r +1 C /f r C → C is an equivalence.Consider the functor gr r +1 ◦⊤ : b r +1 C → C next. Then ∆ ◦ ι r +1 defines a functorin the opposite direction which satisfies gr r +1 ◦⊤ ◦ ∆ ◦ ι r +1 ≃ id. There also existsa natural transformation id → ∆ ◦ ι r +1 ◦ gr r +1 ◦⊤ since gr r +1 ◦⊤ ≃ gr r +1 ◦⊥ , whichinduces a natural equivalence on b r +1 C /b r C by similar reasoning to the above.We thus have a commutative diagram b r +1 C /b r C f r +1 C /f r CC ⊤ gr r +1 ◦⊤ gr r +1 Since gr r +1 ◦ ⊤ and gr r +1 are equivalences, ⊤ is also an equivalence.In particular, we obtain a map of split cofiber sequences U loc ( b r C ) U loc ( b r +1 C ) U loc ( C ) U loc ( f r C ) U loc ( f r +1 C ) U loc ( C ) U loc ( ⊤ ) U loc ( ⊤ ) id By induction, we conclude that U loc ( b r C ) → U loc ( f r C ) is an equivalence for all r . (cid:3) Corollary.
There exists a natural equivalence Γ C ≃
Ω fib( U loc ( B C /B q C ) U loc ( ⊤ ) −−−−−→ U loc ( F C /F q C )) . Proof.
Consider the map of cofiber sequences U loc ( B q C ) U loc ( B C ) U loc ( B C /B q C ) U loc ( F q C ) U loc ( F C ) U loc ( F C /F q C ) U loc ( ⊤ ) U loc ( ⊤ ) U loc ( ⊤ ) Since U loc ( B C ) U loc ( ⊤ ) −−−−−→ U loc ( F C ) is an equivalence by Lemma 3.10, we obtain anatural equivalenceΩ fib( U loc ( B C /B q C ) U loc ( ⊤ ) −−−−−→ U loc ( F C /F q C )) ≃ fib( U loc ( B q C ) U loc ( ⊤ ) −−−−−→ U loc ( F q C )) . The claim follows by combining this with Lemma 3.7. (cid:3)
Definition.
Define the shift ( − )[1] : b C → b C as the functor induced by thefunctor f C → f C arising from the map of posets N → N , i max { , i − } .The next lemma is straightforward.3.13. Lemma.
The cofiber of the canonical natural transformation ( − )[1] → id b C takes values in the essential image of the diagonal functor ∆ : f q C → b q C . Recall that u : F C → f C denotes the forgetful functor.3.14. Proposition.
The functor π ◦ u ◦ ⊤ : B C /B t C → C is an equivalence.Proof.
It suffices to prove that π ◦ ⊤ : b C /b t C → C is an equivalence. The functor∆ ◦ ι : C → b C /b t C provides a right-inverse to π ◦ ⊤ .Note that gr ∆ ιπ ( X ⊤ ) is zero in all but the lowest degree. Therefore, the trans-formation τ ⊤ : ⊤ → ι ◦ π ◦ ⊤ and the zero transformation 0 : ⊥ → ι ◦ π ◦ ⊤ in-duce a natural transformation ( − )[1] → ∆ ◦ ι ◦ π ◦ ⊤ . In the resulting zig-zagid ← ( − )[1] → ∆ ◦ ι ◦ π ◦ ⊤ , the left-hand transformation becomes an equivalence LGEBRAIC K –THEORY OF STABLE ∞ –CATEGORIES VIA BINARY COMPLEXES 9 in b C /b t C by Lemma 3.13. Since the cofiber of the right-hand transformation liesin b t C , this transformation also becomes an equivalence in b C /b t C .Since ∆ ◦ ι is a right-inverse to π ◦ ⊤ , the claim follows. (cid:3) Remark.
By interchanging the roles of ⊤ and ⊥ , Proposition 3.14 also showsthat the functor π ◦ u ◦ ⊥ : B C /B b C → C is an equivalence.3.16.
Corollary.
The functor ⊤ : B C /B t C → F C /F q C is an equivalence.Proof. Since π is an equivalence by Lemma 3.6 and π ◦ ⊤ is an equivalence byProposition 3.14, the claim follows. (cid:3) Corollary.
There exists a natural equivalence U loc ( B t C /B q C ) ≃ fib( U loc ( B C /B q C ) U loc ( ⊤ ) −−−−−→ U loc ( F C /F q C )) . Proof.
Consider the map of cofiber sequences U loc ( B t C /B q C ) U loc ( B C /B q C ) U loc ( B C /B t C )fib (cid:18) U loc ( B C /B q C ) U loc ( ⊤ ) −−−−−→ U loc ( F C /F q C ) (cid:19) U loc ( B C /B q C ) U loc ( F C /F q C ) id U loc ( ⊤ ) Since U loc ( ⊤ ) : U loc ( B C /B t C ) → U loc ( F C /F q C ) is an equivalence by Corollary 3.16,the claim follows. (cid:3) Combining Corollary 3.11 and Corollary 3.17 we obtain the following result.3.18.
Corollary.
There is a natural equivalence Γ C ≃ Ω U loc ( B t C /B q C ) . Lemma.
Let X be a binary complex, i.e. an object of B C . Then [ X ⊤ ,k ] = [ X ⊥ ,k ] ∈ K ( C ) for all k ∈ N .Proof. This follows by an easy induction since gr( X ⊤ ) ≃ gr( X ⊥ ). (cid:3) Definition.
Denote by C χ the full subcategory of C spanned by those objects X satisfying [ X ] = 0 ∈ K ( C ).By Lemma 3.19, the functor π ◦ ⊥ restricts to a functor ⊥ χ : B t C → C χ whichvanishes on B q C .3.21. Proposition.
The functor ⊥ χ : B t C /B q C → C χ induced by π ◦ ⊥ is an equiv-alence. The proof of Proposition 3.21 relies on the following construction.3.22.
Construction.
Let X ∈ b C be a binary complex supported on [0 , r ] satisfying[colim X ⊤ ] = 0 ∈ K ( C ), and hence also [colim X ⊥ ] = 0 ∈ K ( C ) by Lemma 3.19.Let k ≥ r . Choose objects A , B , S ∈ C which fit into cofiber sequences A a −→ X ⊤ k ⊕ S b −→ B and A a ′ −→ S b ′ −→ B. These exist by virtue of Lemma 2.12.Define C ∈ f C as the filtered object0 → · · · → → A a −→ X ⊤ k ⊕ S → → → . . . , where A sits in degree k − X ⊤ k ⊕ S occupies degree k . Define a second filtered object Y ⊤ as X ⊤ → · · · → X ⊤ k − → X ⊤ k − ⊕ A X ⊤ ( k − ≤ k ) ⊕ a ′ −−−−−−−−−−→ X ⊤ k ⊕ S → → . . . We observe that gr( Y ⊤ ) and gr( X ⊥ ⊕ C ) are canonically equivalent. Hence, thegiven data combine to a new binary complex which we denote by µ ( X ). Note that µ ( X ) depends on the choice of k and the cofiber sequences A → X k ⊕ S → B and A → S → B .Moreover, there is a canonical morphism m : X → µ ( X ) such that m ⊥ is givenby the inclusion X ⊥ → X ⊥ ⊕ C , and such that m ⊤ is given by the inclusion of adirect summand up to degree k . Proof of Proposition 3.21.
It suffices to show that ⊥ χ : b t C /b q C → C χ is an equiv-alence. By taking vertical Verdier quotients in the commutative square of exactfunctors b q C b b C b t C b C we obtain an exact functor i : b t C /b q C → b C /b b C . This functor fits into the commu-tative square of exact functors b t C /b q C b C /b b CC χ C i ⊥ χ π ◦⊥ j We make the following claims:(1) ⊥ χ is essentially surjective.(2) i is fully faithful.Since these claims in conjunction with Remark 3.15 imply that ⊥ χ is an equivalence,it suffices to prove the claims.Let us first show claim (1). For any X ∈ C χ , apply Construction 3.22 to ∆( ι ( X ))to obtain a preimage of X under ⊥ χ .For claim (2), we rely on [NS, Theorem I.3.3] to reduce the claim to showingthat the canonical map (induced by the inclusion of indexing categories)colim Z → Y ∈ b q C /Y Map b t C ( X, cofib( Z → Y )) → colim Z → Y ∈ b b C /Y Map b C ( X, cofib( Z → Y ))is an equivalence. Since b t C → b C is fully faithful, it suffices to show that theinclusion b q C /Y ⊆ b b C /Y is cofinal. Using [Lur09, Theorem 4.1.3.1], this in turn canbe reduced to showing that F := b q C /Y × b b C /Y ( b b C /Y ) ( Z → Y ) / is weakly contractible for all Z → Y ∈ b b C /Y . In fact, we claim that F is filtered.Let K be a finite simplicial set, and suppose we are given a map f : K → F .Then f corresponds to a diagram ∆ ⋆ K → b b C /Y with the following properties: • The restriction to ∆ ⋆ ∅ classifies the object Z → Y . • The restriction to K factors via b q C /Y .Since b b C admits finite colimits, there is a universal cone b f ′ : (∆ ⋆ K ) ⊲ → b b C /Y .Let W ∈ b b C denote the object classified by the cone point of b f ′ . Choose k ∈ N such that both Y and W are supported on [0 , k ]. Now apply Construction 3.22 to LGEBRAIC K –THEORY OF STABLE ∞ –CATEGORIES VIA BINARY COMPLEXES 11 W to obtain µ ( W ). We claim that the dashed arrow in the following diagram canbe filled in such that the resulting triangle commutes: W µ ( W ) Y m To construct the required 2–simplex in b C , we rely on Remark 3.4.For any filtered object V , let V ≤ k denote the restriction of V along the inclusion[0 , k ] → N . By construction, we find commutative diagrams ⊤⊥ ( W ) ≤ k ⊤⊥ ( µ ( W )) ≤ k ⊤⊥ ( Y ) ≤ km for ⊤⊥ ∈ {⊤ , ⊥} since ⊤⊥ ( W ) ≤ k is a direct summand in ⊤⊥ ( µ ( W )) ≤ k . These diagramsextend essentially uniquely to commutative diagrams in f C since ⊤⊥ ( Y )( i ) ≃ i > k . By definition, the chosen equivalences of graded objects patch together toyield a 2–simplex in b C .It follows that we can replace the cone point W by µ ( W ) and still obtain adiagram b f : (∆ ⋆ K ) ⊲ → b b C /Y . Since µ ( W ) ∈ b q C , this corresponds to a diagram K ⊲ → F , so F is filtered.This proves that the inclusion b q C /Y ⊆ b b C /Y is cofinal; thus, we have shownclaim (2). (cid:3) Proof of Theorem 1.1.
Combining Corollary 3.18 and Proposition 3.21, we obtainthe following sequence of natural equivalences:Γ
C ≃ Ω U loc ( B t C /B q C ) ≃ Ω U loc ( C χ ) . Since every object in C is a retract of an object in C χ , it follows that U loc ( C χ ) ≃U loc ( C ). This proves the theorem. (cid:3) Since the diagonal F q → B q defines a natural transformation of endofunctorson Cat ex ∞ , the construction Γ C can be iterated. Any word W of length n over thealphabet { F q , B q } defines a stable ∞ –category W C . Letting the word W vary, the2 n possible choices of W assemble into a commutative cube of dimension n . Bytaking the n –fold iterated cofiber of this cube, we obtain a motive Γ n C . To makesense of the naturality of the assignment C 7→ Γ n C , we choose, once and for all, onepreferred order of taking cofibers, say reading the word W from left to right.3.23. Corollary.
For all n ≥ there is a natural equivalence Γ n C ≃ Ω n U loc ( C ) . Proof.
The proof is by induction, with the case n = 1 being Theorem 1.1. Byinduction hypothesis, and using Theorem 1.1 again, we haveΓ n +1 C ≃ cofib(Γ n F q C Γ n ∆ −−−→ Γ n B q C ) ≃ cofib(Ω n U loc ( F q C ) U loc (∆) −−−−−→ Ω n U loc ( B q C )) ≃ Ω n Γ C≃ Ω n +1 U loc ( C ) . (cid:3) Corollary.
For all n ≥ there are natural equivalences K (Γ n C ) ≃ Ω n K ( C ) and K (Γ n C ) ≃ Ω n τ ≥ n K ( C ) . Proof.
Using Corollary 3.23, the statement about non-connective K –theory is im-mediate because K is a localizing invariant. The claim about connective K –theoryfollows by taking connective covers. (cid:3) From Corollary 3.24, we can now deduce a more algebraic description of higher K –groups resembling [Gra12, Corollary 7.4].3.25. Proposition.
Let C be an idempotent complete stable ∞ –category and let n ≥ . Then π n K ( C ) ∼ = π K (Γ n C ) admits the following presentation.It is the abelian group generated by equivalence classes [ X ] of objects in ( B q ) n C ,subject to the relations (1) [ X ] = [ X ′ ] + [ X ′′ ] whenever there exists a cofiber sequence X ′ → X → X ′′ in ( B q ) n C ; (2) [ X ] = 0 if X lies in the essential image of some diagonal functor ( B q ) k F q ( B q ) n − k − C ∆ −→ ( B q ) n C , ≤ k ≤ n − . By abuse of notation, we will refer to the above group also by K (Γ n C ) .Proof. Consider first the case n = 1. Since idempotent completeness can be charac-terized by the existence of certain colimits in C (cf. [Lur09, Section 4.4.5]), it followsfrom [Lur09, Corollary 5.1.2.3] that F q C is idempotent complete. As gr : F q C → L N C preserves colimits, B q C is idempotent complete by [Lur09, Lemma 5.4.5.5].This allows us to identify K ( F q C ) ∼ = π K ( F q C ) and K ( B q C ) ∼ = π K ( B q C ).Since the diagonal functor ∆ : F q C → B q C admits a retraction (by ⊤ or ⊥ ), weobtain a split exact sequence of abelian groups0 → K ( F q C ) → K ( B q C ) → π K (Γ C ) → . By Corollary 3.23, we have π K ( C ) ∼ = π K (Γ C ), and the claim follows immediately.The general case follows by considering cubes of higher dimensions. (cid:3) Shortening binary complexes
Our next goal is to show that the explicit description of K n in Proposition 3.25includes a large number of superfluous generators and relations. To make this state-ment precise, we introduce the following variations of the categories F q C , B q C andthe motive Γ C . In this section, we always assume that C is idempotent complete.4.1. Definition.
Let r ≥
0. Denote by F qr C ⊆ F q C the full subcategory of boundedfiltrations supported on [0 , r ]. Define B qr C as the pullback of the diagram F qr C gr −→ r M i =0 C gr ←− F qr C , and set Γ r C := cofib( U loc ( F qr C ) U loc (∆) −−−−−→ U loc ( B qr C )) ∈ M loc . The evident inclusion functors induce a map Γ r C → Γ C . The key result of thissection, Proposition 4.3 below, states that the induced homomorphism K (Γ r C ) → K (Γ C ) admits a natural section for r = 7.For convenience, we regard Γ r C as the cofiber of the map U loc ( f qr C ) U loc (∆) −−−−−→U loc ( b qr C ).Note that the complexes in f C start with a zero object in degree 0. In thissection we will suppress this zero and write complexes starting with degree 1.Let X ∈ b q C . By Lemmas 2.12 and 3.19 we can choose objects A k , B k , S k fittinginto cofiber sequences A k → X ⊤ k ⊕ S k → B k and A k → X ⊥ k ⊕ S k → B k for each k ≥
3. If X ⊤ k and X ⊥ k are both trivial, also choose A k , B k and S k to be trivial. LGEBRAIC K –THEORY OF STABLE ∞ –CATEGORIES VIA BINARY COMPLEXES 13 Let f ⊤ k : B k → Σ A k and f ⊥ k : B k → Σ A k be the induced maps. Define Y to be thecomplex with Y ⊤ := X ⊤ X ⊤ X ⊤ . . . B Σ A . . . ⊕ f ⊥ and Y ⊥ := X ⊥ X ⊥ X ⊥ . . . B Σ A . . . ⊕ f ⊤ By construction, there is a canonical equivalence gr( Y ⊤ ) ≃ gr( Y ⊥ ).Similarly, define Y k for k ≥ Y ⊤ k := Σ − B k A k . . . X ⊤ k X ⊤ k +1 . . . B k +1 Σ A k +1 . . . Σ − f ⊥ k ⊕⊕ f ⊥ k +1 and Y ⊥ k := Σ − B k A k . . . X ⊥ k X ⊥ k +1 . . . B k +1 Σ A k +1 . . . Σ − f ⊤ k ⊕⊕ f ⊤ k +1 It follows again from the choice of f ⊤ k and f ⊥ k that we have canonical equivalencesgr( ⊤ ( Y k )) ≃ gr( ⊥ ( Y k )). Note that Y k is an object of b q C .4.2. Proposition. In K (Γ C ) we have [ X ] = X k ≥ [ Y k ] . In particular, K (Γ C ) → K (Γ C ) is surjective.Proof. We denote the morphisms X ⊤ k → X ⊤ k +1 by x ⊤ k . Let X ⊤ k +1 /X ⊤ k := cofib( x ⊤ k )and X ⊥ k +1 /X ⊥ k := cofib( x ⊥ k ). We have a morphism X [1] → Y with top componentgiven as follows and bottom component given analogously. The induced map onthe underlying graded is trivial.0 X ⊤ X ⊤ X ⊤ X ⊤ . . .X ⊤ X ⊤ X ⊤ ⊕ B Σ A . . . x ⊤ x ⊤ ( x ⊤ , x ⊤ x ⊤ x ⊤ x ⊤ ( x ⊤ ,
0) 0+ f ⊥ By inspection, we see that the fiber has top componentΣ − X ⊤ −→ Σ − X ⊤ /X ⊤ −→ Σ − X ⊤ /X ⊤ ⊕ Σ − B ⊕ Σ − f ⊥ −−−−−−→ X ⊤ ⊕ A x ⊤ +0 −−−−→ X ⊤ → . . . and bottom componentΣ − X ⊥ −→ Σ − X ⊥ /X ⊥ −→ Σ − X ⊥ /X ⊥ ⊕ Σ − B ⊕ Σ − f ⊤ −−−−−−→ X ⊥ ⊕ A x ⊥ +0 −−−−→ X ⊥ → . . . Note that the fiber splits as the direct sum of the diagonal of the complexΣ − X ⊤ −→ Σ − X ⊤ /X ⊤ −→ Σ − X ⊤ /X ⊤ → → . . . and the complex F with top component0 → → Σ − B , Σ − f ⊥ ) −−−−−−−→ X ⊤ ⊕ A x ⊤ +0 −−−−→ X ⊤ → X ⊤ → . . . and bottom component0 → → Σ − B , Σ − f ⊤ ) −−−−−−−→ X ⊥ ⊕ A x ⊥ +0 −−−−→ X ⊥ → X ⊥ → . . . We conclude that [ X [1]] = [ F ] + [ Y ] ∈ K (Γ C ). By Lemma 3.13, we have [ X ] =[ X [1]] and we can shift F down. The claim now follows by induction on the lengthof the support of X . (cid:3) Proposition.
Sending [ X ] to P k ≥ [ Y k ] yields a natural splitting of the naturalmap K (Γ C ) → K (Γ C ) . Proof.
We first show that P k ≥ [ Y k ] is independent of the choices of A k , B k and S k . Let k ≥ , − B k − B k Σ − X ⊤ k ⊕ Σ − X ⊥ k ⊕ Σ − S Σ − B k X ⊤ k ⊕ A k Σ − X ⊤ k +1 /X ⊤ k ⊕ A k ⊕ Σ − B k +1 X ⊤ k ⊕ A k X ⊤ k +1 ⊕ B k +1 X ⊤ k +1 ⊕ A k +1 X ⊤ k +1 Σ A k +1 X ⊤ k +2 ⊕ B k +2 X ⊤ k +2 ⊕ B k +2 A k +2 Σ A k +2
00 0 0 , Σ − f ⊥ k )0 (0 , Σ − f ⊥ k )(0 , Σ − f ⊥ k ) x ⊤ k ⊕ , − f ⊥ k +1 ) x ⊤ k +0 x ⊤ k +0 0+ f ⊥ k +1 x ⊤ k +1 ⊕ x ⊤ k +1 +00+ f ⊥ k +2 f ⊥ k +2 Note that the 2-cells on the right are all trivial, but this is not true for the 2-cellson the left. Interchanging ⊤ and ⊥ in (4.4) defines a second cofiber sequence ofbounded filtrations supported on [0 , b q C where the map on the underlying graded objects is trivial. Takingthe fiber of this map yields an object F k in b q C whose top component is given bythe left hand column in (4.4) and whose bottom component is given analogously.Note that the right-hand column in the two versions of (4.4) is Y k , while thefiber F k defines the sum of a shifted copy of Y k +1 and a binary complex containedin the essential image of the diagonal functor. Using Lemma 3.13, it follows that LGEBRAIC K –THEORY OF STABLE ∞ –CATEGORIES VIA BINARY COMPLEXES 15 [ Y k ] + [ Y k +1 ] ∈ K (Γ C ) is equal to the class of the binary acyclic complex definedby the middle column. Now it is enough to observe that the middle column isindependent of the choice of A k +1 , B k +1 and S k +1 .The independence of A , B and S follows in the same way.This shows that the assignment X P k ≥ [ Y k ] gives a well-defined map(4.5) ob b q C → K (Γ C ) , which evidently sends equivalent objects to the same class.Let X → X ′ → X ′′ be a cofiber sequence in b q C . Consider the cofiber sequences X ⊤ k → ( X ′ ) ⊤ k → ( X ′′ ) ⊤ k and X ⊥ k → ( X ′ ) ⊥ k → ( X ′′ ) ⊥ k , k ∈ N , as elements in K ( EC ).Since [ X ⊤ k ] = [ X ⊥ k ] and [( X ′′ ) ⊤ k ] = [( X ′′ ) ⊥ k ] in K ( C ) by Lemma 3.19, we concludefrom Example 2.9 that[ X ⊤ k → ( X ′ ) ⊤ k → ( X ′′ ) ⊤ k ] = [ X ⊥ k → ( X ′ ) ⊥ k → ( X ′′ ) ⊥ k ] ∈ K ( EC ) . From Lemma 2.12, it follows that there are cofiber sequences A k → A ′ k → A ′′ k , B k → B ′ k → B ′′ k and S k → S ′ k → S ′′ k fitting into cofiber sequences of cofibersequences as follows: A k X ⊤ k ⊕ S k B k A ′ k ( X ′ ) ⊤ k ⊕ S ′ k B ′ k A ′′ k ( X ′′ ) ⊤ k ⊕ S ′′ k B ′′ k and A k X ⊥ k ⊕ S k B k A ′ k ( X ′ ) ⊥ k ⊕ S ′ k B ′ k A ′′ k ( X ′′ k ) ⊥ k ⊕ S ′′ k B ′′ k Using these for the construction of the objects Y k , Y ′ k , Y ′′ k of b q as above, we obtaincofiber sequences Y k → Y ′ k → Y ′′ k . Hence (4.5) induces a well-defined homomor-phism K ( b q C ) → K (Γ C ).If X ⊤ ≃ X ⊥ , we can make our choices such that f ⊤ k ≃ f ⊥ k for all k ≥ Y ⊤ k ≃ Y ⊥ k for all k ≥
2. Therefore, the map induces a well-defined homomorphism K (Γ C ) → K (Γ C ). It is a split of the natural map K (Γ C ) → K (Γ C ) byProposition 4.2. (cid:3) The preceding Propositions 4.2 and 4.3 prove Theorem 1.2. We obtain the fol-lowing generalization to higher algebraic K -theory.4.6. Theorem.
The canonical map K (Γ n C ) → K (Γ n C ) is a surjection and thecanonical map K (Γ n C ) → K (Γ n C ) ∼ = K n ( C ) admits a natural section.Proof. We argue by induction. Proposition 4.2 and Proposition 4.3 prove the case n = 1, which is the start of the induction. The induction step is analogous to[Gra12, Remark 8.1], [KW, Proof of Theorem 1.4] and Corollary 3.23 above.We will only describe the induction for the existence of the natural section, theinduction for the surjectivity statement is completely analogous.In order to extend Proposition 4.3 from K to higher K –groups, we requirethe additional observation that Γ and Γ r commute. This can be morally seen bypermuting the two factors in N × N , but the formal argument is rather lengthy.Hence we will first complete the proof using this claim before giving the formalargument.The map K (Γ n C ) → K (Γ Γ n − C ) admits a natural section by Proposition 4.3because it is a natural retract of the homomorphism K (Γ B n − C ) → K (Γ B n − C ).Since Γ and Γ commute, it suffices to show that K (Γ n − Γ C ) → K (Γ n C ) ad-mits a natural section. Since this map is in turn a natural retract of the map K (Γ n − B q C ) → K (Γ n − B q C ), this follows from the induction assumption. What is left to do is to provide an argument why Γ and Γ may be permuted.Fix r ∈ N . Let W be a word of length n over the alphabet { B q , B qr } , and let σ ∈ S n be a permutation. We claim that there is a canonical equivalence W C ≃ W σ C ,where W σ denotes the word W permuted according to σ .Recall that the natural transformation gr : F q → L N C is obtained via pullbackwith a map of posets γ : N δ → N [1] . Letting N (0) := N [1] and N (1) := N δ , define for x = ( x , . . . , x n ) ∈ { , } n N ( x ) := n Y i =1 N ( x i ) . Consider the functor N : [1] n → { posets } , x N ( x )induced by γ . Then σ induces a natural isomorphism N ∼ = −→ σ ∗ N to the diagram ofposets obtained by permuting the coordinates according to σ .Let Fun( N , C ) : (∆ ) n → Cat ex ∞ denote the induced n –cube of stable ∞ –categories.Then we obtain an induced equivalence of functorsFun( σ ∗ N , C ) ∼ −→ Fun( N , C ) , which contains as a full subfunctor those cubes in which we restrict to F q and F qr at the appropriate places (according to the original choice of word W ).Let g Fun( N , C ) : (sd ∆ ) n → Cat ex ∞ denote the functor obtained from Fun( N , C ) byfurther precomposing with the map induced by the map of simplicial sets sd ∆ → ∆ which sends the endpoints of the subdivided 1–simplex to 0 and the subdivisionpoint to 1.Then W C is the limit of g Fun( N , C ), while W σ C can be obtained as the limit of g Fun( σ ∗ N , C ). Since we have seen that the diagrams g Fun( N , C ) and g Fun( σ ∗ N , C )are equivalent, we obtain the desired equivalence W C ≃ W σ C . Let W ′ denote the word over the alphabet { F q , F qr } obtained from W by replacing B with F . Then there is an evident diagonal transformation W ′ C ∆ −→ W C whichfits into a commutative square W ′ C W C W ′ σ C W σ C ∆ ≃ ≃ ∆ Applying U loc and taking horizontal cofibers proves that V C ≃ V σ C for any word V over the alphabet { Γ , Γ r } . In particular, Γ and Γ commute. (cid:3) Infinite products
This section is devoted to the proof of Theorem 1.3. The proof of Theorem 1.3for connective K -theory is almost verbatim the same as the proof of [KW, Theo-rem 4.1].5.1. Lemma.
The functor K : Cat ex ∞ → Ab commutes with infinite products.Proof. Let {C i } i ∈ I be a family of stable ∞ –categories. The natural comparisonmap K ( Q i ∈ I C i ) → Q i ∈ I K ( C i ) is obviously surjective. Injectivity follows fromLemma 2.12. (cid:3) LGEBRAIC K –THEORY OF STABLE ∞ –CATEGORIES VIA BINARY COMPLEXES 17 The next lemma, whose proof is similar to the argument in the proof of [KW,Theorem 1.2], shows that Verdier quotients are compatible with the formation ofproducts.5.2.
Lemma.
Let {D i → C i → C i / D i } i ∈ I be a family of Verdier sequences in Cat ex ∞ .Then there is a natural equivalence Y i ∈ I C i / Y i ∈ I D i ≃ Y i ∈ I C i / D i . Proof.
Consider the commutative diagram of stable ∞ –categories and exact func-tors Q i ∈ I C i Q i ∈ I C i / Q i ∈ I D i Q i ∈ I C i / D iℓℓ ′ f Since the localization functor ℓ is essentially surjective, so is f . Therefore, it sufficesto show that f is fully faithful. Using [NS, Theorem I.3.3] to compute mappingspaces in the localization and referring to [BSS, Lemma 3.10] for the fact thatfiltered colimits distribute over products in spaces, we conclude thatMap Q i ∈ I C i / Q i ∈ I D i (( X i ) i , ( Y i ) i ) ≃ colim (( Z i ) i → ( Y i ) i ∈ ( Q i ∈ I D i ) / ( Yi ) i Map Q i ∈ I C i (( X i ) i , cofib(( Z i ) i → ( Y i ) i )) ≃ colim (( Z i ) i → ( Y i ) i ∈ ( Q i ∈ I D i ) / ( Yi ) i Y i ∈ I Map C i ( X i , cofib( Z i → Y i )) ≃ Y i ∈ I colim Z i → Y i ∈ ( D i ) /Yi Map C i ( X i , cofib( Z i → Y i )) ≃ Y i ∈ I Map C i / D i ( X i , Y i ) ≃ Map Q i ∈ I C i / D i ( X i , Y i ) , so f is also fully faithful. (cid:3) Lemma.
Let {C i } i ∈ I be a family of stable ∞ –categories. The canonical functor Idem( Y i ∈ I C i ) → Y i ∈ I Idem( C i ) is an equivalence.Proof. The canonical functor Q i ∈ I C i → Q i ∈ I Idem( C i ) exhibits Q i ∈ I Idem( C i ) asan idempotent completion of Q i ∈ I C i in the sense of [Lur09, Definition 5.1.4.1]:Since idempotent completeness amounts to the existence of certain colimits ([Lur09,Section 4.4.5]) and colimits in a product category can be computed componentwise, Q i ∈ I Idem( C i ) is idempotent complete; moreover, every object in Q i ∈ I Idem( C i ) isa retract of an object in Q i ∈ I C i because this is true for each individual component. (cid:3) Proposition.
Let {C i } i ∈ I be a family of stable ∞ –categories. The comparisonmap π n K ( Y i ∈ I C i ) → Y i ∈ I π n K ( C i ) is an isomorphism for all n ∈ Z . Proof.
By Lemma 5.3 and the fact that K is a localizing invariant, we may assumewithout loss of generality that all C i are idempotent complete.The case n = 0 is provided by Lemma 5.1. For n ≥
1, we consider the commu-tative diagram K (Γ n Q i ∈ I C i ) Q i ∈ I K (Γ n C i ) K (Γ n Q i ∈ I C i ) Q i ∈ I K (Γ n C i ) K (Γ n Q i ∈ I C i ) Q i ∈ I K (Γ n C i ) ϕ in which the vertical homomorphisms are given by the section of Theorem 4.6 fol-lowed by the homomorphism induced by the canonical maps Γ n C i → Γ n C i . Inparticular, Theorem 4.6 together with Corollary 3.24 tells us that this diagramexhibits the comparison map π n K ( Q i ∈ I C i ) → Q i ∈ I π n K ( C i ) as a retract of themiddle horizontal homomorphism.Since F q ( Q i ∈ I C i ) ≃ Q i ∈ I F q C i and L i =0 ( Q i ∈ I C i ) ≃ Q i ∈ I L i =0 C i , we see that B q ( Q i ∈ I C i ) ≃ Q i ∈ I B q C i because limits commute with each other. We concludethat Γ ( Y i ∈ I C i ) ≃ Y i ∈ I Γ C i . Now it is immediate from Lemma 5.1 that ϕ is an isomorphism, which implies theclaim for n ≥ n <
0. Recall from Section 2.2 that π − n K ( C ) isnaturally isomorphic to K (Idem(Σ n C )) for n ≥ Q i ∈ I Ind ℵ ( C i ) κ and Ind ℵ ( Q i ∈ I C i ) κ have trivial K –theory since theyadmit infinite coproducts. Hence using Lemma 5.2, it is easy to see that K ( Y i ∈ I Σ n C i ) ≃ K (Σ n Y i ∈ I C i ) , and the claim follows by another application of Lemma 5.1. (cid:3) Proof of Theorem 1.3.
The theorem is an immediate consequence of Proposition 5.4and [Lur, Remark 1.4.3.8]. The claim about connective K –theory follows by apply-ing τ ≥ ; since τ ≥ is a right adjoint, it preserves products. (cid:3) Proof of Theorem 1.4.
By [BGT13, Theorem 7.13], we have for every stable ∞ –category A and compact idempotent complete stable ∞ –category B a natural equiv-alence of spectraMap( U add ( B ) , U add ( A )) ≃ K (Fun ex ( B , Idem( A ))) . LGEBRAIC K –THEORY OF STABLE ∞ –CATEGORIES VIA BINARY COMPLEXES 19 Let now {C i } i ∈ I be any family of stable ∞ –categories, and let B be a compact,idempotent complete stable ∞ –category. Then we haveMap( U add ( B ) , U add ( Y i ∈ I C i )) ≃ K (Fun ex ( B , Idem( Y i ∈ I C i ))) ( ∗ ) ≃ K (Fun ex ( B , Y i ∈ I Idem( C i ))) ≃ K ( Y i ∈ I Fun ex ( B , Idem( C i ))) ( ∗∗ ) ≃ Y i ∈ I K (Fun ex ( B , Idem( C i ))) ≃ Y i ∈ I Map( U add ( B ) , U add ( C i ))where ( ∗ ) follows from Lemma 5.3 and ( ∗∗ ) follows from Theorem 1.3. Since M add is a localization of Pre Sp ((Cat perf ∞ ) ω ) by [BGT13, Remark 6.8], it is generated bythe images of compact idempotent complete stable ∞ –categories under U add . Thisverifies the universal property of the product. Hence U add ( Y i ∈ I C i ) ≃ Y i ∈ I U add ( C i ) . (cid:3) Remark.
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Rheinische Friedrich-Wilhelms-Universit¨at Bonn, Mathematisches Institut,Endenicher Allee 60, 53115 Bonn, Germany
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