Cyclic homology for bornological coarse spaces

We define Hochschild and cyclic homologies for bornological coarse spaces: for a fixed field $k$ and group $G$, these are lax symmetric monoidal functors $\mathcal{X}HH_{k}^G$ and $\mathcal{X}HC_{k}^G$ from the category of equivariant bornological coarse spaces $G\mathbf{BornCoarse}$ to the cocomplete stable $\infty$-category of chain complexes $\mathbf{Ch}_\infty$. We relate these equivariant coarse homology theories to coarse algebraic $K$-theory $\mathcal{X} K^G_{k}$ and to coarse ordinary homology $\mathcal{X} H^G$ by constructing a trace-like natural transformation $\mathcal{X} K_{k}^G\to \mathcal{X} H^G$ that factors through coarse Hochschild (or cyclic) homology. We further compare the forget-control map for coarse Hochschild homology with the associated generalized assembly map.


Introduction
Coarse geometry is the study of metric spaces from a large-scale point of view [Roe93,Roe96,Roe03]. A new axiomatic and homotopic approach to coarse geometry and coarse homotopy theory has been recently developed by Bunke and Engel [BE16]. In this set-up, the main objects are called bornological coarse spaces [BE16, Def. 2.5] (for example, every metric space is a bornological coarse space in a canonical way). In the equivariant setting, if G denotes a group, G-bornological coarse spaces are bornological coarse spaces with a G-action by automorphisms [BEKW17, Def. 2.1]. Among various invariants of G-bornological coarse spaces we are interested in equivariant coarse homology theories, i.e., functors E : GBornCoarse → C from the category of G-bornological coarse spaces GBornCoarse to a cocomplete stable ∞-category C, satisfying some additional axioms: coarse invariance, flasqueness, coarse excision and u-continuity [BEKW17,Def. 3.10]. Examples of coarse homology theories arise as coarsifications of locally finite homology theories. Among others, there are coarse versions of ordinary homology and of topological K-theory [BE16], of equivariant algebraic K-homology and of topological Hochschild homology [BEKW17,BC17], and of Waldhausen's A-theory [BKW18].
Hochschild and cyclic homologies have classically been defined as homology invariants of algebras, and have been then extended to invariants of dg-algebras, schemes, additive categories and exact categories [Kel99,McC94]. The aim of the paper is to give a concrete construction of coarse homology theories defining Hochschild and cyclic homology for bornological coarse spaces. These coarse homology theories can be defined abstractly by using a universal equivariant coarse homology theory [BC17]. However, we choose to give a more concrete definition that might be more suitable for computations (see, e.g., the construction of the natural transformation to coarse ordinary homology, Theorem 4.8). We now explain the main results of the paper.
Let k be field and let G be a group. We denote by C HH * and C HC * the chain complexes computing Hochschild homology and cyclic homology (of k-algebras) respectively. The Gbornological coarse space G can,min denotes a canonical bornological coarse space associated to the group G (see Example 1.2). Let Ch ∞ be the ∞-category of chain complexes. The following is a combination of Theorem 3.11, Proposition 3.14 and Proposition 3.15: Theorem A. There are lax symmetric monoidal functors X HH G k : GBornCoarse → Ch ∞ and X HC G k : GBornCoarse → Ch ∞ satisfying the following properties: (i) X HH G k and X HC G k are G-equivariant coarse homology theories; (ii) there are equivalences of chain complexes X HH G k ( * ) C HH * (k) and X HC G k ( * ) C HC * (k) between the evaluations of X HH G k and X HC G k at the one point bornological coarse space { * }, endowed with the trivial G-action, and the chain complexes computing Hochschild and cyclic homology of k; (iii) there are equivalences X HH G k (G can,min ) C HH * (k[G]; k) and X HC G k (G can,min ) C HC * (k[G]; k) of chain complexes between the evaluations at the G-bornological coarse space G can,min and the chain complexes computing Hochschild and cyclic homology of the k-algebra k [G].
The construction of the functors X HH G k and X HC G k uses a cyclic homology theory for dgcategories that satisfies certain additive and localizing properties in the sense of Tabuada [Tab07]. This is Keller's cone construction Mix : dgcat → Mix, for dg-categories [Kel99], which is a functor from the category dgcat of small dg-categories to Kassel's category Mix of mixed complexes [Kas87]. Hochschild and cyclic homologies for dg-categories are then defined in terms of mixed complexes, consistently with the classical definitions for k-algebras [Kas87]. We consider the functor (with values to the category of small k-linear categories Cat k ) V G k : GBornCoarse → Cat k , that associates to every G-bornological coarse space X a suitable k-linear category V G k (X) of G-equivariant X-controlled (finite dimensional) k-vector spaces [BEKW17,Def. 8.3]; a k-linear category is a dg-category in a standard way. We prove that the resulting functor Mix loc (see Definition 3.1) with values in the cocomplete stable ∞-category of mixed complexes Mix ∞ is a coarse homology theory (Theorem 3.2). Coarse Hochschild X HH G k and coarse cyclic homology X HC G k are then defined by post-composition of the Hochschild and cyclic homology functors for mixed complexes with the functor X Mix G k (see Definition 3.10).
Let Sp be the ∞-category of spectra. Besides giving an explicit construction, our main motivation is a better understanding of the spectra-valued equivariant coarse algebraic Khomology X K G k : GBornCoarse → Sp [BEKW17, Def. 8.8]. Algebraic K-theory comes equipped with trace maps (e.g., the Dennis trace map from algebraic K-theory of rings to Hochschild homology, or the refined version, the cyclotomic trace, from the algebraic K-theory spectrum to the topological cyclic homology spectrum) and these trace maps have been of fundamental importance in the understanding of algebraic K-theory [BHM93,DGM13]. Inspired by the classical case, we define trace maps from equivariant coarse algebraic K-homology to equivariant coarse Hochschild and cyclic homology and from equivariant coarse Hochschild and cyclic homology to equivariant coarse ordinary homology X H G : GBornCoarse → Sp (see Proposition 4.10, Theorem 4.8 and Proposition 4.9): (1) The classical Dennis trace map induces a natural transformation of equivariant coarse homology theories: coarse homology theories, which induces an equivalence of spectra when evaluated at the one point space { * }.
By composition of these two natural transformations, we get a natural transformation from equivariant coarse algebraic K-homology to equivariant coarse ordinary homology. This last coarse homology theory is defined in terms of equivariant locally finite controlled maps X n+1 → k (see Definition 1.6) and it is easier to compute then coarse algebraic K-homology. We believe that the study of this transformation can be useful for the understanding and detection of coarse K-theory classes.
Conventions. We freely employ the language of ∞-categories. More precisely, we model ∞-categories as quasi-categories [Cis,Lur09,Lur14]. When not otherwise specified, G will denote a group, k a field, ⊗ the tensor product over k. Without further comments, we always consider an additive category as a dg-category in the canonical way (Example 2.3).
Structure of the paper. In Section 1 we review the basic definitions in coarse homotopy theory: bornological coarse spaces, coarse homology theories and the category of controlled objects. In Section 2, we introduce the (cocomplete stable ∞-category) of mixed complexes and Keller's definition of cyclic homology. In Section 3 we define the functors X Mix G k , X HH G k and X HC G k and we prove that they are equivariant coarse homology theories. In the last section Section 4, we construct the natural transformations from coarse algebraic K-homology to coarse Hochschild homology and from coarse Hochschild homology to coarse ordinary homology.

Equivariant coarse homotopy theory
The main purpose of this section is to recollect notations and definitions describing the category GBornCoarse of G-equivariant bornological coarse spaces and of G-equivariant coarse homology theories. We will freely use the terminology of [ 1.1. Equivariant bornological coarse spaces. A bornology on a set X is a subset B ⊆ P(X) of the power set of X that is closed under taking subsets and finite unions, and such that X = ∪ B∈B B.
A coarse structure on a set X is a subset C ⊆ P(X × X) which contains the diagonal ∆ X := {(x, x) ∈ X × X | x ∈ X} and is closed under taking subsets, finite unions, inverses, and compositions. The elements of C are called entourages. If U is an entourage of a coarse space X and B is any subset of X, the U-thickening of B is the subset of X: A bornology B and a coarse structure C on a set X are compatible if every controlled thickening of every bounded set B ∈ B is bounded.
Definition 1.1. [BE16, Definition 2.5] A bornological coarse space is a triple (X, C, B) given by a set X, a bornology B and a coarse structure C on X, such that B and C are compatible.
Morphisms of bornological coarse spaces are maps for which pre-images of bounded sets are bounded and images of entourages are entourages. The category of bornological coarse spaces is denoted by BornCoarse.
Let G be a group acting by automorphisms on a bornological coarse space X. The G-action on X induces a G-action on the set of entourages of X. Let C G be the partially ordered subset of C consisting of the set-wise G-fixed entourages. A G-bornological coarse space [BEKW17, Definition 2.1] is a bornological coarse space (X, C, B) equipped with a G-action by automorphisms such that the set of invariant entourages C G is cofinal in C. We denote by GBornCoarse the category of G-bornological coarse spaces and G-equivariant, proper controlled maps.
is the bornology generated by the d-bounded balls B(x, r). The family of subsets U r := {(x, y) | d(x, y) ≤ r)}, for every r ≥ 0, generates a coarse structure that depends on the metric d denoted by C d . Then, the triple (X, C d , B d ) is a bornological coarse space. (ii) Let G be a group, B min be the minimal bornology on its underlying set and let C can := {G(B × B) | B ∈ B min } be the coarse structure on G generated by the G-orbits. The space G can,min := (G, C can , B min ) is a G-bornological coarse space. (iii) Let X be a G-bornological coarse space and let Z be a G-invariant subset of X. We define the induced coarse structure and bornology on Z by restriction: is a G-bornological coarse space and the inclusion Z → X is a morphism of G-bornological coarse spaces. (iv) Let X be a G-bornological coarse space and let U be a G-invariant entourage of X. We define the induced coarse structure C U as the coarse structure on X generated by U . This is compatible with the bornology B, and X U := (X, C U , B) is a G-bornological coarse space.
1.2. Equivariant coarse homology theories. Equivariant coarse homology theories are functors from the category of G-bornological coarse spaces to a stable cocomplete ∞-category C (e.g. the ∞-category of chain complexes Ch or the ∞-category of spectra Sp), satisfying additional axioms: coarse invariance, flasqueness, coarse excision and u-continuity, that we now recall.
Let f 0 , f 1 : (X, C, B) → (X , C , B ) be morphisms between bornological coarse spaces. We say that f 0 and f 1 are close to each other if the image of the diagonal (f 0 , f 1 )(∆ X ) is an entourage of X . A morphism f : (X, C, B) → (X , C , B ) is an equivalence of bornological coarse spaces if there exists an inverse g : (X , C , B ) → (X, C, B) such that the compositions g • f and f • g are close to the identity maps. In this case, the spaces X and X are called coarsely equivalent. Two morphisms between G-bornological coarse spaces are close to each other if they are close as morphisms between the underlying bornological coarse spaces.
admits a morphism f : X → X such that: (i) f is close to the identity map; (ii) for every entourage U , the subset k∈N (f k × f k )(U ) is an entourage of X; (iii) for every bounded set B in X there exists k such that f k (X) ∩ GB = ∅. (1) A big family Y = (Y i ) i∈I on X is a filtered family of subsets of X satisfying the following: An equivariant big family is a big family consisting of G-invariant subsets. Let Z be a subset of X. If Y is a big family on X, then the intersection Z ∩ Y := (Z ∩ Y i ) i∈I is a big family on Z.
Let Y = (Y i ) i∈I be a filtered family of G-invariant subsets of X. If E : GBornCoarse → C is a functor with values in a cocomplete ∞-category C, we define the value of E at the family Y as the filtered colimit E(Y) := colim i∈I E(Y i ). There is an induced map from E(Y) to E(X).
Definition 1.5. [BEKW17, Definition 3.10] Let G be a group and let GBornCoarse be the category of G-bornological coarse spaces. Let C be a cocomplete stable ∞-category. A G-equivariant C-valued coarse homology theory is a functor E : GBornCoarse −→ C with the following properties: i. Coarse invariance: E sends equivalences X → X of G-bornological coarse spaces to equivalences E(X) → E(X ) of C; ii. Flasqueness: if X is a flasque G-bornological coarse space, then E(X) 0; iii. Coarse excision: E(∅) 0, and for every equivariant complementary pair (Z, Y) on X, is a push-out square; iv. u-continuity: for every G-bornological coarse space (X, C, B), the canonical morphisms Examples of (equivariant) coarse homology theories are coarse ordinary homology (1.4) and coarse topological K-theory [BE16], coarse algebraic K-theory (Definition 1.20) and coarse topological Hochschild homology [BEKW17,BC17], coarse Hochschild and cyclic homology (Theorem 3.11).
Let X be a G-bornological coarse space, n ∈ N a natural number, B a bounded set of X, and x = (x 0 , . . . , x n ) a point of X n+1 . We say that x meets B if there exists an index i ∈ {0, . . . , n} such that x i belongs to B. If U is an entourage of X, we say that x is U -controlled if, for each i and j in {0, . . . , n}, the pair (x i , x j ) belongs to U .
An n-chain c on X is a function c : X n+1 → Z; its support supp(c) is defined as the set of points for which the function c is non-zero: We say that an n-chain c is U -controlled if every point x of supp(c) is U -controlled. The chain c is locally finite if, for every bounded set B, the set of points in supp(c) which meet B is finite. An n-chain c : X n+1 → Z is controlled if it is locally finite and U -controlled for some entourage U of X.
Definition 1.6. Let X be a bornological coarse space. Then, for n ∈ N, X C n (X) denotes the free abelian group generated by the locally finite controlled n-chains on X.
We will also represent n-chains as formal sums that are locally finite and U -controlled. The boundary map ∂ : X C n (X) → X C n−1 (X) is defined as the alternating sum ∂ := i (−1) i ∂ i of the face maps ∂ i (x 0 , . . . , x n ) := (x 0 , . . . ,x i , . . . , x n ). The graded abelian group X C * (X), endowed with the boundary operator ∂ extended linearly to X C * (X), is a chain complex [BE16, Sec. 6.3]. When X is a G-bornological coarse space, we let X C G n (X) be the subgroup of X C n (X) given by the locally finite controlled n-chains that are also Ginvariant. The boundary operator restricts to X C G * (X), and (X C G * (X), ∂) is a subcomplex of (X C * (X), ∂).
If f : X → Y is a morphism of G-bornological coarse spaces, then we consider the map on the products X n → Y n sending (x 0 , . . . , x n ) to (f (x 0 ), . . . , f (x n )). It extends linearly to a map The same can be done for a base ring k instead of Z.
Example 1.8. Assume that X is the one point space and the base ring of coefficients is k. Then, the chain complex X C * (X) has one free generator in each dimension, and the boundary maps are either the null map or the identity, depending on the degree. The coarse homology groups are 0 in positive and negative degree and the base ring k in degree 0.
1.4. The category of controlled objects. The goal of this subsection is to recall the definition of the additive category where the main results of this paragraph are given.
Let G be a group and let X be a G-bornological coarse space.
Remark 1.9. The bornology B(X) on X defines a poset with the partial order induced by subset inclusion; hence, B(X) can be seen as a category.
Recall that an additive category is a category enriched on abelian groups, with a zero object and all finite biproducts. Let A be an additive category with strict G-action. For every element g in G and every functor F : B(X) → A, let gF : B(X) → A denote the functor sending a bounded set B in B(X) to the A-object g(F (g −1 (B))) (and defined on morphisms B ⊆ B as the induced morphism of A (gF )(B ⊆ B ) : gF (g −1 (B)) → gF (g −1 (B ))).
If η : F → F is a natural transformation between two functors F, F : B(X) → A, we denote by gη : gF → gF the induced natural transformation between gF and gF .
consisting of a functor M : B(X) → A and a family ρ = (ρ(g)) g∈G of natural isomorphisms ρ(g) : M → gM , satisfying the following conditions: ( is a push-out; (3) for all B in B(X) there exists a finite subset F of B such that the inclusion induces an isomorphism M (F ) (4) for all elements g, g in G we have the relation ρ(gg ) = gρ(g ) • ρ(g), where gρ(g ) is the natural transformation from gM to gg M induced by ρ(g ).
Notation 1.11. If (M, ρ) is an X-controlled A-object and x is an element of X, we will often write M ( (1.1) of a bounded subset B of X is bounded and U -thickenings preserve the inclusions of bounded sets; we get a functor by post-composition. By using these structure maps we define the abelian group of G-equivariant controlled morphisms from (M, ρ) to (M , ρ ) as the colimit Let X be a G-bornological coarse space and let A be an additive category with strict G-action. The category V G A (X) is the category of G-equivariant Xcontrolled A-objects and G-equivariant controlled morphisms.
Let k be a field. When A is the category of finite dimensional k-vector spaces, then we denote by V G k (X) the associated category of G-equivariant X-controlled (finite dimensional) k-modules.
for every bounded set B in B and defined on morphisms in the canonical way. For every g in G, the family of transformation f * ρ = ((f * ρ)(g)) g∈G is given by the natural isomorphisms is a G-invariant entourage of X and the morphism: the functor from the category of G-bornological coarse spaces to the category of small additive categories obtained in this way.
Remark 1.15. If A is a k-linear category, then the functor V G A : GBornCoarse → Add refines to a functor V G A : GBornCoarse → Cat k from the category of G-bornological coarse spaces to the category of small k-linear categories.
The following properties of the functor V G A are shown in [BEKW17]: which is the identity on objects as the definition of equivariant X controlled A-objects does not depend on the coarse structure. Moreover, the category V G A (X U ) can be seen as a subcategory of V G A (X). On the other hand, every controlled morphism in Lemma 1.17. [BEKW17,Lemma 8.11] Let f, g : X → X be two morphisms of G-bornological coarse spaces. If f and g are close to each other, then they induce naturally isomorphic functors . Let A be an additive category and denote by ⊕ its biproduct.
Definition 1.18. An additive category A is called flasque if it admits an endofunctor S : A → A and a natural isomorphism id A ⊕ S ∼ = S.
We conclude with the definition of coarse algebraic K-homology: Definition 1.20. [BEKW17, Def. 8.8] Let G be a group and let A be an additive category with strict G-action. The G-equivariant coarse algebraic K-homology associated to A is the K-theory of the additive category of A-controlled objects: When A is the category of finite dimensional k-vector spaces, we denote by KX G k the associated K-theory functor. The properties of the functor V G A reviewed above are used in order to prove the following: Theorem 1.21. [BEKW17, Thm. 8.9] Let G be a group and let A be an additive category with strict G-action. Then, the functor KAX G is a G-equivariant Sp-valued coarse homology theory.

Keller's cyclic homology for dg-categories
In this section we recall Keller's construction of cyclic homology for dg-categories [Kel99]. As Keller's construction uses also Kassel's mixed complexes [Kas87], we first recall some properties of dg-categories and mixed complexes and then we define Keller's Hochschild and cyclic homology for dg-categories.
2.1. Dg-categories. Our main references in this subsection are [Kel06,Toë11]. We let k be a commutative ring and ⊗ the tensor product over k. We start with the definition of a dg-algebra: A left dg-module M over a dg-algebra A is a left graded module M = ⊕ p∈Z M p endowed with a differential d (of the same degree as the differential of A) such that d(ma) = d(m)a+(−1) p md(a) for every m ∈ M p and a ∈ A. A morphism of dg-modules is a homogeneous morphism of degree 0 of the underlying graded modules commuting with the differentials. The category of dg-modules over the dg-algebra A and morphisms of dg-modules is denoted by A-Mod.
A dg-category over k is a category enriched on (the category of) chain complexes of k-modules. We spell it out: Definition 2.2. A small differential graded category A (shortly, a dg-category) consists of the following data: • a small set of objects obj(A) (denoted A as well); of chain complexes satisfying the usual unit and composition conditions. We denote by dgcat k the category of small dg-categories (over k) and dg-functors.
Example 2.3. An additive category A is a dg-category in a canonical way: for every object A in A, Hom A (A, A) is a chain complex concentrated in degree 0. We denote by (2.1) ι : Add → dgcat the functor from the category of small additive categories to the category of small dg-categories that sends an additive category to the corresponding dg-category.
If A is a dg-category, then the dg-category A op defined as the category with the same objects as A and morphisms Hom A op (A, B) : = Hom A (B, A), is a dg-category. Let Ch dg (k) be the dg-category of chain complexes over the dg-algebra k.
There is a suitable category of dg-modules over a dg-category A, whose objects are the dg-modules over A and whose morphisms are the natural transformations of dg-functors Remark 2.6. The category of dg-modules (over a dg-algebra or a dg-category) admits two Quillen model structures where the weak equivalences are the object-wise quasi-isomorphisms of dg-modules; these are the injective and the projective model structure induced from the injective and projective model structure on chain complexes, respectively. We remark that the category of dg-modules over a dg-algebra, equipped with the projective model structure (hence the fibrations are the object-wise epimorphisms), is a combinatorial model category; see, for example [Coh13, Rem. 2.14].
If A is a dg-category, we can define an associated derived category: Definition 2.7. [Kel06, Sec. 3.2] The derived category D(A) of a dg-category A is the localization of the category of dg-modules over A at the class of quasi-isomorphisms.
The objects of D(A) are the dg-modules over A and the morphisms are obtained from morphisms of dg-modules by inverting the quasi-isomorphisms. It is a triangulated category with shift functor induced by the 1-translation and triangles coming from short exact sequences of complexes. Let in the sense of Verdier.
2.2. The ∞-category of mixed complexes. In this subsection we describe the (cocomplete stable ∞-)category of unbounded mixed complexes. When the differentials are clear from the context, we refer to a mixed complex (C, b, B) by its underlying k-module C.
Let Λ be the dg-algebra over the field k generated by an indeterminate of degree 1, with 2 = 0 and differential (of degree −1) d( ) = 0. Mixed complexes are nothing but dg-modules over the dg-algebra Λ: Remark 2.11. [Kas87] The category Mix of mixed complexes is equivalent (in fact, isomorphic) to the category of left dg Λ-modules, which we denote by Λ-Mod. In fact, a mixed complex (C, b, B) yields a differential graded left Λ-module whose underlying differential graded module is (C, b) and where the multiplication · c is defined by · c := B(c). Morphisms of mixed complexes correspond to morphisms of dg-Λ-modules. We denote by L : Mix → Λ-Mod the functor sending a mixed complex to the associated Λ-dg-module and by R : Λ-Mod → Mix its inverse functor.
The category of dg-k-modules admits a combinatorial model structure (the projective model structure, see Remark 2.6), whose weak equivalences are the objects-wise quasi-isomorphisms of dg-modules (Definition 2.5). In the language of mixed complexes this translates as follows: Remark 2.13. Quasi-isomorphisms of mixed complexes correspond to quasi-isomorphisms of Λ-dg-modules, i.e., the functors L and R of Remark 2.11 preserve quasi-isomorphisms.
We now introduce the ∞-category Mix ∞ of mixed complexes. We recall that, if C is an ordinary category and W denotes a collection of morphisms of C, then N(C)[W −1 ] is the Analogously, the ∞-category Λ-Mod ∞ is defined as the localization of the category Λ-Mod of dg-Λ-modules at the class W of quasi-isomorphisms of dg-Λ-modules: Proposition 2.15. The ∞-category Mix ∞ is a cocomplete stable ∞-category.  We remark that this definition agrees with the usual definition of Hochschild and cyclic homology of algebras [Kas87].
2.3. Keller's cyclic homology. In this subsection we recall Keller's Hochschild and cyclic homology for dg-categories [Kel99]. Let k be a commutative ring with identity and let A be a k-algebra. Then, one can associate to A a cyclic module Z * (A) [Goo85] (i.e., a cyclic object in the category of k-modules) defined in degree n as the (n + 1)-th tensor product of A over k. In the same way, one can construct a cyclic module out of an additive category A [McC94, Def. 2.1.1]. We present these constructions in the more general setting of dg-categories.
Definition 2.18. [Kel99] Let C be a small dg-category over k. The additive cyclic nerve of C is the cyclic k-module defined by: where the sum runs over all the objects (C 0 , C 1 , . . . , C n ) in C n+1 . The face and degeneracy maps, and the cyclic action, are defined as follows: We get a covariant functor from the category of small dg-categories over k to the category of cyclic k-modules. To every cyclic k-module, we can then associate a mixed complex as we now describe.
Let M be a cyclic k-module. If d i and s i denote the i-th face and i-th degeneracy maps of M respectively and t n+1 denotes the cyclic operator in degree n, then we let b : M n → M n−1 be the alternating sum Thanks to the work of Keller, we know that this functor enjoys many useful properties, among others agreement, additivity and localization [Kel99]. As we work in the context of ∞-categories, we will spell them out in this language.
From now on we assume that k is a field. The ∞-category of small of dg-categories dgcat k,∞ := N(dgcat k )[W − (1) it sends equivalences of small dg-categories to equivalences of mixed complexes; (2) it commutes with filtered colimits; (3) it sends short exact sequences A → B → C of dg-categories to cofiber sequences of Mix ∞ . Moreover, if A is a k-algebra, there is an equivalence of mixed complexes where projA is the additive category of finitely generated projective modules.
Observe that the functor Mix preserves filtered colimits, hence the functor loc •Mix preserves filtered colimits because the localization preserves filtered colimits as well, and by commutativity of the diagram also Mix • loc. By Proposition 2.15, the ∞-category Mix ∞ is stable and cocomplete and cofiber sequences of Mix ∞ [Lur14, Def. 1.1.1.6] are detected in its homotopy category, i.e., in D(Λ). We observe here that Keller's theorem holds in a more general setting (for more general rings and for exact categories). However, we only need these properties in the context of additive categories. Moreover, in such context, Keller

Equivariant coarse Hochschild and cyclic homology
For a fixed base field k and group G, we define equivariant coarse Hochschild homology X HH G k and cyclic homology X HC G k versions of the classical Hochschild and cyclic homology of k-algebras. This will be achieved by first studying an intermediate equivariant coarse homology theory X Mix G k with values in the ∞-category of mixed complexes. We then provide some comparison results.
3.1. The equivariant coarse homology theory X Mix G k . Let k be a field, Cat k the category of small k-linear categories, V G k : GBornCoarse → Cat k the functor of Remark 1.15, let Mix : dgcat k → Mix be the functor of Definition 2.20, ι : Cat k → dgcat k the functor of Example 2.3 and loc the localization functor loc : Mix → Mix ∞ .
Definition 3.1. We denote by X Mix G k the following functor to the ∞-category of mixed complexes.
The main result of the section is the following theorem: Theorem 3.2. The functor Proof. The category Mix ∞ is stable and cocomplete by Proposition 2.15. The functor X Mix G k satisfies coarse invariance (see Proposition 3.3), vanishing on flasque spaces (see Proposition 3.4), u-continuity (see Proposition 3.5) and coarse excision (see Theorem 3.6), i.e., the axioms describing an equivariant coarse homology theory of Definition 1.5.
We now prove that the functor X Mix G k satisfies the axioms of Definition 1.5. Proposition 3.3. The functor X Mix G k : GBornCoarse → Mix ∞ satisfies coarse invariance. Proof. If f : X → Y is a coarse equivalence of G-bornological coarse spaces, then it induces a natural equivalence f * : V G k (X) → V G k (Y ) by Lemma 1.17. Keller's functor Mix sends equivalences of dg-categories to equivalences of mixed complexes by Theorem 2.21 (1). Hence, the functor f * induces the equivalence X Mix G k (X) i.e., the functor X Mix G k is coarse invariant. Recall the definition of flasque spaces Definition 1.3.
Proposition 3.4. The functor X Mix G k : GBornCoarse → Mix ∞ vanishes on flasque spaces. Proof. By Lemma 1.19, the category V G k (X) is a flasque category, hence there exists an endofunctor S : is equivalent to the 0-morphism and that X Mix G k (X) 0.
Proposition 3.5. The functor X Mix G k : GBornCoarse → Mix ∞ is u-continuous. Proof. Let X be a G-bornological coarse space, and let C G be the poset of G-invariant controlled sets. By Remark 1.16, there is an equivalence V G k (X) colim U ∈C G V G k (X U ) of k-linear categories, hence of dg-categories. The functor Mix commutes with filtered colimits, and we get the equivalence in Mix ∞ , which shows that the functor X Mix G k is u-continuous. Theorem 3.6. The functor X Mix G k : GBornCoarse → Mix ∞ satisfies coarse excision. Before giving the proof of this theorem we first need some more terminology.
Definition 3.7. [Kas15] A full additive subcategory A of an additive category U is a Karoubifiltration if every diagram X → Y → Z in U, with X, Z ∈ A, admits an extension By [Kas15, Lemma 5.6], this definition is equivalent to the classical one [Kar70,CP97]. If A is a Karoubi-filtration of U, we can construct a quotient category U/A. Its objects are the objects of U, and the morphisms sets are defined as follows: where the relation identifies pairs of maps U → V whose difference factorizes through an object of A.
Let X be a G-bornological coarse space and let Y = (Y i ) i∈I be an equivariant big family on X (see Definition 1.4). The bornological coarse space Y i is a subspace of X with the induced bornology and coarse structure. The inclusion . Lemma 3.8. [BEKW17, Lemma 8.14] Let Y be an equivariant big family on the G-bornological coarse space X. Then, the full additive subcategory V G k (Y) of V G k (X) is a Karoubi filtration. Let X be a G-bornological coarse space, and (Z, Y) be an equivariant complementary pair. Consider the functor  Proof of Theorem 3.6. Let X be a G-bornological coarse space, and let (Z, Y) be an equivariant complementary pair on X. By Lemma 3 are Karoubi filtrations and yield the following sequences of k-linear categories:  [Cap19,Rem. 3.3.12]), Karoubi filtrations induce short exact sequences of dg-categories, hence by Theorem 2.21 we get cofiber sequences of mixed complexes. The inclusion Z → X induces a commutative diagram (where the rows are the obtained cofiber sequences) where a * is the map induced by a : . By Lemma 3.9, the functor a yields an equivalence of categories, hence of mixed complexes and the left square is a co-Cartesian square in Mix ∞ .
In order to conclude the proof, we recall that X Mix G k (Y) is defined as the filtered colimit The functor Mix commutes with filtered colimits of dg-categories, hence we have the equivalence ) and the same holds for Z ∩ Y. By using these identifications, we obtain the co-Cartesian square in Mix ∞ meaning that X Mix G k satisfies coarse excision. 3.2. Coarse Hochschild and cyclic homology. In this subsection we define equivariant coarse Hochschild and cyclic homology. The functors forget : Mix → Ch (2.5), sending a mixed complex to the underlying chain complex, and Tot(B−) : Mix → Ch (2.6), sending a mixed complex to the total complex of its associated bicomplex, send quasi-isomorphisms of mixed complexes to quasi-isomorphisms of chain complexes, hence they descend to functors between the localizations.
Definition 3.10. Let k be a field, G a group and Ch ∞ the ∞-category of chain complexes. The G-equivariant coarse Hochschild homology X HH G k (with k-coefficients) is the G-equivariant Ch ∞ -valued coarse homology theory defined as composition of the functor X Mix G k of Definition 3.1 and of the functor forget (2.5). The composition involving composition with the functor Tot(B−) (2.6) is G-equivariant coarse cyclic homology.
The definitions are justified by the following: Theorem 3.11. The functors X HH G k : GBornCoarse → Ch ∞ and X HC G k : GBornCoarse → Ch ∞ are G-equivariant Ch ∞ -valued coarse homology theories.
Proof. By Theorem 3.2, the functor X Mix G k : GBornCoarse → Mix ∞ is an equivariant coarse homology theory and satisfies coarse invariance, coarse excision, u-continuity, and vanishing on flasques. The functors forget : Mix ∞ → Ch ∞ and the functor Tot(B−) : Mix ∞ → Ch ∞ commute with filtered colimits and send cofiber sequences to cofiber sequences. The two compositions with X Mix G k satisfy coarse invariance, coarse excision, u-continuity, and vanishing on flasques, hence the functors X HH G k and X HC G k are equivariant coarse homology theories. The category of mixed complexes has a natural symmetric monoidal structure induced by tensor products between the underlying chain complexes [Kas87]. As tensor products of mixed complexes preserve equivalences, by [Hin16,Prop. 3 [CT12]). This implies that coarse Hochschild and cyclic homologies are lax symmetric monoidal functors: Proposition 3.12. The functors X HH G k and X HC G k admit lax symmetric monoidal refinements: 3.3. Comparison results. In this subsection, we compare equivariant coarse Hochschild homology with the classical version of Hochschild homology for k-algebras. Furthermore, we show that the forget-control map for coarse Hochschild homology is equivalent to the associated generalized assembly map.
Notation 3.13. Let A be a k-algebra. We denote by Let { * } be the one point bornological coarse space, endowed with a trivial G-action.
Proposition 3.14. There are equivalences of chain complexes X HH k ( * ) C HH * (k; k) and X HC k ( * ) C HC * (k; k) between the coarse Hochschild (cyclic) homology of the point and the classical Hochschild (cyclic) homology of k.
Proof. By Theorem 2.21, the mixed complex Mix(A) associated to a k-algebra A is equivalent to the mixed complex associated to the k-linear category of finitely generated projective A-modules. When X is a point endowed with a trivial G-action and k is a field, the k-linear category V k (X) is isomorphic to the category Vect f.d.
Mix(k), which is enough to prove the statement.
Let G be a group. By Example 1.2, there is a canonical G-bornological coarse space G can,min = (G, C can , B min ) associated to it. Let X be a G-set and let X min,max denote the G-bornological coarse space with minimal coarse structure and maximal bornology. Let G be a group, H a subgroup of G and endow the set G/H with the minimal coarse structure and the maximal bornology. The previous calculation can be extended to G-bornological coarse spaces of the form (G/H) min,max ⊗ G can,min : Remark 3.16. Let G be a group, H a subgroup of G and endow the set G/H with the minimal coarse structure and the maximal bornology; then, we get an equivalence of chain complexes: X HH G k ((G/H) min,max ⊗ G can,min ) C HH * (k[H]; k); the same holds for equivariant coarse cyclic homology.
One of the main applications of coarse homotopy theory is within the studying of assembly map conjectures. We conclude this subsection with a comparison result between the forgetcontrol maps for equivariant coarse Hochschild and cyclic homology and the associated assembly maps. Recall  induced by A i . We use the following notation: and ((M 0 , ρ 0 ), . . . , (M n , ρ n )) a tuple of objects of V G k (X). Let (x 0 , . . . , x n ) be a point of X n+1 . The symbol denotes the linear operator (A 0 • · · · • A n )|(x 0 , . . . , x n ) : M 0 (x n ) → M 0 (x n ) defined as the composition which is a finite dimensional k-vector space.
Let X be a G-bornological coarse space and let X C n (X) be the k-linear vector space generated by the locally finite controlled n-chains on X (see Definition 1.6).
Definition 4.3. We let ϕ n : CN n (V G k (X)) → X C n (X) be the map defined on elementary tensors as and extended linearly. Proof. In order to prove that ϕ n (A 0 ⊗ . . . ⊗ A n ) is locally finite and controlled we show that its support supp(ϕ n (A 0 ⊗ . . . ⊗ A n )) (1.2) is locally finite and that there exists an entourage U of X such that every x = (x 0 , . . . , x n ) in supp(ϕ n (A 0 ⊗ . . . ⊗ A n )) is U -controlled. We first observe that the operators A i : (M i+1 , ρ i+1 ) → (M i , ρ i ) are U i -controlled for some entourage U i of X. By Definition 1.12, A i is given by a natural transformation of functors satisfying an equivariance condition. For every point x in X, A i restricts to a morphism where the direct sum has only finitely many non-zero summands. Let K be a bounded set of X. The set of points x n ∈ K for which M 0 (x n ) is non-zero is finite (as a consequence of Definition 1.13). For such a fixed x n , there are only finitely many points x n−1 ∈ U n [K] such that the corresponding map A xn,x n−1 n The set U n [K] is a bounded set of X, the morphism A n−1 : M n → M n−1 is U n−1 -controlled and we can repeat the same argument for A n−1 , hence for each A i . This implies that the n-chain is locally finite because, for the given bounded set K, we have found only finitely tuples (x 0 , . . . , x n ) in the support of ϕ n (A 0 ⊗ . . . ⊗ A n ) that meet K.
The chain is also U -controlled, where U is the entourage U := U 0 • · · · • U n of X.
This is achieved by using that the trace map is additive, invariant under cyclic permutations and that the morphisms A i • A i+1 factor through all the points x i of X (that give contribution zero up to finitely many). For a complete proof we refer to [Cap19, Prop. 4.3.6].
If M is a cyclic module, by Remark 2.19, we get a mixed complex. The chain complex X C G (X) is also a mixed complex with the differential B = 0.
Lemma 4.7. The chain map ϕ : CN(V G k (X)) → X C G (X) of Definition 4.3 extends to a map ϕ : Mix(V G k (X)) → X C G (X) that is a morphism of mixed complexes. Proof. The proof is a simple computation and uses the definition of the operator B (2.8) and that the trace is invariant under cyclic permutations. See also [Cap19, Lemma 4.3.7].
We can now construct the natural transformation Φ X HH G k : X HH G k → X C G : Theorem 4.8. The map ϕ extends to natural transformations Φ X HH G k : X HH G k → X C G . and Φ X HC G k : X HC G k → n∈N X C G of G-equivariant Ch ∞ -valued coarse homology theories.
Proof. The map ϕ : (CN(V G k (X)), d) → (X C G (X), ∂) of Definition 4.3 is a chain map by Proposition 4.6. Let f : X → Y be a morphism of G-equivariant bornological coarse spaces. Consider the induced chain map X C G (f ) : X C G (X) → X C G (Y ) and the induced functor f * = V G k (f ) : V G k (X) → V G k (X ). By functoriality of the additive cyclic nerve, f * induces a morphism CN(f * ) : CN * (V G k (X)) → CN * (V G k (Y )) of cyclic modules (hence, a chain map between the underlying chain complexes as well).
The diagram ϕn is commutative. The map ϕ extends to the associated mixed complexes by Lemma 4.7 and this extension preserves the commutative diagram (of associated mixed complexes). After localization and application of the forgetful functor (recall the definition of equivariant coarse Hochschild homology in terms of X Mix G k , Definition 3.10), the map ϕ yields a natural transformation of equivariant coarse homology theories Φ X HH G k : X HH G k → X C G . To every mixed complex C, we associate the chain complex Tot(BC) (2.4) defined by Tot n (BC) = i≥0 C n−2i with differential d(c n , c n−2 , . . . ) = (bc n + Bc n−2 , . . . ). By Lemma 4.7, we conclude that the map ϕ extends to a chain map on the total complex as well, and to a natural transformation of coarse homology theories where the sum is indexed on the natural numbers because the (mixed complex associated to) the additive cyclic nerve of V G k (X) is positively graded. The following result implies that the transformation Φ X HH k : X HH k → X C is non-trivial: Proposition 4.9. If X is the one point space { * }, then the transformation Φ X HH k : X HH k ( * ) → X C( * ) induces an equivalence of chain complexes.
Proof. Let c : { * } n+1 → k be an n-chain in X C n ( * ); we identify this chain with the element c ∈ k that is its image. Let ι n : X C n ( * ) → CN n (V k ( * ))) be the map sending c to the element (·c) ⊗ (·1 k ) ⊗ . . . ⊗ (·1 k ). This extends to a chain map that gives a section of the trace map, i.e., ϕ • ι = id.
As coarse Hochschild homology and coarse ordinary homology of the point are both isomorphic to the Hochschild homology of the ground field k (by Example 1.8 and Proposition 3.14), we get equivalences of chain complexes X HH k ( * ) C HH * (k) X C k ( * ). By using these equivalences and the section ϕ • ι = id, we obtain that, when X is the one point space, the transformation Φ X HH k induces an equivalence of chain complexes.
By applying the Eilenberg-MacLane correspondence EM (1.3), we now assume that equivariant coarse Hochschild and cyclic homology take values in the ∞-category Sp of spectra. induced by the Dennis trace maps from algebraic K-theory to Hochschild homology.
In particular, when X is the G-bornological coarse space G can,min , the induced map