aa r X i v : . [ m a t h . K T ] M a y A universal coarse K -theory Ulrich Bunke ∗ Denis-Charles Cisinski † May 18, 2017
Abstract
In this paper, we construct an equivariant coarse homology theory with values inthe category of non-commutative motives of Blumberg, Gepner and Tabuada, withcoefficients in any small additive category. Equivariant coarse K -theory is obtainedfrom the latter by passing to global sections. The present construction extends jointwork of the first named author with Engel, Kasprowski and Winges by promotingcodomain of the equivariant coarse K -homology functor to non-commutative mo-tives. Contents K -theory of additive categories . . . . . . . . . . . . . . . . . . 72.3 Properties of the universal K -theory for additive categories . . . . . . . . . 12 X -controlled A -objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 The universal equivariant coarse homology theory . . . . . . . . . . . . . . 213.4 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.5 Functors on the orbit category . . . . . . . . . . . . . . . . . . . . . . . . . 23 References 27 ∗ Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, 93040 Regensburg, [email protected] † Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, 93040 Regensburg, [email protected] Introduction
In this paper we introduce a universal K -theory-like equivariant coarse homology theoryassociated to an additive category.Equivariant coarse homology theories have been introduced in [BEKWb] as the basicobjects of equivariant coarse homotopy theory. Equivariant coarse homology theoriesgive rise to Bredon-type equivariant homology theories by a construction which we willexplain in the Remark 1.5 below. Such a presentation of equivariant homology theo-ries plays an important role in the proofs of isomorphism conjectures, see e.g. [CP95],[BFJR04],[BLR08] .As an example of an equivariant coarse homology theory, in [BEKWb] we constructed anequivariant coarse K -homology functor K X G A associated to an additive category A . It isdefined as a composition of functorsK X G A : G BornCoarse V G A → Add K → Sp . (1.1)In this formula G BornCoarse denotes the category of G -bornological coarse spaces, and V G A is a functor which associates to a G -bornological coarse space X the additive categoryof equivariant X -controlled A -objects V G A ( X ). The functor K is a non-connective K -theory functor from additive categories to spectra [PW85].In [BGT13] Blumberg, Gepner, and Tabuada interpret algebraic K -theory as a localizinginvariant of small stable ∞ -categories. They construct a universal localizing invariant U loc : Cat ex ∞ → M loc , (1.2)where Cat ex ∞ denotes the ∞ -category of small stable ∞ -categories.The goal of the present paper is to construct a factorization of K X G A over U loc . To findsuch a factorization is very much in spirit of the ideas of Balmer-Tabuada [BT08] andmight be helpful in an approach to the “Mother Isomorphism Conjecture”. It also leedsto further new examples of equivariant coarse homology theories, see Example 1.2.In Section 2 we construct the functor Ch b ( − ) ∞ : Add → Cat ex ∞ which sends a smalladditive category A to the small stable ∞ -category Ch b ( A ) ∞ obtained from the boundedchain complexes over A by inverting homotopy equivalences. We then analyse the com-position UK : Add Ch b ( − ) ∞ → Cat ex ∞ U loc → M loc . In Section 3 we first review bornological coarse spaces and define the category
BornCoarse .We furthermore recall the notion of an equivariant coarse homology theory, and the defi-nition of the categories of X -controlled A -objects V G A ( X ). Then we continue and define,in analogy to (1.1), the functorUK X G A : G BornCoarse V G A → Add UK → M loc . These are only some papers of a vast list we will not try to review here. K -homology functor (1.1)by K X G A ( − ) ≃ map M loc ( U loc ( S ω ∞ ) , UK X G A ( − )) , (1.3)where S ω ∞ denotes the small stable ∞ -categories of compact spectra.Our main theorem is: Theorem 1.1 (Theorem 3.7) . The functor UK X G A is an equivariant coarse homologytheory. The arguments for the proof are very similar to the case of usual K -theory and essentiallycopied from [BEKWb].From the universal coarse homology theory UK X G A associated to A one can derive newcoarse homology theories which have not been considered so far. Example 1.2.
An example is topological Hochschild homology THH. In [BGT13] topo-logical Hochschild homology was characterized as a localizing invariantTHH :
Cat ex ∞ → Sp . By the universal property of the morphism (1.2) there is an essentially unique factorization
Cat ex ∞U loc $ $ ■■■■■■■■■ THH / / Sp M loc THH ∞ < < ②②②②②②②② . This allows to define an equivariant coarse homology theoryTHH X G A : G BornCoarse UK X G A → M loc THH ∞ → Sp . In Section 3.5 we package the coarse homology functors UK X H A for the subgroups H of G together into a functorUK X A : G BornCoarse → Fun ( Orb ( G ) op , M loc ) . We want to stress the following point. In order to define the functor UK X A (Definition3.16) it is not necessary to know how to define equivariant coarse homology homologytheories. One only needs the non-equivariant version UK X A . Equivariance is built inusing the coinduction functor which right-Kan extends G -equivariant objects to functorson Orb ( G ) op . This idea might be helpful in other cases where the non-equivariant versionof a coarse homology is already known, while the details of a construction of an equivariantversions are not yet fixed. The construction of the equivariant homology theories using thecoinduction functor plays an important role in the proof of the descent principle discussedin [BEKWa].Our main result here, besides of the construction of UK X A , is:3 heorem 1.3 (Theorem 3.17) . UK X A is an equivariant homology theory. Remark 1.4.
The descent principle itself does not seem to apply to the universal invari-ants. The main additional input of the proof of the descent principle is the fact shownby Pedersen [Car95] that the connective algebraic K -theory functor K ≥ : Add → Sp ≥ preserves products. This result extends to the non-connective K -theory, but probably notto the functor UK. Remark 1.5.
By Elmendorff’s theorem the category
Fun ( Orb ( G ) op , Spc ) can be consid-ered as the home of G -equivariant homotopy theory. Bredon-type G -equivariant C -valuedhomology theories are then represented by objects in Fun ( Orb ( G ) , C ), see Davis-L¨uck[DL98]. The values of the functor UK X A must not be confused with such homology the-ories since UK X A takes values in contravariant functors instead of covariant ones. TheDavis-L¨uck type equivariant homology theories can be recovered as follows.To every set S we can associate a bornological coarse space S min,max given by S with theminimal coarse and the maximal bornological structures. This gives a functor T : Set → BornCoarse , S S min,max , and hence a functor T G : Fun ( BG,
Set ) → G BornCoarse . The compositionUK G A : Orb ( G ) → Fun ( BG,
Set ) T G → G BornCoarse UK X G A → M loc is a Bredon-type M loc -valued equivariant homology theory. The equivariant algebraic K associated to an additive category as considered in [DL98], [BLR08], [BFJR04] can nowbe recovered in the style of (1.3) byK G A ( − ) ≃ map M loc ( U loc ( S ω ∞ ) , UK G A ( − )) . Indeed this follows from (1.3) and the equivalence K G A ( − ) ≃ K X G A ◦ T G which we checkin [BEKWb]. Acknowledgement: A great part of this work is a side product of the collaboration with A.Engel, D. Kasprovski, Ch. Winges and M. Ullmann on various projects in equivariantcoarse homotopy theory. The authors were supported by the SFB 1085 (DFG).
Let
Cat ex ∞ denote the ∞ -category of small stable ∞ -categories and exact functors.4 efinition 2.1.
1. A morphism u : A → B in Cat ex ∞ is a Morita equivalence if it induces an equivalenceof Ind -completions
Ind ( u ) : Ind ( A ) → Ind ( B ) .2. An exact sequence in Cat ex ∞ is a commutative square of the form A i / / (cid:15) (cid:15) B (cid:15) (cid:15) / / C in which the morphism i is fully faithful, such that the induced functor B / A → C is a Morita equivalence. Remark 2.2.
For a stable ∞ -category A , the Ind -completion can be identified withthe ∞ -category of exact functors from the opposite of A to the stable ∞ -category ofspectra. The idempotent completion Idem( A ) of A can be defined as the full subcategoryof compact objects in the Ind -completion of A . Moreover, the triangulated category Ho ( Ind ( A )) is compactly generated (more precisely, the objects of A form a generatingfamily of compact generators), and the triangulated category Ho (Idem( A )) is canonicallyequivalent to the idempotent completion (or karoubianization) of the category Ho ( A ) (inthe classical sense). Therefore, one can characterize Morita equivalences as the exactfunctors A → B inducing an equivalence of ∞ -categories Idem( A ) → Idem( B ).Following [BGT13] a functor Cat ex ∞ → C with C a presentable stable ∞ -category is calleda localizing invariant if it inverts Morita equivalences, sends exact sequences to fibresequences, and preserves filtered colimits.In [BGT13] a universal localizing invariant U loc : Cat ex ∞ → M loc has been constructed. The non-connective K -theory of small stable ∞ -categories is alocalizing invariant. It can be derived from the universal invariant as follows. Let S ω ∞ be the small stable ∞ -category of compact spectra and A be a small stable ∞ -category.Then by [BGT13, Thm. 1.3] the non-connective K -theory spectrum of A is given by K ( A ) ≃ map M loc ( U loc ( S ω ∞ ) , U loc ( A )) . (2.1) Remark 2.3.
There is also a connective K -theory spectrum K W ald ( A ), which is theconnective cover of K ( A ), and whose value at π is the usual Grothendieck group of thetriangulated category Ho (Idem( A )). Definition 2.4.
1. A stable ∞ -category A is flasque if there exists an exact functor S : A → A suchthat there exists an equivalence id A ⊕ S ≃ S . . A delooping of a stable ∞ -category A is a collection of exact sequences of ∞ -categories of the form S n ( A ) / / (cid:15) (cid:15) F n ( A ) (cid:15) (cid:15) / / S n +1 ( A ) for all non negative integer n , such that F n ( A ) is flasque for all n , together with aMorita equivalence A → S ( A ) . Given a delooping of a stable ∞ -category A as in the definition above, we have thefollowing commutative squares of spectra: K W ald ( S n ( A )) / / (cid:15) (cid:15) K W ald ( F n ( A )) (cid:15) (cid:15) / / K W ald ( S n +1 ( A )) . (2.2)Since K W ald ( F n ( A )) ≃
0, these squares define canonical maps K W ald ( S n ( A )) → Ω( K W ald ( S n +1 ( A ))) (2.3)and hence maps Ω n ( K W ald ( S n ( A ))) → Ω n +1 ( K W ald ( S n +1 ( A ))) . (2.4) Proposition 2.5.
Given any choice of delooping of a stable ∞ -category A , there is acanonical equivalence of spectra colim n ≥ Ω n ( K W ald ( S n ( A ))) ≃ K ( A ) . Furthermore, for any non negative integer n , there is a canonical isomorphism K (Idem( S n ( A ))) ∼ = π − n ( K ( A )) , where K = π ◦ K W ald denotes the Grothendieck group functor.Proof.
We have cofiber sequences of the form K ( S n ( A )) / / (cid:15) (cid:15) K ( F n ( A )) (cid:15) (cid:15) / / K ( S n +1 ( A )) , whence equivalences K ( S n ( A )) → Ω( K ( S n +1 ( A ))) (2.5)from which we deduce an equivalence K ( A ) ≃ −→ colim n ≥ Ω n ( K ( S n ( A ))) . (2.6)6y naturality of the map K W ald → K , we get the commutative square K W ald ( A ) / / (cid:15) (cid:15) colim n ≥ Ω n ( K W ald ( S n ( A ))) (cid:15) (cid:15) K ( A ) ≃ / / colim n ≥ Ω n ( K ( S n ( A )))of which any inverse of the bottom horizontal map gives a map colim n ≥ Ω n ( K W ald ( S n ( A ))) → K ( A ) (2.7)Since K W ald ( S n ( A )) is the connective cover of K ( S n ( A )), the canonical map from K W aldj ( S n ( A ))to K j ( S n ( A )) ∼ = K j − n ( A ) is an isomorphism for all non negative integers j . Since theformation of stable homotopy groups commutes with small filtered colimits, we havecanonical isomorphisms of groups for any integer i : π i (cid:0) colim n ≥ Ω n ( K W ald ( S n ( A ))) (cid:1) ∼ = colim n ≥ π i + n ( K W ald ( S n ( A ))) ∼ = colim n ≥− i π i + n ( K W ald ( S n ( A ))) ∼ = colim n ≥− i π i + n ( K ( S n ( A ))) ∼ = colim n ≥ π i + n ( K ( S n ( A ))) ∼ = π i (cid:0) colim n ≥ Ω n ( K ( S n ( A ))) (cid:1) ∼ = K i ( A ) . Therefore, map (2.7) is a stable weak homotopy equivalence. K -theory of additive categories An additive category A can be considered as a Waldhausen category whose weak equiva-lences are the isomorphisms and whose cofibrations are the inclusions of direct summands.The connective K -theory K W ald ( A ) is defined as the Waldhausen K -theory of this Wald-hausen category (this agrees with Quillen’s definition, up to a functorial equivalence).This construction can be refined in order to produce a non-connective K -theory spectrum K ( A ) such that π i ( K ( A )) ∼ = π i ( K W ald ( A )) for all integers i , with i ≥ i ≥ A isidempotent complete); see [PW85, Sch06]. In Proposition 2.20 below we provide an alter-native description as a specialization of a universal K -theory for additive categories.We consider the categories Add and
RelCat of small additive categories and relativecategories. We have a functor( Ch b ( − ) , W h ) : Add → RelCat (2.8)which sends an additive category A to the relative category ( Ch b ( A ) , W h ) of boundedchain complexes and homotopy equivalences. We have a localization functor L : RelCat → Cat ∞ (2.9)7hich sends a relative category ( C , W ) to the localization C [ W − ]. For an additivecategory A we will use the notation Ch b ( A ) ∞ := L ( Ch b ( A ) , W h ) . Remark 2.6.
Assume that we model ∞ -categories by quasi-categories. Then a modelfor the localization functor is given by( C , W ) N (( L HW C ) fib ) , where L HW C is the hammock localization producing a simplicial category from a relativecategory, ( − ) ( fib ) is the fibrant replacement in simplicial categories, and N is the coherentnerve functor which sends fibrant simplicial categories to quasi categories.As explained for instance in [CP97, Section 4], the category Ch b ( A ) has the structure ofa cofibration category where1. weak equivalences are homotopy equivalences,2. cofibrations are morphisms of chain complexes which are degree-wise split injections.Note that if A → B is a cofibration in Ch b ( A ), then the quotient A/B exists. In thiscase we say that A → B → A/B is a short exact sequence.Let ℓ : Ch b ( A ) → Ch b ( A ) ∞ (2.10)denote the canonical morphism. Proposition 2.7. If A is a small additive category, then Ch b ( A ) ∞ is a small stable ∞ -category. Moreover,1. (2.10) preserves the null object.2. (2.10) sends short exact sequences to cofiber sequences.Proof. Let W ch denote the subset of W h of homotopy equivalences which are in additioncofibrations. Lemma 2.8.
The morphism of relative categories ( Ch b ( A ) , W ch ) → ( Ch b ( A ) , W h ) in-duces an equivalence L ( Ch b ( A ) , W ch ) → Ch b ( A ) ∞ of ∞ -categories.Proof. Since Ch b ( A ) is a cofibration category, by Ken Brown’s Lemma every morphism f : A → B in Ch b ( A ) has a factorization A f / / i (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ B r p p ✔⑦✐ C p, ≃ ? ? ⑦⑦⑦⑦⑦⑦⑦⑦ , i is a cofibration, p is a weak equivalence, and furthermore r is a cofibration suchthat p ◦ r = id B . If f is in addition a weak equivalence, then so are i and r . Soevery functor from Ch b ( A ) to an infinity category which sends the elements of W ch toequivalences also sends the elements of W h to equivalences. This implies the assertion inview of the universal property of the functor (2.10).By [Lur09, Thm. 2.2.0.1 and Prop. 2.2.4.1] and Lemma 2.8 the mapping spaces of the ∞ -category Ch b ( A ) ∞ are represented by the simplicial mapping sets of the hammocklocalization ( L HW ch Ch b ( A )) fib , see Remark 2.6. More precisely, for objects A and B of Ch b ( A ) we have a canonical homotopy equivalence Map Ch b ( A ) ∞ ( ℓ ( A ) , ℓ ( B )) ≃ Map ( L HWch Ch b ( A )) fib ( A, B ) . (2.11)The homotopy type of the simplicial mapping sets in the hammock localization can becomputed using a method first introduced by Dwyer and Kan, and further studied byWeiss in [Wei99, Rem. 1.2]. For two objects A and B of Ch b ( A ) we consider the followingcategory M ( A, B ):1. The objects are pairs ( f, s ), where f : A → C is just a morphism and s : B → C atrivial cofibration.2. A morphism u : ( f, s ) → ( f ′ , s ′ ) is a commutative diagram of the form C u (cid:15) (cid:15) A f ♣♣♣♣♣♣ f ′ & & ◆◆◆◆◆◆ B s g g ◆◆◆◆◆◆ s ′ x x ♣♣♣♣♣♣ C ′ .
3. The composition is defined in the obvious way.Then by [Wei99, Rem. 1.2] we have a canonical weak equivalence of simplicial sets N ( M ( A, B )) ≃ Map ( L HWch Ch b ( A )) fib ( A, B ) . (2.12)We now consider a pushout square in Ch b ( A ) of the form A i (cid:15) (cid:15) a / / A ′ i ′ (cid:15) (cid:15) B b / / B ′ (2.13)in which the map i is a cofibration. By virtue of [Wei99, Thm. 2.1], for any object E in Ch b ( A ), the commutative square of simplicial sets N ( M ( A, E )) N ( M ( A ′ , E )) o o N ( M ( B, E )) O O N ( M ( B ′ , E )) O O o o (2.14)9s homotopy Cartesian. This implies, by (2.12), that Map ( L HWch Ch b ( A )) fib ( A, B )( A, E ) Map ( L HWch Ch b ( A )) fib ( A, B )( A ′ , E ) o o Map ( L HWch Ch b ( A )) fib ( A, B )( B, E ) O O Map ( L HWch Ch b ( A )) fib ( A, B )( B ′ , E ) O O o o (2.15)is a homotopy Cartesian square of Kan complexes. Consequently, by virtue of [Lur09,Thm. 4.2.4.1] (and possibly [Lur09, Remark A.3.3.13]), the square (2.13) is sent by ℓ toa pushout square in Ch b ( A ) ∞ .Similarly, for every object A of Ch b ( A ) the simplicial set N ( M ( A, ℓ sends the null complex to aterminal object in Ch b ( A ) ∞ .Note that the opposite of an additive category is again an additive category. Appyingwhat precedes to the opposite category of A , we see that ℓ sends 0 also to an initial objectin Ch b ( A ) ∞ , and that any pullback square of the form (2.13) in which the map b is adegree-wise split surjection is sent to a pull-back square in Ch b ( A ) ∞ .The explicit description of the mapping spaces in Ch b ( A ) ∞ given by (2.11) and (2.12)implies that, up to equivalence, all maps of Ch b ( A ) ∞ come from maps of Ch b ( A ). Inparticular, by virtue of [Lur09, Cor. 4.4.2.4] and its dual version, we have proved thatthe ∞ -category Ch b ( A ) ∞ has finite limits and finite colimits, and that 0 = ℓ (0) is a nullobject.Since, up to equivalence, any morphism in Ch b ( A ) ∞ is the image under ℓ of a cofibration,any cofiber sequence of Ch b ( A ) ∞ is the image under ℓ of a pushout square of the form A i / / (cid:15) (cid:15) B p (cid:15) (cid:15) / / C in which i is a cofibration. But such a square also is a pullback square, with p a degree-wise split surjection, and therefore, the functor ℓ sends such a square to a fiber sequence.In other words, any cofiber sequence in Ch b ( A ) ∞ is a fiber sequence. Replacing A byits opposite category, we see that we also proved the converse: any fiber sequence in Ch b ( A ) ∞ is a cofiber sequence. In other words, the ∞ -category Ch b ( A ) ∞ is stable in thesense of [Lur14, Def. 1.1.1.9], and we have proved the proposition. Remark 2.9.
For an additive category A the category Ch b ( A ) has a natural dg -enrichedrefinement Ch b ( A ) dg . By [Lur14, Thm. 1.3.1.10] we can thus form a small ∞ -category N dg ( Ch b ( A ) dg ). It is easy to check that this category is pointed by the zero complex,has finite colimits (sums and push-outs exist), and that the suspension is represented bythe shift. Consequently, N dg ( Ch b ( A ) dg ) is a stable ∞ -category. In fact, [Lur14, 1.3.2.10]asserts this for the dg -nerve N dg ( Ch ( A ) dg ) of the dg -category of not necessarily bounded10hain complexes. We can consider N dg ( Ch b ( A ) dg ) as a full subcategory of N dg ( Ch ( A ) dg )which is stable under finite colimits.By the universal property of the localization functor (2.9) the natural morphism Ch b ( A ) → N dg ( Ch b ( A ) dg ) factorizes through a morphism Ch b ( A ) ∞ → N dg ( Ch b ( A ) dg ) . (2.16)This morphism induces an equivalence of homotopy categories Ho ( Ch b ( A ) ∞ ) ≃ → Ho ( N dg ( Ch b ( A ) dg )) . Proposition 2.7 is now equivalent to the fact that (2.16) is an equivalence of ∞ -categories.Let A be an object of Ch b ( A ). Corollary 2.10.
We have the relation Σ( ℓ ( A )) ≃ ℓ ( A [1]) .Proof. Let A be an object of Ch b ( A ). By Proposition 2.7, the functor ℓ sends the shortexact sequence A → Cone ( id A ) → A [1]to the cofiber sequence ℓ ( A ) / / (cid:15) (cid:15) ℓ ( Cone ( id A )) (cid:15) (cid:15) / / ℓ ( A [1])Since 0 → Cone ( id A ) is a chain homotopy equivalence, this cofiber sequence exhibits ℓ ( A [1]) as the suspension of ℓ ( A ) in Ch b ( A ) ∞ . Proposition 2.11.
The functor Ch b ( − ) ∞ has a natural factorization Cat ex ∞ (cid:15) (cid:15) Add / / Ch b ( − ) ∞ ♥♥♥♥♥♥ Cat ∞ . Proof.
Exactness of functors and stability of ∞ -categories are detected in Ho ( Cat ∞ ). Welet E be be the non-full subcategory of Ho ( Cat ∞ ) consisting of stable ∞ -categories andexact functors. Then we have a pull-back Cat ex ∞ / / (cid:15) (cid:15) Cat ∞ (cid:15) (cid:15) E / / Ho ( Cat ∞ ) .
11y Proposition 2.7, the functor Ch b ( − ) ∞ takes values in stable ∞ -categories. We mustshow that it send a morphism f : A → A ′ between additive categories to an exactfunctor. By [Lur14, Cor. 1.4.2.14] we must show that Ch b ( f ) ∞ preserves the zero objectand suspension. It clearly preserves the zero object (represented by the zero complex).By Corollary 2.10 in the domain and target of Ch b ( f ) ∞ the suspension is represented bythe shift and Ch b ( f ) clearly preserves the shift, Ch b ( f ) ∞ also preserves suspension. Definition 2.12.
We define the functor
UK :
Add → M loc as the composition
UK :
Add Ch b ( − ) ∞ → Cat ex ∞ U loc → M loc . K -theory for additive categories Recall that a morphism between additive categories is a functor between the underlyingcategories which preserves finite coproducts. It is an equivalence if the underlying mor-phism of categories is an equivalence. More generally, it is a Morita equivalence if itinduces an equivalence of idempotent completions. Finally, recall the similar Definition2.1 (in conjuction with Remark 2.2) for morphisms between stable ∞ -categories.Let A be an additive category. Lemma 2.13.
The morphism A → Idem( A ) induces a Morita equivalence Ch b ( A ) ∞ → Ch b (Idem( A )) ∞ . Proof.
In view of Definition 2.1 and Remark 2.2 we must show that the induced morphism c : Idem( Ch b ( A ) ∞ ) → Idem( Ch b (Idem( A )) ∞ )is an equivalence. We first observe that for any additive category B the natural morphism B → Idem( B ) induces a fully faithful embedding Ch b ( B ) ∞ → Ch b (Idem( B )) ∞ . This isobvious if we use the description of the ∞ -categories of chain complexes via dg -nerves asdiscussed in Remark 2.9. We now have a commuting diagram Ch b ( A ) ∞ / / (cid:15) (cid:15) Ch b (Idem( A )) ∞ i (cid:15) (cid:15) Idem( Ch b ( A ) ∞ ) c / / Idem( Ch b (Idem( A )) ∞ ) . Since the vertical morphisms and the upper horizontal morphism are fully-faithful, so is c . It remains to show that c is essentially surjective. We note that the morphism i identi-fies Ch b (Idem( A )) ∞ with the smallest full stable subcategory of Idem( Ch b (Idem( A )) ∞ )containing the objects in the image of the compositionIdem( A ) [0] → Ch b (Idem( A )) ℓ → Ch b (Idem( A )) ∞ . (2.17)12n view of the commuting diagram Ch b (Idem( A )) ∞ i * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ Idem( A ) (2.17) ♥♥♥♥♥♥♥♥♥♥♥♥♥ Idem( ℓ ◦ [0]) ( ( PPPPPPPPPPPP
Idem( Ch b (Idem( A )) ∞ )Idem( Ch b ( A ) ∞ ) c ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ every object in the image of the composition i ◦ (2.17) also belongs to the essential imageof c . Hence c is essentially surjective and we have shown the lemma Remark 2.14.
A particular case of a result of Balmer and Schlichting [BS01, Thm. 2.8]states that, for any small idempotent complete additive category A , the triangulatedcategory Ho ( Ch b ( A ) ∞ ) is idempotent complete. Applying this result to i and the proofof Lemma 2.13 to c we conclude that for any small additive category A , the naturalfunctors Idem( Ch b ( A ) ∞ ) c → Idem( Ch b (Idem( A )) ∞ ) i ← Ch b (Idem( A )) ∞ are equivalences of ∞ -categories. Lemma 2.15.
The functor
UK :
Add → M loc sends Morita equivalences of additivecategories to equivalences in M loc .Proof. Since the functor U loc sends Morita equivalences of stable ∞ -categories to equiva-lences of spectra, it is sufficient to prove that the functor A Ch b ( A ) ∞ sends Moritaequivalences of additive categories to Morita equivalences of ∞ -categories.First of all, this functor sends equivalences of additive categories to equivalences of ∞ -categories because the functor A Ch b ( A ) preserves equivalences of categories, andbecause the localization functor (2.9) sends equivalences of (relative) categories to equiv-alences of ∞ -categories.For an additive category A the morphism A → Idem( A ) is a Morita equivalence of addi-tive categories. By Lemma 2.13 we get a Morita equivalence Ch b ( A ) ∞ → Ch b (Idem( A )) ∞ of stable ∞ -categories.Any Morita equivalence f : A → A ′ between additive categories fits into a commutativesquare of the form A f / / (cid:15) (cid:15) A ′ (cid:15) (cid:15) Idem( A ) Idem( f ) / / Idem( A ′ )in which the functor Idem( f ) is an equivalence of categories. By the observations madeabove the functor Ch b ( − ) ∞ sends the two vertical and the lower horizontal morphism toequivalences. This readily implies that the map UK( f ) is an equivalence from Ch b ( A ) ∞ to Ch b ( A ′ ) ∞ . 13et A be an additive category. Definition 2.16.
A full additive subcategory C of A is a Karoubi-filtration if every dia-gram X → Y → Z in A with X, Z ∈ C admits an extension X (cid:15) (cid:15) / / Y / / ∼ = (cid:15) (cid:15) ZU incl / / U ⊕ U ⊥ pr / / U O O with U ∈ C . In [Kas15, Lemma 5.6] it is shown that Definition 2.16 is equivalent to the standarddefinition of a Karoubi filtration as considered in [CP97].A morphism in A is called completely continuous (c.c.) if it factorizes over a morphismin C . We can form a new additive category A / C with the same objects as A and themorphisms Hom A / C ( A, A ′ ) := Hom A ( A, A ′ ) /c.c. . Assume that C is a Karoubi filtration in an additive category A . Lemma 2.17.
The sequence of stable ∞ -categories Ch b ( C ) ∞ → Ch b ( A ) ∞ → Ch b ( A / C ) ∞ is exact.Proof. As observed in Lemma 2.13, for any additive category B , the natural functor Ch b ( B ) ∞ → Ch b (Idem( B )) ∞ is a Morita equivalence of stable ∞ -categories. Further-more the canonical functor Idem( A / C ) → Idem(Idem( A ) / Idem( C )) is an equivalence ofcategories, because the functor Idem (being a left adjoint) sends pushouts of additive cate-gories to pushouts in the category of idempotent complete additive categories. Therefore,the canonical functor Ch b ( A / C ) ∞ → Ch b (Idem( A ) / Idem( C )) ∞ is a Morita equivalence of stable ∞ -categories.All this means that, without loss of generality, it is sufficient to prove the case where C is idempotent complete (since we may replace A and C by their idempotent completionsat will). By [BGT13, Prop. 5.15] it suffices to show that, under this extra assumption on C , Ho ( Ch b ( C ) ∞ ) → Ho ( Ch b ( A ) ∞ ) → Ho ( Ch b ( A / C ) ∞ )is exact. In view of Example [Sch04, Ex. 1.8] this is shown in [Sch04, Prop. 2.6].14ssume that C is a Karoubi filtration in an additive category A . Proposition 2.18.
We have the following fibre sequence in M loc . UK( C ) → UK( A ) → UK( A / C ) Proof.
This immediately follows from Lemma 2.17, Definition 2.12, and the fact that U loc sends exact sequences to fibre sequences. Definition 2.19.
An additive category A is called flasque if there exists an endo-functor S : A → A such that id A ⊕ S ≃ S . Let A be a small additive category. By K ( A ), we denote Schlichting’s non-connective K -theory spectrum. Its stable homotopy groups in non-negative degrees i agree withQuillen’s K -theory groups of Idem( A ) denoted above by K W aldi (Idem( A )), and with the K -theory groups defined by Pedersen and Weibel in negative degrees.Let A be a small additive category. Proposition 2.20.
There is a canonical equivalence of spectra K ( A ) ≃ map M loc ( U loc ( S ω ∞ ) , UK( A )) . Proof.
Using Lemma 2.13 and the fact that U loc sends Morita equivalences to equiva-lences, Formula (2.1) expresses map M loc ( U loc ( S ω ∞ ) , UK( A )) as the non-connective K -theoryspectrum K ( Ch b (Idem( A )) ∞ ). By virtue of [BGT13, Cor. 7.12] the connective cover of K ( Ch b (Idem( A )) ∞ ) agrees with Waldhausen’s K-theory of Ch b (Idem( A )), which in turnsagrees with Quillen’s K -theory of Idem( A ) by the Gillet-Waldhausen theorem (as shownin [TT90]). There is a construction, due to Karoubi, and recalled in [PW85, Example5.1], which associates functorially, to any small additive category A , a flasque additivecategory C ( A ), together with an additive full embedding A → C ( A ) which turns A intoa Karoubi filtration of C ( A ). The suspension S ( A ) of A is then defined as the quotient C ( A ) / A . Iterating this construction, we see with Lemma 2.17 that we have a deloopingof the stable ∞ -category Ch b ( A ) ∞ , of the form Ch b ( S n ( A )) ∞ / / (cid:15) (cid:15) Ch b ( C ( S n ( A ))) ∞ (cid:15) (cid:15) / / Ch b ( S n +1 ( A ) ∞ )This delooping exists at the level of Waldhausen categories Ch b ( S n ( A )) / / (cid:15) (cid:15) Ch b ( C ( S n ( A ))) (cid:15) (cid:15) / / Ch b ( S n +1 ( A ))15y naturality of the comparison maps K W ald ( Ch b ( A )) → K W ald ( Ch b ( A )) (where the lefthand denotes the usual Waldhausen construction associated with the cofibration category Ch b ( A )), we obtain a natural morphism of spectra: colim n ≥ Ω n ( K W ald ( Ch b ( S n ( A ))) → colim n ≥ Ω n ( K W ald ( Ch b ( S n ( A ))) ∞ ) . By virtue of Proposition 2.5, the codomain of this map is the non-connective spectrum K ( Ch b ( A ) ∞ ). The domain of this map is precisely Schlichting’s definition [Sch06, Def.8] of the non-connective K -theory of A . Since both side agree canonically on connectivecovers, to prove that this comparison map is a stable weak homotopy equivalence, it issufficient to prove that it induces an isomorphism on stable homotopy groups in nonpositive degree. But this map induces an isomorphism on π for any A . Therefore,replacing A by S n ( A ), for positive integers n , we see that it induces an isomorphism indegree − n , because both sides are canonically isomorphic to K − n ( A ) = K (Idem( S n ( A ))) ∼ = K (Idem( Ch b ( S n ( A ))))(this is proved in Theorem [Sch06, Theorem 8, p. 124] for the left-hand side, and followsfrom the last assertion of Proposition 2.5 for the right-hand side).Let A be an additive category. Proposition 2.21. If A is flasque, then UK( A ) ≃ .Proof. Let B be an additive category. We define a new additive category D ( B ) by:1. An object of B is a triple ( B, B , B ) of an object B of B with two subobjects suchthat the induced morphism B ⊕ B → B is an isomorphism.2. A morphism ( B, B , B ) → ( B ′ , B ′ , B ′ ) is a morphism B → B ′ preserving thesubobjects.We have a sequence B i → D ( B ) p → B where the first morphism i sends the object B in B to the object ( B, B,
0) of D ( B ),and the second morphism p sends the object ( B, B , B ) of D ( B ) to the object B of B .The morphism i is fully faithful and determines a Karoubi filtration on D ( B ), and themorphism p is the projection onto the quotient. It has a left-inverse j : B → D ( B ) givenby B ( B, , B ).We finally have a functor s : D ( B ) → B which sends ( B, B , B ) to B .Let f, g : A → B be two morphisms between additive categories. Then we can definea morphism ( f, g ) : A → D ( B ) by A ( f ( A ) ⊕ g ( A ) , f ( A ) , g ( A )) which is uniquelydetermined by the choice of representatives of the sums. We now consider the composition h : A ( f,g ) → D ( B ) s → B . Note that M loc is stable so that it makes sense to add morphisms. We haveUK( f ) , UK( g ) , UK( h ) ∈ π ( Map M loc (UK( A ) , UK( B ))) . emma 2.22. We have
UK( h ) ≃ UK( f ) + UK( g ) .Proof. We have a fibre sequence UK( B ) UK( i ) → UK( D ( B )) UK( p ) → UK( B ) which is split byUK( j ). We have a commutative diagramUK( A ) UK( f )+UK( g ) ' ' UK( f ) ⊕ UK( g ) / / UK( B ) ⊕ UK( B ) UK( i ) ◦ pr +UK( j ) ◦ pr ≃ / / (cid:15) (cid:15) ✤✤✤ UK( D ( B )) UK( s ) / / UK( B )UK( A ) UK( h ) UK( f,g ) / / UK( B ⊕ B ) UK( i,j ) / / UK( D ( B )) UK( s ) / / UK( B ) . The dashed arrow is induced by the two embeddings B → B ⊕ B and the universalproperty of the sum.Let S : A → A be the endo-functor such that id A ⊕ S ≃ S . We get the equivalenceUK( id A ) + UK( S ) ≃ UK( S ) and this implies UK( id A ) ≃
0, hence UK( A ) ≃ Proposition 2.23.
The functor UK preserves filtered colimits.Proof. The functor (2.8) clearly preserves filtered colimits. The localization functor (2.9)preserves filtered colimits. In order to see this we could use the model given in Remark2.6 and observe that hammock localization preserves filtered colimits, a filtered colimit ofKan-complexes is again a Kan complex, and the coherent nerve functor preserves filteredcolimits, and finally that a filtered colimit of quasi-categories is a filtered colimit in the ∞ -category Cat ∞ . The inclusion Cat ex ∞ → Cat ∞ preserves filtered colimits. And finally,by definition the universal localizing U ∞ invariant preserves filtered colimits. Following [BE16] a bornological coarse space is a triple ( X, C , B ) of a set X with a coarsestructure C and a bornology B which are compatible in the sense that every coarse thicken-ing of a bounded set is again bounded. A morphism f : ( X, C , B ) → ( X ′ , C ′ , B ′ ) betweenbornological coarse spaces is a controlled and proper map. We thus have a category BornCoarse of bornological coarse spaces and morphisms.For a group G a G -bornological coarse space is a bornological coarse space ( X, C , B ) withan action of G by automorphisms such that its set of G -invariant entourages C G is cofinal17n C . A morphism between G -bornological coarse spaces is an equivariant morphism ofbornological coarse spaces. In this way we have defined a full subcategory G BornCoarse ⊆ Fun ( BG,
BornCoarse )of G -bornological coarse spaces and equivariant morphisms.The category G BornCoarse has a symmetric monoidal structure which we denote by ⊗ .It is defined by ( X, C , B ) ⊗ ( X ′ , C ′ , B ′ ) := ( X × X ′ , ChC × C ′ i , BhB × B ′ i ) , where G acts diagonally on the product set, ChC × C ′ i denotes the coarse structure gen-erated by all products U × U ′ of entourages of X and X ′ , and BhB × B ′ i denotes thebornology generated by all products B × B ′ of bounded subsets of X and X ′ .In [BE16] we axiomatized the notion of a coarse homology theory. The equivariant casewill be studied throughly in [BEKWb]. In the following definition and the text below wedescribe the axioms.Let C be a stable cocomplete ∞ -category and consider a functor E : BornCoarse → C . Definition 3.1. E is an equivariant C -valued coarse homology theory if it satisfies:1. coarse invariance.2. excision.3. vanishing on flasques.4. u -continuity. The detailed description of these properties is as follows:1. (coarse invariance) We require that for every G -bornological coarse space X theprojection { , } max,max ⊗ X → X induces an equivalence E ( { , } max,max ⊗ X ) → E ( X ) . Here for a set S we let S max,max denote the set S equipped with the maximal coarseand bornological structures.2. (excision) A big family in a G -bornological coarse spaces X is a filtered family ( Y i ) i ∈ I of invariant subsets of X such that for every i in I and entourage U of X there exists j in I such that U [ Y i ] ⊆ Y j . We define E ( Y ) := colim i ∈ I E ( Y i ). We furthermore set E ( X, Y ) := Cofib( E ( Y ) → E ( X )) .
18 complementary pair ( Z, Y ) consists of an invariant subset Z of X and a big familysuch that there exists i in I such that Z ∪ Y i = X . Note that Z ∩ Y := ( Z ∩ Y i ) i ∈ I is a big family in Z . Excision then requires that the canonical morphism E ( Z, Z ∩ Y ) → E ( X, Y )is an equivalence. Here all subsets of X are equipped with the induced bornologicalcoarse structure.3. A G -bornological coarse space X is flasque if it admits an equivariant selfmap f : X → X (we say that f implements flasquenss) such that ( f × id )( diag X ) is anentourage of X , for every entourage U of X , the subset S n ∈ N ( f n × f n )( U ) of X × X is again an entourage of X , and for every bounded subset B of X there exists an n in N such that f n ( X ) ∩ Γ B = ∅ . Vanishing on flasques requires that E ( X ) ≃ X is flasque.4. For every invariant entourage U of X we define a new bornological coarse space X U .It has the underlying bornological space of X and the coarse structure Ch U i . Thecondition of u -continuity requires that the natural morphisms X U → X induce anequivalence colim U ∈C G E ( X U ) ≃ → E ( X ) . X -controlled A -objects In this section we associate in a functorial way to every additive category and bornolog-ical coarse space a new additive category of controlled objects and morphisms. Thisconstruction is taken from [BEKWb].
Definition 3.2.
A subset of a bornological coarse space is called locally finite if its inter-section with every bounded subset is finite.
Let A be an additive category and ˆ A denote its Ind -completion. We consider A as a fullsubcategory of ˆ A .Let X be a G -bornological coarse space. Definition 3.3.
An equivariant X -controlled A -object is a triple ( M, φ, ρ ) , where1. M is an object of ˆ A ,2. φ is a measure on X with values in the idempotents on M admitting images,3. ρ is an action of G on M ,such that: . For every bounded subset B of X we have M ( B ) ∈ A .2. For every subset Y of X and element g of G we have φ ( gY ) = ρ ( g ) φ ( Y ) ρ ( g ) − . (3.1)
3. The natural map M x ∈ X M ( { x } ) → M (3.2) is an isomorphism.4. The subset supp ( M, φ, ρ ) := { x ∈ X | M ( { x } ) = 0 } of X is locally finite. Here we write M ( Y ) for the choice of an image of φ ( Y ). Remark 3.4.
The measure φ appearing in Definition 3.3 is defined on all subsets of X and is finitely additive. For every two subsets Y and Z it satisfies φ ( Y ) φ ( Z ) = φ ( Y ∩ Z ).In particular, we have φ ( Y ) φ ( Z ) = φ ( Z ) φ ( Y ).Let ( M, φ, ρ ) and ( M ′ , φ ′ , ρ ′ ) be two equivariant X -controlled A -objects and A : M → M ′ be a morphism in ˆ A . Using the isomorphisms (3.2) for every pair of points ( x, y ) in X we obtain a morphism A x,y : M ( { y } ) → M ′ ( { x } ) between objects of A . We say that A iscontrolled if the set supp ( A ) := { ( x, y ) ∈ X × X | A x,y = 0 } is an entourage of X . Definition 3.5.
A morphism A : ( M, φ, ρ ) → ( M ′ , φ ′ , ρ ′ ) between two equivariant X -controlled A -objects is a morphism A : M → M ′ in ˆ A which intertwines the G -actionsand is controlled. We let V G A ( X ) denote the category of equivariant X -controlled A -objects and morphisms.It is again an additive category. If f : X → X ′ is an equivariant morphism between G -bornological coarse spaces, then we define a morphism between additive categories V G A ( f ) : V G A ( X ) → V G A ( X ′ ) , ( M, φ, ρ ) f ∗ ( M, φ, ρ ) := (
M, φ ◦ f − , ρ ) , A f ∗ ( A ) := A .
In the way we have defined a functor V G A : G BornCoarse → Add . If G is the trivial group, then we omit G from the notation.20 .3 The universal equivariant coarse homology theory Let A be an additive category. Definition 3.6.
We define the functor UK X G A := UK ◦ V G A : G BornCoarse → M loc . Theorem 3.7.
The functor UK X G A is an equivariant coarse homology theory.Proof. In the following four lemmas we verify the conditions listed in Definition 3.1. Wekeep the arguments sketchy as more details are given in [BEKWb].
Lemma 3.8. UK X G A is coarsely invariant.Proof. If X is a G -bornological coarse space, then the projection p : { , } max,max ⊗ X → X induces an equivalence of additive categories V G A ( { , } max,max ⊗ X ) → V G A ( X ). Usingthis fact the Lemma follows from Lemma 2.15.Let i : X → { , } max,max ⊗ X denote the inclusion given by the element 0 of { , } .Then p ◦ i = id X . We now construct an isomorphism u : V G A ( i ) ◦ V G A ( p ) → id . Onthe object ( M, φ, ρ ) of V G A ( { , } max,max ⊗ X ) it is given by u ( M,φ,ρ ) := id M : M → M . This map defines an isomorphism of equivariant { , } max,max ⊗ X -controlled A -objects ( M, φ ◦ p − ◦ i − , ρ ) and ( M, φ, ρ ). One easily checks the conditions for a naturaltransformation.
Lemma 3.9. UK X G A is excisive.Proof. Let X be a G -bornological coarse space and Y := ( Y i ) i ∈ I be a big family on X .Then V G A ( Y ) := colim i ∈ I V G A ( Y i )is a Karoubi-filtration in V G A ( X ). Since UK commutes with filtered colimits we have anequivalence UK X G A ( Y ) ≃ UK( V G A ( Y )) . By Proposition 2.18 we get a fibre sequenceUK X G A ( Y ) → UK X G A ( X ) → UK( V G A ( X ) / V G A ( Y )) . We conclude that UK X G A ( X, Y ) ≃ UK( V G A ( X ) / V G A ( Y )) . (3.3)Let now ( Z, Y ) be a complementary pair. Then we get a similar sequenceUK X G A ( Z ∩ Y ) → UK X G A ( Z ) → UK( V G A ( Z ) / V G A ( Z ∩ Y )) . One now checks that there is a canonical equivalence of additive categories ψ : V G A ( Z ) / V G A ( Z ∩ Y ) → V G A ( X ) / V G A ( Y )21nduced by the inclusion i : Z → X . We define an inverse φ : V G A ( X ) / V G A ( Y ) → V G A ( Z ) / V G A ( Z ∩ Y )as follows:1. On objects φ sends ( M, φ, ρ ) to ( M ( Z ) , φ ( Z ∩ − ) , ρ | M ( Z ) ).2. On morphisms φ sends the class [ A ] of A : ( M, φ, ρ ) → ( M ′ , φ ′ , ρ ′ ) to the class of φ ′ ( Z ) Aφ ( Z ) : M ( Z ) → M ′ ( Z ).One now observes that there is a canonical isomorphism φ ◦ ψ → id V G A ( Z ) / V G A ( Z ∩Y ) . Furthermore the inclusions M ( Z ) → M induce an isomorphism ψ ◦ φ → id V G A ( X ) / V G A ( Y ) . Using the equivalence (3.3) twice we see that the natural morphismUK X G A ( Z, Z ∩ Y ) → UK X G A ( X, Y )is an equivalence. Lemma 3.10. UK X G A vanishes on flasques.Proof. Assume that X is a flasque G -bornological coarse space. We claim that then V G A ( X ) is a flasque additive category so that the Lemma follows from Proposition 2.21.Let f : X → X be the selfmap implementing flasqueness. We can then define an exactfunctor S : V G A ( X ) → V G A ( X )by S ( M, φ, ρ ) := (cid:16) M n ∈ N M, ⊕ n ∈ N φ ◦ ( f n ) − , ⊕ n ∈ N ρ (cid:17) . One checks that S is well-defined and that V G A ( f ) ◦ S ≃ id V G A ( X ) ⊕ S .
There is an isomorphism v : V G A ( f ) → id V G A ( X ) which is given on the object ( M, φ, ρ ) by v ( M,φ,ρ ) := id M : ( M, φ ◦ f − , ρ ) → ( M, φ, ρ ) . Consequently, S ≃ id V G A ( X ) ⊕ S as required. Lemma 3.11. UK X G A is u -continuous. roof. Let X be a G -bornological coarse space. Then by an inspection of the definitionswe have an isomorphism of additive categories V G A ( X ) ∼ = colim U ∈C G V G A ( X U ) . The lemma now follows from Proposition 2.23.
In the section we verify an additional continuity property of the coarse homology UK X G A .The property plays an important role in the comparison of assembly and forget controlmaps discussed in [BEKWb].Let X be a G -bornological coarse space. Definition 3.12.
A filtered family of invariant subsets ( Y i ) i ∈ I of X is called trapping ifevery locally finite subset of X is contained in Y i some index i in I . Definition 3.13.
An equivariant C -valued coarse homology theory E is continuous if forevery trapping exhaustion ( Y i ) i ∈ I of a G -bornological coarse space X the natural morphism colim i ∈ I E ( Y i ) → E ( X ) is an equivalence. Proposition 3.14.
The coarse homology theory UK X G A is continuous.Proof. Let X be a G -bornological coarse space. If ( M, φ, ρ ) is an equivariant X -controlled A -object, then its support supp ( M, φ, ρ ) is locally finite.Let ( Y i ) i ∈ I be a trapping exhaustion of X . Then there exists i in I such that ( M, φ, ρ ) issupported on Y i . Consequently we have an isomorphism colim i ∈ I V G A ( Y i ) ∼ = V G A ( X ) . By Proposition 2.23 we now get colim i ∈ I UK X G A ( Y i ) ≃ UK X G A ( X ). In this section we package the equivariant homology theories UK X H A for all subgroups H of G together into one object. We furthermore give an alternative construction of theequivariant coarse homology theories from the non-equivariant coarse homology theory.The orbit category Orb ( G ) of G is the category of transitive G -sets and equivariant maps.Every object of the orbit category is isomorphic to one of the from G/H with the left G -action for a subgroup H of G . In particular we have the object G with the left action.The right action of G induces an isomorphism of groups Aut
Orb ( G ) ( G ) ∼ = G op . Hence we23ave a functor i : BG → Orb ( G ) op which sends the unique object of BG to G . If C is apresentable ∞ -category, then we have an adjunctionRes : Fun ( Orb ( G ) op , C ) ⇆ Fun ( BG, C ) : Coind , where Res is the restriction along i .For every subgroup H of G we have an evaluation Fun ( Orb ( G ) op , C ) → C , E E ( H ) at the object G/H . The family of all these evaluations detects limits, colimits, andequivalences in
Fun ( Orb ( G ) op , C ).We consider the relative category ( Add , W ), where W are the equivalences of addi-tive categories. Applying the localization functor (2.9) we then obtain an ∞ -category Add ∞ := L ( Add , W ). We furthermore have a morphism ℓ : Add → Add ∞ where wesecretly identify ordinary categories with ∞ -categories using the nerve functor.A marked additive category is a pair ( A , M ) of an additive category A and a collection M of isomorphisms in A (called marked isomorphisms) which is closed under compositionand taking inverses. A morphism between marked additive categories ( A , M ) → ( A ′ , M ′ )is an exact functor A → A ′ which in addition sends M → M ′ . We get a category Add + of small marked additive categories and morphisms. A marked isomorphism u : F → F ′ between two functors F, F ′ : ( A , M ) → ( A ′ , M ′ ) between marked additive categories is anisomorphism of functors such that for every object A of A the isomorphism u A : F ( A ) → F ( A ′ ) is a marked isomorphism.A morphism between marked additive categories is called a marked equivalence if it isinvertible up to marked isomorphism. We consider the relative category ( Add + , W + ),where W + are the marked equivalences. We define Add + ∞ := L ( Add + , W + ) and let ℓ + : Add + → Add + ∞ denote the canonical localization functor.The functor F : Add + → Add which forgets the marking induces a commuting square
Add + F / / ℓ + (cid:15) (cid:15) Add ℓ (cid:15) (cid:15) Add + ∞ F ∞ / / Add ∞ (3.4)of ∞ -categories.In [BEKWa, Sec. 7] it is shown that Add + ∞ is a presentable ∞ -category. Consequentlywe have an adjunctionRes : Fun ( Orb ( G ) op , Add + ∞ ) ⇆ Fun ( BG,
Add + ∞ ) : Coind . xample 3.15. We consider a marked additive category A + with an action of G , i.e., anobject of Fun ( BG,
Add + ). Then we can form the object Coind( ℓ + ( A + )) in Fun ( Orb ( G ) op , Add + ∞ ).In order to understand this object we calculate the evaluation Coind( ℓ + ( A + )) ( H ) for a sub-group H of G . As a first step we haveCoind( ℓ + ( A + )) ( H ) ≃ lim BH ℓ + ( A + ) . We now analyse the right-hand side. We consider the following marked additive categoryˆ A + ,H :1. The objects of ˆ A + ,H are pairs ( M, ρ ), where M is an object of A + and ρ = ( ρ ( h )) h ∈ H is a collection of marked isomorphisms ρ ( h ) : M → h ( M ) such that h ( ρ ( h ′ )) ◦ ρ ( h ) = ρ ( h ′ h ) for all pairs of elements h ′ , h in H .2. A morphism A : ( M, ρ ) → ( M ′ , ρ ′ ) in ˆ A + ,H is a morphism A : M → M ′ in A + suchthat h ( A ) ◦ ρ ( h ) = ρ ′ ( h ) ◦ A for every h in H . The composition of morphisms isinduced by the composition in A + .3. The marked isomorphisms in ˆ A + ,H are the isomorphisms given by marked isomor-phisms in A + .In [BEKWa, Sec. 7] it is shown that we have a natural equivalence lim BH ℓ + ( A + ) ≃ ℓ + ( ˆ A + ,H ) . As a final result we get the equivalenceCoind( ℓ + ( A + )) ( H ) ≃ ℓ + ( ˆ A + ,H ) . (3.5)In follows from (3.4), the universal property of the localization ℓ , and Lemma 2.15 thatwe have a factorization Add + F / / ℓ + $ $ ❏❏❏❏❏❏❏❏❏❏ Add UK / / ℓ $ $ ❏❏❏❏❏❏❏❏❏❏❏ M loc Add + ∞ F ∞ / / Add ∞ UK ∞ : : ✈✈✈✈✈✈✈✈✈✈ . (3.6)We refine the functor V A : BornCoarse → Add described in Section 3.2 to a functor V + A : BornCoarse → Add + by defining V + A ( X ) := ( V A ( X ) , M ) , where the set of marked isomorphisms M is the set of diag ( X )-controlled isomorphismsin V A ( X ). 25 efinition 3.16. We define the functor UK X A : G BornCoarse → Fun ( Orb ( G ) op , M ∞ ) as the composition G BornCoarse → Fun ( BG,
BornCoarse ) V + A → Fun ( BG,
Add + ) ℓ + → Fun ( BG,
Add + ∞ ) Coind → Fun ( Orb ( G ) op , Add + ∞ ) F ∞ → Fun ( Orb ( G ) op , Add ∞ ) UK ∞ → Fun ( Orb ( G ) op , M loc ) Theorem 3.17. UK X A is an equivariant coarse homology theory.Proof. Since the conditions listed in Definition 3.1 concern equivalences or colimits andthe collection of evaluations UK X A ( H ) : G BornCoarse → M loc detect equivalences andcolimits it suffices to show thatUK X A ( H ) : G BornCoarse → M loc are equivariant coarse homology theories for all subgroups H of G .We now note that if E : H BornCoarse → C is a H -equivariant C -valued coarse homologytheory, then E ◦ Res GH : G BornCoarse → C is a G -equivariant C -valued coarse homologytheory, where Res GH denotes the functor which restricts the action from G to H .The theorem now follows from the following Lemma and Theorem 3.7 (applied to thesubgroups H in place of G ). Lemma 3.18.
We have an equivalence of functors from G BornCoarse to M loc UK X A ( H ) ≃ UK X H A ◦ Res GH . Proof.
Let X be a G -bornological coarse space. By (3.5) we have an equivalenceCoind( ℓ + ( V + A ( X ))) ( H ) = ℓ + ( ˆ V + A (Res GH ( X )) H ) . Hence we have an equivalenceUK X A ( H ) ( X ) ≃ UK ∞ ( F ∞ ( ℓ + ( ˆ V + A (Res GH ( X )) H )))) (3.6) ≃ UK( F ( ˆ V + A (Res GH ( X )) H )) . (3.7)By a comparison of the explicit description of ˆ V A (Res GH ( X )) H given in Example 3.15 withthe definition of V H A (Res GH ( X )) given in Section 3.2, and using that (3.1) expresses exactlythe condition that ρ ( g ) is diag ( X )-controlled (i.e., marked), one now gets a canonicalisomorphism (actually an equality) of additive categories F ( ˆ V + A (Res GH ( X )) H ) ∼ = V H A (Res GH ( X )) . In view of Definition 3.6 we have an equivalenceUK( F ( ˆ V + A (Res GH ( X )) H )) ≃ UK X H A (Res GH ( X )) . Together with (3.7) it yields the equivalenceUK X A ( H ) ( X ) ≃ UK X H A (Res GH ( X )) . as desired. 26 eferences [BE16] U. Bunke and A. Engel. Homotopy theory with bornological coarse spaces. https://arxiv.org/abs/1607.03657 , 07 2016.[BEKWa] U. Bunke, A. Engel, D. Kasprowski, and Ch. Winges. The descent principle.in preparation.[BEKWb] U. Bunke, A. Engel, D. Kasprowski, and Ch. Winges. Equivariant coarsehomotopy thery. in preparation.[BFJR04] A. Bartels, T. Farrell, L. Jones, and H. Reich. On the isomorphism conjecturein algebraic K -theory. Topology , 43(1):157–213, 2004.[BGT13] A. J. Blumberg, D. Gepner, and G. Tabuada. A universal characterization ofhigher algebraic K -theory. Geom. Topol. , 17(2):733–838, 2013.[BLR08] A. Bartels, W. L¨uck, and H. Reich. The K -theoretic Farrell–Jones conjecturefor hyperbolic groups. Invent. math. , 172:29–70, 2008. arXiv:math/0701434.[BS01] P. Balmer and M. Schlichting. Idempotent completion of triangulated cate-gories.
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