Abstract
The goal of the paper is to define Hochschild and cyclic homology for bornological coarse spaces, i.e., lax symmetric monoidal functors \({{\,\mathrm{\mathcal {X}HH}\,}}_{}^G\) and \({{\,\mathrm{\mathcal {X}HC}\,}}_{}^G\) from the category \(G\mathbf {BornCoarse}\) of equivariant bornological coarse spaces to the cocomplete stable \(\infty \)category \(\mathbf {Ch}_\infty \) of chain complexes reminiscent of the classical Hochschild and cyclic homology. We investigate relations to coarse algebraic Ktheory \(\mathcal {X}K^G_{}\) and to coarse ordinary homology \({{\,\mathrm{\mathcal {X}H}\,}}^G\) by constructing a tracelike natural transformation \(\mathcal {X}K_{}^G\rightarrow {{\,\mathrm{\mathcal {X}H}\,}}^G\) that factors through coarse Hochschild (and cyclic) homology. We further compare the forgetcontrol map for \({{\,\mathrm{\mathcal {X}HH}\,}}_{}^G\) with the associated generalized assembly map.
Introduction
Coarse geometry is the study of metric spaces from a largescale point of view [27,28,29,30]. A new axiomatic and homotopic approach to coarse geometry and coarse homotopy theory has been recently developed by Bunke and Engel [3]. In this setup, the main objects are called bornological coarse spaces [3, Definition 2.5], and every metric space is a bornological coarse space in a canonical way. In the equivariant setting, if G is a group, Gbornological coarse spaces are bornological coarse spaces with a Gaction by automorphisms [4, Definition 2.1]. Among various invariants of Gbornological coarse spaces we are interested in equivariant coarse homology theories, i.e., functors
from the category \(G\mathbf {BornCoarse}\) of Gbornological coarse spaces to a cocomplete stable \(\infty \)category \(\mathbf {C}\), satisfying some additional axioms: coarse invariance, flasqueness, coarse excision and ucontinuity [4, Definition 3.10]. Examples of coarse homology theories arise as coarsifications of locally finite homology theories [3]. Among other theories, there are coarse versions of ordinary homology and of topological Ktheory [3], of equivariant algebraic Khomology and of topological Hochschild homology [2, 4], and of Waldhausen’s Atheory [6].
Classically, Hochschild and cyclic homologies have been defined as homology invariants of algebras [22], then extended to invariants of dgalgebras, schemes, additive categories and exact categories [21, 26]. The aim of the paper is twofold: we construct coarse homology theories defining Hochschild and cyclic homology for bornological coarse spaces and then we study their relations to coarse algebraic Ktheory and coarse ordinary homology. We remark that these coarse homology theories can be abstractly defined by using a universal equivariant coarse homology theory constructed by Bunke and Cisinski [2]. However, we choose to provide a more concrete construction with the hope that it might be more suitable for computations (see, e.g., the application to the construction of the natural transformation to coarse ordinary homology, Theorem 4.8). We now describe the main results of the paper.
Let k be field and let G be a group. We denote by \(C_{*}^{\mathrm {HH}}\) and \(C_{*}^{\mathrm {HC}}\) the chain complexes computing Hochschild homology and cyclic homology (of kalgebras) respectively. The Gbornological coarse space \(G_{\mathrm {can,min}}\) denotes a canonical bornological coarse space associated to the group G (see Example 1.2). Let \(\mathbf {Ch}_\infty \) be the \(\infty \)category of chain complexes. The following is a combination of Theorem 3.11, Propositions 3.12, 3.14, Proposition 3.15:
Theorem A
There are lax symmetric monoidal functors
satisfying the following properties:

(i)
\(\mathcal {X}\mathrm {HH}_{k}^G\) and \(\mathcal {X}\mathrm {HC}_{k}^G\) are Gequivariant coarse homology theories;

(ii)
there are equivalences of chain complexes
$$\begin{aligned} \mathcal {X}\mathrm {HH}_{k}^G(*)\simeq C_{*}^{\mathrm {HH}}(k) \quad \text { and } \quad \mathcal {X}\mathrm {HC}_{k}^G(*)\simeq C_{*}^{\mathrm {HC}}(k) \end{aligned}$$between the evaluations of \({{\,\mathrm{\mathcal {X}HH}\,}}_k^G\) and \({{\,\mathrm{\mathcal {X}HC}\,}}_k^G\) at the onepoint bornological coarse space \(\{*\}\), endowed with the trivial Gaction, and the chain complexes computing Hochschild and cyclic homology of k;

(iii)
there are equivalences
$$\begin{aligned} {{\,\mathrm{\mathcal {X}HH}\,}}_k^G(G_{\mathrm {can,min}})\simeq C_{*}^{{{\,\mathrm{HH}\,}}}(k[G];k) \quad \text { and }\quad {{\,\mathrm{\mathcal {X}HC}\,}}_k^G(G_{\mathrm {can,min}})\simeq C_{*}^{{{\,\mathrm{HC}\,}}}(k[G];k) \end{aligned}$$of chain complexes between the evaluations at the Gbornological coarse space \(G_{\mathrm {can,min}}\) and the chain complexes computing Hochschild and cyclic homology of the kalgebra k[G].
The construction of the functors \({{\,\mathrm{\mathcal {X}HH}\,}}_k^G\) and \({{\,\mathrm{\mathcal {X}HC}\,}}_k^G\) uses a cyclic homology theory for dgcategories that satisfies certain additive and localizing properties in the sense of Tabuada [35]. This is Keller’s cone construction
for dgcategories [21], defined as a functor from the category \(\mathbf {dgcat}\) of small dgcategories to Kassel’s category \(\mathbf {Mix}\) of mixed complexes [19]. Hochschild and cyclic homologies for dgcategories are then defined in terms of mixed complexes, consistently with the classical definitions for kalgebras [19]. We also consider the functor (with values in the category of small klinear categories \(\mathbf {Cat}_{k}\))
that associates to every Gbornological coarse space X a suitable klinear category \(V_k^G(X)\) of Gequivariant Xcontrolled (finitedimensional) kvector spaces [4, Definition 8.3]; a klinear category is a dgcategory in a standard way. We prove that the following functor
(see Definition 3.1) with values in the cocomplete stable \(\infty \)category of mixed complexes \(\mathbf {Mix}_{\infty }\) is a coarse homology theory (Theorem 3.2). Coarse Hochschild \({{\,\mathrm{\mathcal {X}HH}\,}}_k^G\) and coarse cyclic homology \({{\,\mathrm{\mathcal {X}HC}\,}}_k^G\) are then defined by postcomposition of the Hochschild and cyclic homology functors for mixed complexes with the functor \({{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}_k^G\) (see Definition 3.10).
Let \(\mathbf {Sp}\) be the \(\infty \)category of spectra. The main reason of defining coarse versions of Hochschild and cyclic homology is to relate them to (the \(\mathbf {Sp}\)valued) equivariant coarse algebraic Khomology \(\mathcal {X}K_{k}^G:G\mathbf {BornCoarse}\rightarrow \mathbf {Sp}\) [4, Definition 8.8], because classically algebraic Ktheory comes equipped with trace maps (e.g., the Dennis trace map from algebraic Ktheory of rings to Hochschild homology, or the refined version, the cyclotomic trace, from the algebraic Ktheory spectrum to the topological cyclic homology spectrum) to cyclic homology theories, and these trace maps have been of fundamental importance in its understanding [5, 13]. Inspired by the classical case, we define trace maps to equivariant coarse Hochschild and cyclic homology and from equivariant coarse Hochschild and cyclic homology to equivariant coarse ordinary homology \({{\,\mathrm{\mathcal {X}H}\,}}^G:G\mathbf {BornCoarse}\rightarrow \mathbf {Sp}\) (see Propositions 4.9, 4.10, Theorem 4.8):
Theorem B

(1)
The classical Dennis trace map induces a natural transformation of equivariant coarse homology theories:
$$\begin{aligned} \mathrm {K}\mathcal {X}_{k}^G\rightarrow {{\,\mathrm{\mathcal {X}HH}\,}}_{k}^G; \end{aligned}$$ 
(2)
There is a natural transformation
$$\begin{aligned} \Phi _{{{\,\mathrm{\mathcal {X}HH}\,}}_{k}^G}:{{\,\mathrm{\mathcal {X}HH}\,}}_{k}^G\rightarrow {{\,\mathrm{\mathcal {X}H}\,}}^G_{} \end{aligned}$$of Gequivariant coarse homology theories, which induces an equivalence of spectra when evaluated at the onepoint space \(\{*\}\).
By composition, we get a natural transformation,
that factors through coarse Hochschild homology. The advantage of this transformation is that equivariant coarse ordinary homology \({{\,\mathrm{\mathcal {X}H}\,}}^G\) is defined in terms of equivariant locally finite controlled maps \( X^{n+1}\rightarrow k\) (see Definition 1.6) and it might be suitable for computations of coarse Ktheory classes.
We conclude with some applications to assembly maps. One of the main applications of coarse homotopy theory is within the studying of assembly map conjectures. We then provide a comparison result between the forgetcontrol maps for equivariant coarse Hochschild and cyclic homology and the associated assembly maps (see Proposition 3.18).
Structure of the paper
In Sect. 1 we review the basic definitions in coarse homotopy theory: bornological coarse spaces, coarse homology theories and categories of controlled objects. In Sect. 2, we introduce the (cocomplete stable \(\infty \)category) of mixed complexes and Keller’s definition of cyclic homology. In Sect. 3 we define the functors \({{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}_k^G\), \({{\,\mathrm{\mathcal {X}HH}\,}}_k^G\) and \({{\,\mathrm{\mathcal {X}HC}\,}}_k^G\) and we prove that they are equivariant coarse homology theories. In the last Sect. 4, we construct the natural transformations from coarse algebraic Khomology to coarse ordinary homology factoring through coarse Hochschild homology.
Conventions
We freely employ the language of \(\infty \)categories. More precisely, we model \(\infty \)categories as quasicategories [9, 24, 25]. When not otherwise specified, G will denote a group, k a field, \(\otimes \) the tensor product over k. Without further comments, we always consider an additive category as a dgcategory in the canonical way.
Equivariant coarse homotopy theory
The main purpose of this section is to recollect the basic definitions in coarse homotopy theory and the notations needed in Sects. 3 and 4. We describe the category \(G\mathbf {BornCoarse}\) of \(G\)equivariant bornological coarse spaces and the associated \(G\)equivariant coarse homology theories, we give the examples of coarse ordinary homology and coarse algebraic Khomology, together with the properties of the category of controlled objects \(V_k^G(X)\). We refer to [3, Sect. 2] and [4, Sects. 2 and 3] for a comprehensive introduction to (equivariant) coarse homotopy theory.
Equivariant bornological coarse spaces
A bornology on a set X is a subset \(\mathcal {B}\subseteq \mathcal {P}(X)\) of the power set of X that is closed under taking subsets and finite unions, and such that \(X=\cup _{B\in \mathcal {B}} B\). Its elements are called bounded sets.
A coarse structure on a set X is a subset \(\mathcal {C}\subseteq \mathcal {P}(X\times X)\) which contains the diagonal \(\Delta _X:=\{(x,x)\in X\times X\mid x\in X\}\) and is closed under taking subsets, finite unions, inverses, and compositions. The elements of \(\mathcal {C}\) are called entourages. If \(\mathcal {U}\) is an entourage of a coarse space X and B is any subset of X, the \(\mathcal {U}\)thickening of B is the subset of X:
A bornology \(\mathcal {B}\) and a coarse structure \(\mathcal {C}\) on a set X are compatible if for every \(U\in \mathcal {C}\) and every \(B\in \mathcal {B}\) the controlled thickening \(U[B]\) belongs to the family \(\mathcal {B}\).
Definition 1.1
[3, Definition 2.5] A bornological coarse space is a triple \((X,\mathcal {C},\mathcal {B})\) given by a set X, a bornology \(\mathcal {B}\) and a coarse structure \(\mathcal {C}\) on X, such that \(\mathcal {B}\) and \(\mathcal {C}\) are compatible.
Morphisms of bornological coarse spaces are maps such that preimages of bounded sets are bounded sets and images of entourages are entourages. A Gbornological coarse space [4, Definition 2.1] is a bornological coarse space \((X,\mathcal {C},\mathcal {B})\) equipped with a Gaction by automorphisms such that the set of Ginvariant entourages \(\mathcal {C}^G\) is cofinal in \(\mathcal {C}\). We denote by \(G\mathbf {BornCoarse}\) the category of Gbornological coarse spaces and \(G\)equivariant, proper controlled maps. When clear from the context, we shortly write X for denoting a Gbornological coarse space \((X,\mathcal {C},\mathcal {B})\).
Example 1.2

(i)
Let \(G\) be a group, \(\mathcal {B}_{\min }\) be the minimal bornology on its underlying set and let \(\mathcal {C}_{\mathrm {can}}:=\langle \{G(B\times B)\mid B\in \mathcal {B}_{\min }\}\rangle \) be the coarse structure on G generated by the \(G\)orbits. The space \(G_{\mathrm {can,min}}:=(G,\mathcal {C}_{\mathrm {can}},\mathcal {B}_{\min } )\) is a \(G\)bornological coarse space.

(ii)
Let X be a \(G\)bornological coarse space and let Z be a \(G\)invariant subset of X. Then, the triple \(Z_X:=(Z,\mathcal {C}_Z,\mathcal {B}_Z)\) is a \(G\)bornological coarse space, where \(\mathcal {C}_Z:=\{(Z\times Z)\cap U\mid U\in \mathcal {C}\}\) and \(\mathcal {B}_Z:=\{Z\cap B\mid B\in \mathcal {B}\}\).

(iii)
Let \(U\) be a \(G\)invariant entourage of X. If \(\mathcal {C}_U\) denotes the coarse structure on X generated by \(U\), then \(X_U:=(X,\mathcal {C}_U,\mathcal {B})\) is a \(G\)bornological coarse space. Observe that there is a canonical morphism \(X_U\rightarrow X\).
Equivariant coarse homology theories
Let \(f_0,f_1:X\rightarrow X'\) 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 \((f_0,f_1)(\Delta _X)\) is an entourage of \(X'\). A morphism \(f:X\rightarrow X'\) is an equivalence if there exists an inverse \(g:X'\rightarrow X\) such that the compositions \(g\circ f\) and \(f\circ g\) are close to the respective identity maps. Two morphisms between \(G\)bornological coarse spaces are close to each other if they are close as morphisms between the underlying bornological coarse spaces.
Definition 1.3
[4, Definition 3.8] A \(G\)bornological coarse space \((X,\mathcal {C},\mathcal {B})\) is called flasque if it admits a morphism \(f:X\rightarrow X\) such that:

(i)
f is close to the identity map;

(ii)
for every entourage \(U\), the subset \(\bigcup _{k\in \mathbb {N}} (f^k\times f^k)(U)\) is an entourage of X;

(iii)
for every bounded set B in X there exists k such that \(f^k(X)\cap GB=\emptyset \).
Definition 1.4
[4, Definition 3.5 and 3.7] Let \((X,\mathcal {C},\mathcal {B})\) be a Gbornological coarse space.

(1)
A big family \(\mathcal {Y}=(Y_i)_{i\in I}\) on X is a filtered family of subsets of X satisfying the following:
$$\begin{aligned} \forall \, i\in I, \quad \forall \, U\in \mathcal {C}, \, \exists j\in I \text { such that } U[Y_i]\subseteq Y_j \end{aligned}$$An equivariant big family is a big family consisting of \(G\)invariant subsets.

(2)
A pair \((Z,\mathcal {Y})\) consisting of a subset Z of X and of a big family \(\mathcal {Y}\) on X is called a complementary pair if there exists an index \(i\in I\) for which \(Z\cup Y_i=X\). It is an equivariant complementary pair if Z is a \(G\)invariant subset and \(\mathcal {Y}\) is an equivariant big family.
Let Z be a subset of X. If \(\mathcal {Y}\) is a big family on X, then the intersection \(Z\cap \mathcal {Y}:=(Z\cap Y_i)_{i\in I}\) is a big family on Z. If \(E:G\mathbf {BornCoarse}\rightarrow \mathbf {C}\) is a functor with values in a cocomplete \(\infty \)category \(\mathbf {C}\), we define the value of E at the family \(\mathcal {Y}\) as the filtered colimit \(E(\mathcal {Y}):=\mathrm {colim}_{i\in I} E(Y_i)\). There is an induced map from \(E(\mathcal {Y})\) to E(X). Let \(X_U\) be the Gbornological coarse space constructed in Example 1.2.
Definition 1.5
[4, Definition 3.10] Let G be a group and let \(G\mathbf {BornCoarse}\) be the category of Gbornological coarse spaces. Let \(\mathbf {C}\) be a cocomplete stable \(\infty \)category. A \(G\)equivariant \(\mathbf {C}\)valued coarse homology theory is a functor
with the following properties:

i.
Coarse invariance: E sends equivalences \(X\rightarrow X'\) of \(G\)bornological coarse spaces to equivalences \(E(X)\rightarrow E(X')\) of \(\mathbf {C}\);

ii.
Flasqueness: if X is a flasque \(G\)bornological coarse space, then \(E(X)\simeq 0\);

iii.
Coarse excision: \(E(\emptyset )\simeq 0\), and for every equivariant complementary pair \((Z,\mathcal {Y})\) on X, the diagram
is a pushout square;

iv.
ucontinuity: for every \(G\)bornological coarse space \((X,\mathcal {C},\mathcal {B})\), the canonical morphisms \(X_U\rightarrow X\) induce an equivalence \(E(X)\simeq {{\,\mathrm{colim}\,}}_{U\in \mathcal {C}^G}E(X_U)\).
Examples of (equivariant) coarse homology theories are coarse ordinary homology (1.4) and coarse topological Ktheory [3], coarse algebraic Ktheory (Definition 1.19) and coarse topological Hochschild homology [2, 4], coarse Hochschild and cyclic homology.
Coarse ordinary homology
Let X be a Gbornological coarse space, \(n\in \mathbb {N}\) a natural number, B a bounded set of X, and \(x=(x_0,\dots ,x_n)\) a point of \(X^{n+1}\). We say that x meets B if there exists \(i\in \{ 0,\dots ,n \} \) 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, the pair \((x_i,x_j)\) belongs to \(U\).
An nchain c on X is a function \( c:X^{n+1}\rightarrow \mathbb {Z}\); its support \({{\,\mathrm{supp}\,}}(c)\) is defined as the set of points for which the function c is nonzero:
We say that an nchain c is \(U\)controlled if every point x of \({{\,\mathrm{supp}\,}}(c)\) is \(U\)controlled. The chain c is locally finite if, for every bounded set B, the set of points in \({{\,\mathrm{supp}\,}}(c)\) which meet B is finite. An nchain \(c:X^{n+1}\rightarrow \mathbb {Z}\) 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 \(n\in \mathbb {N}\), \(\mathcal {X}C_n(X)\) denotes the free abelian group generated by the locally finite controlled nchains on X.
We will also represent nchains as formal sums \(\sum _{x\in X^{n+1}} c(x) x\) that are locally finite and \(U\)controlled. The boundary map \(\partial :\mathcal {X}C_n(X)\rightarrow \mathcal {X}C_{n1}(X)\) is defined as the alternating sum \(\partial :=\sum _i(1)^i\partial _i\) of the face maps \(\partial _i(x_0,\dots ,x_n):=(x_0,\dots ,\hat{x_i},\dots ,x_n)\). The graded abelian group \(\mathcal {X}C_*(X)\), endowed with the boundary operator \(\partial \) extended linearly to \(\mathcal {X}C_{*}(X)\), is a chain complex [3, Sect. 6.3]. When X is a \(G\)bornological coarse space, we let \(\mathcal {X}C_n^G(X)\) be the subgroup of \(\mathcal {X}C_n(X)\) given by the locally finite controlled nchains that are also \(G\)invariant. The boundary operator restricts to \(\mathcal {X}C_*^G(X)\), and \((\mathcal {X}C_*^G(X),\partial )\) is a subcomplex of \((\mathcal {X}C_*(X),\partial )\).
If \(f:X\rightarrow Y\) is a morphism of Gbornological coarse spaces, then we consider the map on the products \(X^{n+1}\rightarrow Y^{n+1}\) sending \((x_{0},\dots ,x_{n})\) to \((f(x_{0}),\dots ,f(x_{n})) \). It extends linearly to a map \( \mathcal {X}C^G(f):\mathcal {X}C^G_{n}(X)\rightarrow \mathcal {X}C^G_{n}(Y) \) that involves sums over the preimages by f. This describes a functor \(\mathcal {X}C^G:G\mathbf {BornCoarse}\rightarrow \mathbf {Ch}\) with values in the category \(\mathbf {Ch}\) of chain complexes over the integers. The \(\infty \)category \(\mathbf {Ch}_\infty \) of chain complexes is defined as the localization (in the realm of \(\infty \)categories [25, Sect. 1.3.4]) of the nerve of the category \(\mathbf {Ch}\) at the class W of quasiisomorphisms of chain complexes \( \mathbf {Ch}_\infty :={{\,\mathrm{N}\,}}_{}(\mathbf {Ch})[W^{1}]. \) By postcomposing the functor \(\mathcal {X}C^G\) with the functor
(the Eilenberg–MacLane correspondence between chain complexes and spectra [33, Theorem 1.1] or [3, Sect. 6.3]), we get a functor to the \(\infty \)category of spectra
called equivariant coarse ordinary homology:
Theorem 1.7
[4, Theorem 7.3] The functor \({{\,\mathrm{\mathcal {X}H}\,}}^G\) is a Gequivariant \(\mathbf {Sp}\)valued coarse homology theory.
Example 1.8
If X is a point, its coarse homology groups are 0 in positive and negative degree and the base ring k in degree 0.
The category \(V^G_\mathbf {A}(X)\) of controlled objects
The goal of this subsection is to recall the definition of the additive category \(V^G_\mathbf {A}(X)\) of Gequivariant Xcontrolled \(\mathbf {A}\)objects [4, Definition 8.3] and of the functor \( V_\mathbf {A}^G:G\mathbf {BornCoarse}\rightarrow \mathbf {Add}\) sending a Gbornological coarse space to the category \(V^G_\mathbf {A}(X)\). This functor is an essential ingredient in the construction of coarse homology theories like coarse algebraic Khomology and coarse Hochschild and cyclic homology.
Let G be a group and let X be a Gbornological coarse space.
Remark 1.9
The bornology \(\mathcal {B}(X)\) on X defines a poset with the partial order induced by subset inclusion; hence, \(\mathcal {B}(X)\) can be seen as a category.
Let \(\mathbf {A}\) be an additive category with strict Gaction. For every element g in \(G\) and every functor \(F:\mathcal {B}(X)\rightarrow \mathbf {A}\), let \(g F:\mathcal {B}(X)\rightarrow \mathbf {A}\) denote the functor sending a bounded set B in \(\mathcal {B}(X)\) to the \(\mathbf {A}\)object \(g(F(g^{1}(B)))\) (and defined on morphisms \(B\subseteq B'\) as the induced morphism of \(\mathbf {A}\) \((gF)(B\subseteq B'):gF(g^{1}(B))\rightarrow gF(g^{1}(B'))\)).
If \(\eta :F\rightarrow F'\) is a natural transformation between two functors \(F,F':\mathcal {B}(X)\rightarrow \mathbf {A}\), we denote by \(g\eta :g F\rightarrow g F'\) the induced natural transformation between gF and \(gF'\).
Definition 1.10
[4, Definition 8.3] A \(G\)equivariant Xcontrolled \(\mathbf {A}\)object is a pair \((M,\rho )\) consisting of a functor \(M:\mathcal {B}(X)\rightarrow \mathbf {A}\) and a family \(\rho =(\rho (g))_{g\in G}\) of natural isomorphisms \(\rho (g):M\rightarrow g M\), satisfying the following conditions:

(1)
\(M(\emptyset )\cong 0\);

(2)
for all \(B,B'\) in \(\mathcal {B}(X)\), the commutative diagram
is a pushout;

(3)
for all B in \(\mathcal {B}(X)\) there exists a finite subset F of B such that the inclusion induces an isomorphism \(M(F)\xrightarrow {\cong } M(B)\);

(4)
for all elements \(g, g'\) in G we have the relation \(\rho (g g')=g\rho (g')\circ \rho (g)\), where \(g\rho (g')\) is the natural transformation from gM to \(g g' M\) induced by \(\rho (g')\).
Notation 1.11
If \((M,\rho )\) is an Xcontrolled \(\mathbf {A}\)object and x is an element of X, we will often write M(x) instead of \(M(\{x\})\) for the value of the functor M at the bounded set \(\{x\}\) of X.
Let X be a Gbornological coarse space and let \((M,\rho )\) be an equivariant Xcontrolled \(\mathbf {A}\)object. Let B be a bounded set of X and let x be a point in B. The inclusion \(\{x\}\rightarrow B\) induces a morphism \(M(\{x\})\rightarrow M(B)\) of \(\mathbf {A}\). The conditions of Definition 1.10 imply that \(M(\{x\})=0\) for all but finitely many points of B and that the canonical morphism (induced by the universal property of the coproduct in \(\mathbf {A}\))
is an isomorphism. The \(U\)thickening \(U[B]\) (1.1) of a bounded subset B of X is bounded and Uthickenings preserve the inclusions of bounded sets; we get a functor \( U[]:\mathcal {B}(X)\rightarrow \mathcal {B}(X) \).
Definition 1.12
[4, Definition 8.6] Let \((M,\rho )\) and \((M',\rho ')\) be Gequivariant Xcontrolled \(\mathbf {A}\)objects and let \(U\in \mathcal {C}^G(X)\) be a Ginvariant entourage of X. A \(G\)equivariant \(U\)controlled morphism \(\varphi :(M,\rho ) \rightarrow (M',\rho ')\) is a natural transformation
such that \(\rho '(g)\circ \varphi =(g \varphi )\circ \rho (g) \) for all g in \(G\).
The set of Gequivariant \(U\)controlled morphisms \(\varphi :(M,\rho )\rightarrow (M',\rho ')\) is denoted by \({{\,\mathrm{Mor}\,}}_U((M,\rho ),(M',\rho '))\). For every bounded set B of X, the inclusion \(U\subseteq U'\) induces an inclusion \(U[B]\subseteq U'[B]\); this yields a natural transformation of functors \(M'\circ U[] \rightarrow M'\circ U'[]\), hence a map
by postcomposition.
By using these structure maps we define the abelian group of \(G\)equivariant controlled morphisms from \((M,\rho )\) to \((M',\rho ')\) as the colimit
Definition 1.13
[4] Let X be a Gbornological coarse space and let \(\mathbf {A}\) be an additive category with strict Gaction. The category \(V^G_\mathbf {A}(X)\) is the category of \(G\)equivariant Xcontrolled \(\mathbf {A}\)objects and Gequivariant controlled morphisms.
Let k be a field. When \(\mathbf {A}\) is the category of finitedimensional kvector spaces, then we denote by \(V_k^G(X)\) the associated category of Gequivariant Xcontrolled (finitedimensional) kmodules.
Lemma 1.14
[4, Lemma 8.7] The category of equivariant Xcontrolled \(\mathbf {A}\)objects \(V^G_\mathbf {A}(X)\) is additive.
Let \(f:(X,\mathcal {C},\mathcal {B})\rightarrow (X',\mathcal {C}',\mathcal {B}')\) be a morphism of \(G\)bornological coarse spaces. If \((M,\rho )\) is a \(G\)equivariant Xcontrolled \(\mathbf {A}\)object, we consider the functor \( f_* M:\mathcal {B}'\rightarrow \mathbf {A}\) defined by \(f_* M(B'):=M(f^{1}(B'))\) for every bounded set \(B'\) in \(\mathcal {B}'\) and defined on morphisms in the canonical way. For every g in G, the family of transformations \(f_{*}\rho =((f_{*}\rho )(g))_{g\in G}\) is given by the natural isomorphisms \((f_{*}\rho )(g):f_{*}M\rightarrow g(f_{*}M)\) with \(((f_*\rho )(g))(B'):=\rho (g)(f^{1}(B')).\) The pair \(f_*(M,\rho ):=(f_*M,f_*\rho )\) defined in this way is a \(G\)equivariant \(X'\)controlled \(\mathbf {A}\)object [4, Sect. 8.2]. Assume also that \(U\) is an invariant entourage of X and that \(\varphi :(M,\rho ) \rightarrow (M',\rho ')\) is an equivariant \(U\)controlled morphism. Then, the set \(V:= (f \times f)(U)\) is a Ginvariant entourage of \(X'\) and the morphism:
is Vcontrolled. We have just described a functor \( f_* :=V^G_{\mathbf {A}}(f):V^G_\mathbf {A}(X) \rightarrow V^G_\mathbf {A}(X'). \)
We denote by
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 \(\mathbf {A}\) is a klinear category, then the functor \(V_\mathbf {A}^G:G\mathbf {BornCoarse}\rightarrow \mathbf {Add}\) refines to a functor \(V_\mathbf {A}^G:G\mathbf {BornCoarse}\rightarrow \mathbf {Cat}_{k}\) from the category of Gbornological coarse spaces to the category of small klinear categories.
The following properties of the functor \(V^G_\mathbf {A}\) are shown in [4]:
Remark 1.16
Let \((X,\mathcal {C},\mathcal {B})\) be a \(G\)bornological coarse space, \(U\in \mathcal {C}^G\) a \(G\)invariant entourage of X and \(X_U:=(X,\mathcal {C}_U,\mathcal {B})\) the \(G\)bornological coarse space obtained by restriction of the structures. Then, the category \(V_\mathbf {A}^G(X)\) is the filtered colimit
indexed on the poset of \(G\)invariant entourages of X.
Lemma 1.17
[4, Lemma 8.11] Let \(f,g:X\rightarrow X'\) be two morphisms of \(G\)bornological coarse spaces. If f and g are close to each other, then they induce naturally isomorphic functors \(f_*\cong g_*:V^G_\mathbf {A}(X)\rightarrow V_\mathbf {A}^G(X')\).
Let \(\mathcal {A}\) be an additive category and denote by \(\oplus \) its biproduct. Recall that \(\mathcal {A}\) is called flasque if it admits an endofunctor \(S:\mathcal {A}\rightarrow \mathcal {A}\) and a natural isomorphism \(\mathrm {id}_\mathcal {A}\oplus S\cong S\).
Lemma 1.18
[4, Lemma 8.13] If X is a flasque \(G\)bornological coarse space, then the category \(V_\mathbf {A}^G(X)\) of Gequivariant Xcontrolled \(\mathbf {A}\)objects is a flasque category.
We conclude with the definition of coarse algebraic Khomology:
Definition 1.19
[4, Definition 8.8] Let G be a group and let \(\mathbf {A}\) be an additive category with strict Gaction. The Gequivariant coarse algebraic Khomology associated to \(\mathbf {A}\) is the Ktheory of the additive category of \(\mathbf {A}\)controlled objects:
When \(\mathbf {A}\) is the category of finitedimensional kvector spaces, we denote by \(K\mathcal {X}_{k}^G\) the associated Ktheory functor. The properties of the functor \(V_{\mathbf {A}}^G\) reviewed above are used in order to prove the following:
Theorem 1.20
[4, Theorem 8.9] Let G be a group and let \(\mathbf {A}\) be an additive category with strict Gaction. Then, the functor \(K\mathbf {A}\mathcal {X}^G\) is a Gequivariant \(\mathbf {Sp}\)valued coarse homology theory.
Keller’s cyclic homology for dgcategories
In this section we recall Keller’s construction of cyclic homology for dgcategories [21]. We start by recalling some properties of differential graded categories and mixed complexes, we introduce Keller’s construction and then review Keller’s Localization Theorem [21, Theorem 1.5]. Keller defines the cyclic homology of a dgcategory as the cyclic homology of a suitable mixed complex associated to it. We point here that, in the next Sect. 3 and in particular in Definition 3.1, we will need Keller’s cyclic homology in the less general context of additive (klinear) categories. However, for consistency with his language and for sake of completeness, we will state Keller’s definition and results in the broader context of dgcategories.
Dgcategories
In the following, we use the same conventions on differential graded categories and their properties as found in [20]; we refer to the same survey for a general overview on the subject. We recall that a dgcategory over k is a category enriched on (the category of) chain complexes of kmodules and that every additive, or klinear, category, is a dgcategory in a canonical way. We denote by \(\mathbf {dgcat}_k\) the category of small dgcategories (over k) and dgfunctors.
Remark 2.1
The category of dgmodules (over a dgalgebra or a dgcategory) admits two Quillen model structures where the weak equivalences are the objectwise quasiisomorphisms of dgmodules; 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 dgmodules over a dgalgebra, equipped with the projective model structure (hence the fibrations are the objectwise epimorphisms), is a combinatorial model category; see, for example, [10, Remark 2.14].
If \(\mathcal {A}\) is a dgcategory, we can define an associated derived category:
Definition 2.2
[20, Sect. 3.2] The derived category \(\mathcal {D}(\mathcal {A})\) of a dgcategory \(\mathcal {A}\) is the localization of the category of dgmodules over \(\mathcal {A}\) at the class of quasiisomorphisms.
The objects of \(\mathcal {D}(\mathcal {A})\) are the dgmodules over \(\mathcal {A}\) and the morphisms are obtained from morphisms of dgmodules by inverting the quasiisomorphisms. It is a triangulated category with shift functor induced by the 1translation and triangles coming from short exact sequences of complexes.
Let \(\mathcal {A}\) and \(\mathcal {B}\) be two small dgcategories. A dgfunctor \(F:\mathcal {A}\rightarrow \mathcal {B}\) is called a Morita equivalence if it induces an equivalence of derived categories. For a precise definition of Morita equivalences we refer to [20, Sect. 3.8], or [10, Definition 2.29].
Theorem 2.3
[34, Theorem 5.1] The category \(\mathbf {dgcat}_k\) of small dgcategories over k admits the structure of a combinatorial model category whose weak equivalences are the Morita equivalences.
For a description of fibrations and cofibrations we refer to [34, Theorem 5.1], or [20, Theorem 4.1]. We conclude with the definition of short exact sequences of dgcategories:
Definition 2.4
[20, Sect. 4.6] A short exact sequence of dgcategories is a sequence of morphisms \(\mathcal {A}\rightarrow \mathcal {B}\rightarrow \mathcal {C}\) inducing an exact sequence of triangulated categories
in the sense of Verdier.
The \(\infty \)category of mixed complexes
In this subsection we describe the (cocomplete stable \(\infty \))category of unbounded mixed complexes. We follow Kassel’s approach [19].
Definition 2.5
[19, Sect. 1] A mixed complex (C, b, B) is a triple consisting of a \(\mathbb {Z}\)graded kmodule \(C=(C_p)_{p\in \mathbb {Z}}\) together with differentials b and B
of degree \(1\) and 1, respectively, satisfying the following identities:
Morphisms of mixed complexes are given by maps commuting with both the differentials b and B. The category of mixed complexes and morphisms of mixed complexes is denoted by \(\mathbf {Mix}\).
When the differentials are clear from the context, we refer to a mixed complex (C, b, B) by its underlying kmodule C.
Let \(\Lambda \) be the dgalgebra over the field k
generated by an indeterminate \(\epsilon \) of degree 1, with \(\epsilon ^2=0\) and differential (of degree \(1\)) \(d(\epsilon )=0\). Mixed complexes are nothing but dgmodules over the dgalgebra \(\Lambda \):
Remark 2.6
[19] The category \(\mathbf {Mix}\) of mixed complexes is equivalent (in fact, isomorphic) to the category of left \(\Lambda \)dgmodules, which we denote by \(\Lambda \text {}\mathbf {Mod}\). We denote by \(L:\mathbf {Mix}\rightarrow \Lambda \mathbf  \mathbf {Mod}\) the functor sending a mixed complex to the associated \(\Lambda \)dgmodule and by \(R:\Lambda \mathbf  \mathbf {Mod}\rightarrow \mathbf {Mix}\) its inverse functor.
The category of kdgmodules admits a combinatorial model structure (the projective model structure, see Remark 2.1), whose weak equivalences are the objectwise quasiisomorphisms of dgmodules. In the language of mixed complexes this translates as follows:
Definition 2.7
A morphism \((C,b,B)\rightarrow (C',b',B')\) of mixed complexes is called a quasiisomorphism if the underlying bcomplexes are quasiisomorphic via the induced chain map \((C,b)\rightarrow (C',b')\).
Remark 2.8
Quasiisomorphisms of mixed complexes correspond to quasiisomorphisms of \(\Lambda \)dgmodules, i.e., the functors L and R of Remark 2.6 preserve quasiisomorphisms.
If \(\mathbf {C}\) is an ordinary category and W denotes a collection of morphisms of \(\mathbf {C}\), then \({{\,\mathrm{N}\,}}_{}(\mathbf {C})[W^{1}]\) is the \(\infty \)category obtained by the nerve \({{\,\mathrm{N}\,}}(\mathbf {C})\) of \(\mathbf {C}\) by inverting the set of morphisms W (see [9, Definition 7.1.2 and Proposition 7.1.3], [25, Definition 1.3.4.1]).
Definition 2.9
The \(\infty \)category
of mixed complexes is defined as the localization of the (nerve of the) category \(\mathbf {Mix}\) at the class \(W_{\mathrm {mix}}\) of quasiisomorphisms of mixed complexes.
Analogously, the \(\infty \)category \(\Lambda \mathbf  \mathbf {Mod}_\infty \) is defined as the localization of the category \(\Lambda \mathbf  \mathbf {Mod}\) of \(\Lambda \)dgmodules at the class W of quasiisomorphisms of \(\Lambda \)dgmodules:
Proposition 2.10
The \(\infty \)category \(\mathbf {Mix}_\infty \) is a cocomplete stable \(\infty \)category.
Proof
The category \(\Lambda \mathbf  \mathbf {Mod}\) is a (pretriangulated) dgcategory. By applying the dgnerve functor \({{\,\mathrm{N}\,}}_{\text {dg}}\) [25, Constr. 1.3.1.6] we obtain an \(\infty \)category \({{\,\mathrm{N}\,}}_{\text {dg}}(\Lambda \mathbf  \mathbf {Mod})\) [25, Proposition 1.3.1.10]. The dgnerve functor sends pretriangulated dgcategories to stable \(\infty \)categories [14, Theorem 4.3.1], [25, Proposition 1.3.1.10]. The \(\infty \)category \({{\,\mathrm{N}\,}}_{\text {dg}}(\Lambda \mathbf  \mathbf {Mod})\) is stable and its homotopy category can be identified (as a triangulated category) with the derived category \(\mathcal {D}(\Lambda )\) associated to the dgalgebra \(\Lambda \).
The category \(\Lambda \mathbf  \mathbf {Mod}\) is equipped with a combinatorial simplicial model structure by Remark 2.1. By [25, Proposition 1.3.1.17] and by the fact that the simplicial nerve of the simplicial category associated to \(\Lambda \mathbf  \mathbf {Mod}\) is equivalent to the localization \({{\,\mathrm{N}\,}}(\Lambda \mathbf  \mathbf {Mod})[W^{1}]\) (by [25, Remark 1.3.4.16 and Thm. 1.3.4.20] where we also use that the model category \(\Lambda \mathbf  \mathbf {Mod}\) is combinatorial, hence admits functorial factorizations), the two constructions \({{\,\mathrm{N}\,}}(\Lambda \mathbf  \mathbf {Mod})[W^{1}]\) and \({{\,\mathrm{N}\,}}_{\text {dg}}(\Lambda \mathbf  \mathbf {Mod})\) present equivalent \(\infty \)categories. Hence, the \(\infty \)category \(\Lambda \mathbf  \mathbf {Mod}_{\infty }\) is a stable \(\infty \)category. The \(\infty \)category \(\Lambda \mathbf  \mathbf {Mod}_{\infty }\) is also cocomplete by [25, Proposition 1.3.4.22] because the model category \(\Lambda \mathbf  \mathbf {Mod}\) is combinatorial.
The categories \(\mathbf {Mix}\) and \(\Lambda \mathbf  \mathbf {Mod}\) are isomorphic by Remark 2.6 and the functor \(L:\mathbf {Mix}\rightarrow \Lambda \mathbf  \mathbf {Mod}\) and its inverse \(R:\Lambda \mathbf  \mathbf {Mod}\rightarrow \mathbf {Mix}\) preserve quasiisomorphisms by Remark 2.8. This yields an equivalence of \(\infty \)categories
that proves the statement. \(\square \)
Remark 2.11
The homotopy category of the stable \(\infty \)category \(\mathbf {Mix}_\infty \) is canonically equivalent to the derived category \(\mathcal {D}(\Lambda )\) of the dgalgebra \(\Lambda \).
We conclude the subsection with the definition of Hochschild and cyclic homology of mixed complexes. A mixed complex (C, b, B) functorially determines a double chain complex \(\mathcal {B}C\) [22, Sect. 2.5.10] by means of the differentials b and B:
here, the chain complex (C, b) is placed in bidegree \((0,*)\), i.e., \(\mathcal {B}C_{(0,*)}=(C_{*},b)\), and the chain complex \((C[n],b_{C[n]})\), placed in bidegree \((n,*)\), is the chain complex (C, b) shifted by \(n\), hence \(\mathcal {B}C_{(p,q)}=C_{qp}\) for \(p\ge 0\) and \(\mathcal {B}C_{(p,q)}=0\) for \(p<0\). The total chain complex \({{\,\mathrm{Tot}\,}}(\mathcal {B}C)\), functorially associated to the double chain complex \(\mathcal {B}C\), is the chain complex defined in degree n by \({{\,\mathrm{Tot}\,}}_n (\mathcal {B}C)=\bigoplus _{i\ge 0} C_{n2i} \) with differential d acting as follows: \(d(c_n,c_{n2},\dots ):=(bc_n + Bc_{n2},\dots ).\)
Let \(\mathbf {Ch}\) be the category of chain complexes over k. Consider the forgetful functor
sending a mixed complex (M, b, B) to its underlying chain complex (M, b), and the functor
just described above.
Definition 2.12
[19, Sect. 1] Let (C, b, B) be a mixed complex. The Hochschild homology \({{\,\mathrm{HH}\,}}_*(C)\) of (C, b, B) is the homology of the underlying chain complex (C, b). Its cyclic homology \({{\,\mathrm{HC}\,}}_*(C)\) is the homology of the associated chain complex \({{\,\mathrm{Tot}\,}}(\mathcal {B}C)\).
We remark that this definition agrees with the classical definition of Hochschild and cyclic homology of algebras [19, 22].
Keller’s cyclic homology
Let k be a commutative ring with identity and let A be a kalgebra. Then, one can associate to A a cyclic module \(Z_*(A)\) [15] (i.e., a cyclic object in the category of kmodules) 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 \(\mathcal {A}\) [26, Definition 2.1.1]. We present these constructions in the more general setting of dgcategories.
Definition 2.13
[21] Let \(\mathcal {C}\) be a small dgcategory over k. The additive cyclic nerve of \(\mathcal {C}\) is the cyclic kmodule defined by:
where the sum runs over all the objects \((C_0,C_1,\dots ,C_n)\) in \(\mathcal {C}^{n+1}\). The face and degeneracy maps, and the cyclic action, are defined as follows:
where \(\sigma =(\deg f_n)(\deg f_{n1}+\dots +\deg f_0)\).
We get a covariant functor from the category of small dgcategories over k to the category of cyclic kmodules. To every cyclic kmodule M, we can further associate a mixed complex by letting \(b:M_n\rightarrow M_{n1}\) be the alternating sum
of face maps, and by defining the cochain map \(B:M_n\rightarrow M_{n+1}\) as the composition
Here, s denotes the extra degeneracy \(s=(1)^{n+1}t_{n+1}s_n:M_n\rightarrow M_{n+1}\) and \(N:=\sum _{i=0}^n t_{n+1}^i\).
Remark 2.14
Let M be a cyclic module. Then, the triple (M, b, B), where b and B are the differentials (2.6) and (2.7), respectively, is a mixed complex. Morphisms of cyclic modules commute with the face and the degeneracy maps and with the cyclic operators; they yield in this way morphisms of mixed complexes and a functor from the category of cyclic modules to the category of mixed complexes.
Definition 2.15
[21, Definition 1.3] We denote by
the functor from the category of small dgcategories over k to the category of mixed complexes defined as composition of the additive cyclic nerve functor of Definition 2.13 with the functor of Remark 2.14.
Thanks to the work of Keller, we know that this functor enjoys many useful properties, among others agreement, additivity and localization [21]. As we work in the context of \(\infty \)categories, we will spell them out in this language.
From now on we assume that k is a field. The \(\infty \)category of small dgcategories \( \mathbf {dgcat}_{k,\infty }:={{\,\mathrm{N}\,}}(\mathbf {dgcat}_{k})[W_{\text {Morita}}^{1}] \) is the localization at the class \(W_{\text {Morita}}\) of Morita equivalences. By [21, Theorem 1.5], the functor \(\mathrm {Mix}\) of Definition 2.15 sends Morita equivalences of dgcategories to quasiisomorphisms of mixed complexes and descends to a functor
between the localizations. Keller’s Localization Theorem [21, Theorem 1.5] can then be summarized as follows:
Theorem 2.16
[21, Theorem 1.5] Let k be a field. The functor \( \mathrm {Mix}:\mathbf {dgcat}_k\rightarrow \mathbf {Mix}_{\infty }\) satisfies the following:

(1)
it sends equivalences of small dgcategories to equivalences of mixed complexes;

(2)
it commutes with filtered colimits;

(3)
it sends short exact sequences \(\mathcal {A}\rightarrow \mathcal {B}\rightarrow \mathcal {C}\) of dgcategories to cofiber sequences of \(\mathbf {Mix}_{\infty }\).
Moreover, if A is a kalgebra, there is an equivalence of mixed complexes
where \(\mathrm {proj}\,A\) is the additive category of finitely generated projective modules.
Observe that the functor \(\mathrm {Mix}\circ {{\,\mathrm{loc}\,}}\) preserves filtered colimits. By Proposition 2.10, the \(\infty \)category \(\mathbf {Mix}_{\infty }\) is stable and cocomplete and cofiber sequences of \(\mathbf {Mix}_{\infty }\) [25, Definition 1.1.1.6] are detected in its homotopy category, i.e., in \(\mathcal {D}(\Lambda )\). 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 (over a field). Moreover, in this context, Keller’s functor \(\mathrm {Mix}\) is equivalent to the cyclic homology functor constructed by McCarthy [26] (see also [8, Lemma 3.4.4 and Remark 3.4.5]).
Equivariant coarse Hochschild and cyclic homology
For a fixed base field k and group G, we define equivariant coarse Hochschild \( {{\,\mathrm{\mathcal {X}HH}\,}}_{k}^G \) and cyclic homology \( {{\,\mathrm{\mathcal {X}HC}\,}}_k^G \) versions of the classical Hochschild and cyclic homology of kalgebras. This is achieved by first studying an intermediate equivariant coarse homology theory \({{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}_{k}^G\) with values in the \(\infty \)category of mixed complexes. In the definition of \({{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}_{k}^G\), we employ Keller’s functor \(\mathrm {Mix}:\mathbf {dgcat}_{k}\rightarrow \mathbf {Mix}\) in the context of klinear categories, and we apply it to the category \(V_k^G(X)\) of controlled objects; we then define Hochschild/cyclic homology of a bornological coarse space X as Hochschild/cyclic homology of \(\mathrm {Mix}(V_k^G(X))\). We conclude the section with some properties of these homology theories and of the associated assembly maps.
The equivariant coarse homology theory \({{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}_{k}^G\)
Let k be a field, \(\mathbf {Cat}_{k}\) the category of small klinear categories, \(V_{k}^{G}:G\mathbf {BornCoarse}\rightarrow \mathbf {Cat}_k\) the functor of Remark 1.15, let \(\mathrm {Mix}:\mathbf {dgcat}_{k}\rightarrow \mathbf {Mix}\) be the functor of Definition 2.15, \(\iota :\mathbf {Cat}_{k}\rightarrow \mathbf {dgcat}_{k}\) the functor sending a klinear category to its associated dgcategory and \({{\,\mathrm{loc}\,}}\) the localization functor \({{\,\mathrm{loc}\,}}:\mathbf {Mix}\rightarrow \mathbf {Mix}_\infty \).
Definition 3.1
We denote by \({{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}_k^G\) the following functor
from the category of Gbornological coarse spaces to the \(\infty \)category of mixed complexes.
The proof that the functor \({{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}_k^G\) satisfies the axioms of Definition 1.5 describing an equivariant coarse homology theory follows the ideas of coarse algebraic Khomology [4, Sect. 8] and does not require assumptions on G or k. For every Gset X and additive category with Gaction \(\mathbf {A}\), the category \(V_\mathbf {A}^G(X_{\text {min,max}}\otimes G_{\mathrm {can,min}})\) is equivalent to the additive category \(\mathbf {A}*_G X\) [7, Definition 2.1] by [4, Definition 8.21 and Proposition 8.24]. As a consequence, \({{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}_{k}^G(X_{\text {min,max}}\otimes G_{\mathrm {can,min}})\simeq \mathrm {Mix}(\mathbf {A}*_G X)\), and the functor \({{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}_{k}^G\), when applied to such spaces, can be described as the mixed complex of a suitable additive category. In the case of the group G, this result says that the category of controlled objects \(V_k^G(G_{\mathrm {can,min}})\) is equivalent to the category of finitely generated free k[G]modules (see Proposition 3.15), which, together with Proposition 3.14, further justifies the use of Definition 3.1 in the construction of a cyclic homology theory for Gbornological coarse spaces.
The main result of the section is the following theorem:
Theorem 3.2
The functor
is a \(G\)equivariant \(\mathbf {Mix}_\infty \)valued coarse homology theory.
Proof
The category \(\mathbf {Mix}_{\infty }\) is stable and cocomplete by Proposition 2.10. We prove below that the functor \({{{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}}_{k}^G\) satisfies coarse invariance (see Proposition 3.3), vanishing on flasque spaces (see Proposition 3.4), ucontinuity (see Proposition 3.5) and coarse excision (see Theorem 3.6), i.e., the axioms describing an equivariant coarse homology theory. \(\square \)
Proposition 3.3
The functor \({{{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}}_{k}^G :G\mathbf {BornCoarse}\rightarrow \mathbf {Mix}_\infty \) satisfies coarse invariance.
Proof
If \(f:X\rightarrow Y\) is a coarse equivalence of \(G\)bornological coarse spaces, then it induces a natural equivalence \(f_*:V_k^G(X)\rightarrow V_k^G(Y)\) by Lemma 1.17. Keller’s functor \(\mathrm {Mix}\) sends equivalences of dgcategories to equivalences of mixed complexes by Theorem 2.16 (1). Hence, the functor \(f_{*}\) induces an equivalence \({ {{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}}_k^G(X)\xrightarrow {\sim }{ {{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}}_k^G(Y)\) in \(\mathbf {Mix}_{\infty }\), i.e., the functor \({{{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}}_{k}^G\) is coarse invariant. \(\square \)
Recall the definition of flasque spaces in Definition 1.3.
Proposition 3.4
The functor \({{{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}}_{k}^G :G\mathbf {BornCoarse}\rightarrow \mathbf {Mix}_\infty \) vanishes on flasque spaces.
Proof
By Lemma 1.18, the category \(V_k^G(X)\) is a flasque category, hence there exists an endofunctor \(S:V_k^G(X)\rightarrow V_k^G(X)\) such that \(\mathrm {id}_{V_k^G(X)}\oplus S\cong S\). By [32, Theorem 2.3.11] (see also [8, Theorem 3.3.5]), the morphisms \(\mathrm {Mix}(\mathrm {id}) \oplus \mathrm {Mix}(S)\) and \(\mathrm {Mix}(\mathrm {id} \oplus S)\cong \mathrm {Mix}(S)\) are equivalent in \(\mathbf {Mix}_{\infty } \). This means that the morphism
is equivalent to the 0morphism and that \({{{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}}_{k}^G(X)\simeq 0\). \(\square \)
Proposition 3.5
The functor \({{{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}}_{k}^G :G\mathbf {BornCoarse}\rightarrow \mathbf {Mix}_\infty \) is ucontinuous.
Proof
Let X be a \(G\)bornological coarse space, and let \(\mathcal {C}^G\) be the poset of \(G\)invariant controlled sets. By Remark 1.16, there is an equivalence \(V_k^G(X)\simeq {{\,\mathrm{colim}\,}}_{U\in \mathcal {C}^G}V_k^G(X_U) \) of klinear categories, hence of dgcategories. The functor \(\mathrm {Mix}\) commutes with filtered colimits, and we get the equivalence
in \(\mathbf {Mix}_{\infty }\), which shows that the functor \({{{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}}_{k}^G\) is ucontinuous. \(\square \)
Theorem 3.6
The functor \({{{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}}_{k}^G :G\mathbf {BornCoarse}\rightarrow \mathbf {Mix}_\infty \) satisfies coarse excision.
Before giving the proof of this theorem we first need some more terminology.
Definition 3.7
[18] A full additive subcategory \(\mathcal {A}\) of an additive category \(\mathcal {U}\) is a Karoubifiltration if every diagram \( X\rightarrow Y\rightarrow Z \) in \(\mathcal {U}\), with \(X,Z\in \mathcal {A}\), admits an extension
for some object \(A\in \mathcal {A}\).
By [18, Lemma 5.6], this definition is equivalent to the classical one [11, 17]. If \(\mathcal {A}\) is a Karoubifiltration of \(\mathcal {U}\), we can construct a quotient category \(\mathcal {U}/\mathcal {A}\). Its objects are the objects of \(\mathcal {U}\), and the morphisms sets are defined as follows:
where the relation identifies pairs of maps \(U\rightarrow V\) whose difference factors through an object of \(\mathcal {A}\).
Let X be a \(G\)bornological coarse space and let \(\mathcal {Y}=(Y_i)_{i\in I}\) 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 \(Y_i\hookrightarrow X\) induces a functor \(V_k^G(Y_i)\rightarrow V_k^G(X)\) which is injective on objects. The categories \(V_k^G(Y_i)\) and \( V_k^G(\mathcal {Y}):={{\,\mathrm{colim}\,}}_{i\in I} V_k^G(Y_i) \) are full subcategories of \(V_k^G(X)\).
Lemma 3.8
[4, Lemma 8.14] Let \(\mathcal {Y}\) be an equivariant big family on the Gbornological coarse space X. Then, the full additive subcategory \(V_k^G(\mathcal {Y})\) of \(V_k^G(X)\) is a Karoubi filtration.
Let X be a \(G\)bornological coarse space, and let \((Z,\mathcal {Y})\) be an equivariant complementary pair. Consider the functor
induced by the inclusion \(i:Z\rightarrow X\); on objects, it coincides with \(i_*:V_k^G(Z) \rightarrow V_k^G(X)\), but on morphisms it sends the equivalence class [A] of A to the equivalence class \([i_*(A)]\) of \(i_*(A)\).
Lemma 3.9
[4, Proposition 8.15] The functor a in (3.1) is an equivalence of categories.
Proof of Theorem 3.6
Let X be a \(G\)bornological coarse space, and let \((Z,\mathcal {Y})\) be an equivariant complementary pair on X. By Lemma 3.8, \(V_k^G(Z\cap \mathcal {Y})\subseteq V_k^G(Z)\) and \(V_k^G(\mathcal {Y})\subseteq V_k^G(X)\) are Karoubi filtrations and yield the following sequences of klinear categories:
and
By [31, Example. 1.8, Proposition 2.6] (see also [8, Remark 3.3.12]), Karoubi filtrations induce short exact sequences of dgcategories. Hence, Theorem 2.16 gives cofiber sequences of mixed complexes. The inclusion \(Z\hookrightarrow X\) induces a commutative diagram (where the rows are the obtained cofiber sequences)
here \(a_*\) is the map induced by \(a:V_k^G(Z)/V_k^G(Z\cap \mathcal {Y})\rightarrow V_k^G(X)/V_k^G(\mathcal {Y})\) in (3.1). By Lemma 3.9, the functor a yields an equivalence of categories, hence of mixed complexes and the left square is a coCartesian square in \(\mathbf {Mix}_{\infty }\).
In order to conclude the proof, we recall that \({{{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}}_k^G(\mathcal {Y})\) is defined as the filtered colimit \({{{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}}_k^G(\mathcal {Y})={{\,\mathrm{colim}\,}}_i {{{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}}_k^G(Y_i)\) and that \( V_k^G(\mathcal {Y}):={{\,\mathrm{colim}\,}}_{i\in I} V_k^G(Y_i). \) The functor \(\mathrm {Mix}\) commutes with filtered colimits of dgcategories. Hence we have the equivalence \(\mathrm {Mix}(V_k^G(\mathcal {Y}))= \mathrm {Mix}({{\,\mathrm{colim}\,}}_i V_k^G(Y_i))\simeq {{\,\mathrm{colim}\,}}_i \mathrm {Mix}(V_k^G(Y_i))\), and the same holds for \(Z\cap \mathcal {Y}\). By using these identifications, we obtain the coCartesian square in \(\mathbf {Mix}_{\infty }\)
This means that \({{{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}}_{k}^G\) satisfies coarse excision. \(\square \)
Coarse Hochschild and cyclic homology
The functors \(\mathrm {forget}:\mathbf {Mix}\rightarrow \mathbf {Ch}\) in (2.4), sending a mixed complex to the underlying chain complex, and \({{\,\mathrm{Tot}\,}}(\mathcal {B}):\mathbf {Mix}\rightarrow \mathbf {Ch}\) in (2.5), sending a mixed complex to the total complex of its associated bicomplex, send quasiisomorphisms of mixed complexes to quasiisomorphisms of chain complexes. Hence they descend to functors between the localizations.
Definition 3.10
Let k be a field, G a group and \(\mathbf {Ch}_{\infty }\) the \(\infty \)category of chain complexes. The \(G\)equivariant coarse Hochschild homology \({{\,\mathrm{\mathcal {X}HH}\,}}_k^G\) (with kcoefficients) is the \(G\)equivariant \(\mathbf {Ch}_\infty \)valued coarse homology theory
defined as composition of the functor \({{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}_{k}^G\) of Definition 3.1 and of the functor \(\mathrm {forget}\) in (2.4). The composition
involving composition with the functor \({{\,\mathrm{Tot}\,}}(\mathcal {B})\) (2.5) is \(G\)equivariant coarse cyclic homology.
The definitions are justified by the following:
Theorem 3.11
The functors
are \(G\)equivariant \(\mathbf {Ch}_{\infty }\)valued coarse homology theories.
Proof
By Theorem 3.2, the functor \({{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}^G_k:G\mathbf {BornCoarse}\rightarrow \mathbf {Mix}_{\infty }\) is an equivariant coarse homology theory. The functors \(\mathrm {forget}:\mathbf {Mix}_\infty \rightarrow \mathbf {Ch}_\infty \) and the functor \({{\,\mathrm{Tot}\,}}(\mathcal {B}):\mathbf {Mix}_\infty \rightarrow \mathbf {Ch}_\infty \) commute with filtered colimits and send cofiber sequences to cofiber sequences. Hence the two compositions with \({{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}^G_k\) satisfy coarse invariance, coarse excision, ucontinuity, and vanishing on flasques. \(\square \)
The category of mixed complexes has a natural symmetric monoidal structure induced by tensor products between the underlying chain complexes [19]. As tensor products of mixed complexes preserve equivalences, k being a field, we get a symmetric monoidal \(\infty \)category \(\mathbf {Mix}_{\infty }^\otimes :={{\,\mathrm{N}\,}}(\mathbf {Mix}^\otimes )[W_{\mathrm {mix}}^{\otimes ,1}]\rightarrow {{\,\mathrm{N}\,}}(\mathbf {Fin}_*)\) (with monoidal structure induced by the monoidal structure on \(\mathbf {Mix}\) by [16, Proposition 3.2.2]). By [19, Theorem 2.4], the functor \(\mathrm {Mix}\) has a lax symmetric monoidal refinement (see also [12]). This implies that coarse Hochschild and cyclic homologies are lax symmetric monoidal functors:
Proposition 3.12
The functors \({{\,\mathrm{\mathcal {X}HH}\,}}_{k}^G\) and \({{\,\mathrm{\mathcal {X}HC}\,}}_{k}^G\) admit lax symmetric monoidal refinements:
and
where \(\mathbf {Ch}_{\infty }^\otimes \) is the \(\infty \)category of chain complexes with its standard symmetric monoidal structure.
Proof
By [1, Theorem 3.26] and [19, Theorem 2.4], the functor \({{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}_{k}^G\) of Definition 3.1 admits a lax symmetric monoidal refinement
As the functors \(\mathrm {forget}\) in (2.4) and \({{\,\mathrm{Tot}\,}}(\mathcal {B})\) in (2.5) are lax symmetric monoidal, coarse Hochschild and cyclic homology admit lax symmetric monoidal refinements as well. \(\square \)
Comparison results and assembly maps
In this subsection, we compare equivariant coarse Hochschild homology with the classical version of Hochschild homology for kalgebras. Furthermore, we show that the forgetcontrol map for coarse Hochschild homology is equivalent to the associated generalized assembly map.
Notation 3.13
Let A be a kalgebra. We denote by
the chain complexes computing the Hochschild and cyclic homology of the mixed complex \(\mathrm {Mix}(A)\) associated to the cyclic object \(Z_*(A)\) associated to A [15, 22].
Let \(\{*\}\) be the onepoint bornological coarse space, endowed with a trivial Gaction.
Proposition 3.14
There are equivalences of chain complexes
between the coarse Hochschild (cyclic) homology of the point and the classical Hochschild (cyclic) homology of k.
Proof
By Theorem 2.16, the mixed complex \(\mathrm {Mix}(A)\) associated to a kalgebra A is equivalent to the mixed complex associated to the klinear category of finitely generated projective Amodules. When X is a point endowed with a trivial \(G\)action and k is a field, the klinear category \(V_k(X)\) is isomorphic to the category \(\mathbf {Vect}_k^{\mathrm {f.d.}}\) of finitedimensional kvector spaces, i.e., \(\mathrm {Mix}(V_k(\{*\}))\simeq \mathrm {Mix}\big (\mathbf {Vect}_k^{\mathrm {f.d.}}\big )\simeq \mathrm {Mix}(k)\).
\(\square \)
Let G be a group. By Example 1.2, there is a canonical Gbornological coarse space \(G_{\mathrm {can,min}}=(G,\mathcal {C}_{\text {can}},\mathcal {B}_{\min } )\) associated to it.
Proposition 3.15
There are equivalences of chain complexes:
and
between the \(G\)equivariant coarse Hochschild and cyclic homologies of \(G_{\mathrm {can,min}}\) and the classical Hochschild and cyclic homologies of the group algebra \(k[G]\).
Proof
The category \(V_k^G(G_{\mathrm {can,min}})\) of \(G\)equivariant \(G_{\mathrm {can,min}}\)controlled finitedimensional kvector spaces is equivalent to the category \(\mathbf {Mod}^{\mathrm {fg,free}}(k[G])\) of finitely generated free \(k [G]\)modules [4, Proposition 8.24]. By Theorem 2.16, Keller’s mixed complex \(\mathrm {Mix}(\mathbf {Mod}^{\mathrm {fg,free}}(k[G]))\) of the category of finitely generated free k[G]modules is equivalent to the mixed complex associated to the category \(\mathbf {Mod}^{\mathrm {fg,proj}}(k[G])\) of finitely generated projective modules (because they are Morita equivalent dgcategories). Therefore, the result follows from the chain of equivalences of mixed complexes
where the last equivalence is again true by Theorem 2.16. \(\square \)
Let X be a Gset and let \(X_{\mathrm {min,max}}\) denote the Gbornological coarse space with minimal coarse structure and maximal bornology.
Remark 3.16
Let H be a subgroup of G; then, by [4, Proposition 8.24] we get an equivalence of chain complexes:
the same holds for equivariant coarse cyclic homology.
One of the main applications of coarse homotopy theory is the study of assembly maps. 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 the definitions of the cone functor \(\mathcal {O}^\infty _{hlg}\) [4, Definition 10.10], of the forgetcontrol map \(\beta \) [4, Definition 11.10] and of the coarse assembly map \(\alpha \) [4, Definition 10.24]. By [4, Theorem 11.16], the forgetcontrol map for a Gequivariant coarse homology theory E can be compared with the classical assembly map for the associated Gequivariant homology theory \(E\circ \mathcal {O}^\infty _{hlg}:G\mathbf {Top} \rightarrow \mathbf {C}\).
By applying the Eilenberg–MacLane correspondence (1.3), we can assume that the equivariant coarse homology theories \({{\,\mathrm{\mathcal {X}HH}\,}}_{k}^G\) and \({{\,\mathrm{\mathcal {X}HC}\,}}_{k}^G\) are equivariant spectravalued coarse homology theories.
Definition 3.17
Let \(\mathbf {HH}_k^G:=\mathcal {EM}\circ {{\,\mathrm{\mathcal {X}HH}\,}}_k^G\circ \mathcal {O}^\infty _{\text {hlg}}:G\mathbf {Top}\rightarrow \mathbf {Sp}\) be the \(G\)equivariant homology theory associated to equivariant coarse Hochschild homology.
Let \(\mathbf {Fin}\) be the family of finite subgroups of \(G\). The following is a consequence of [4, Theorem 11.16] (see also [8, Proposition 4.2.7]):
Proposition 3.18
The forgetcontrol map \(\beta _{G_{\mathrm {can,min}},G_\mathrm {{max,max}}}\) for \({{\,\mathrm{\mathcal {X}HH}\,}}_{k}^G\) is equivalent to the assembly map \(\alpha _{E_{\mathbf {Fin}}G,G_{\mathrm {can,min}}}\) for the Ghomology theory \(\mathbf {HH}_k^G\).
Furthermore, the assembly map \(\alpha _{E_{\mathbf {Fin}}G,G_{\mathrm {can,min}}}\) for the Ghomology theory \(\mathbf {HH}_k^G\) (hence, the forgetcontrol map \(\beta _{G_{\mathrm {can,min}},G_\mathrm {{max,max}}}\) for \({{\,\mathrm{\mathcal {X}HH}\,}}_{k}^G\)) is split injective by [23, Theorem 1.7].
From coarse algebraic Ktheory to coarse ordinary homology
In this section we define a natural transformation
from equivariant coarse Hochschild homology \({{\,\mathrm{\mathcal {X}HH}\,}}_k^G\) to the \(\mathbf {Ch}_{\infty }\)valued equivariant coarse ordinary homology \({{\,\mathrm{\mathcal {X}H}\,}}^G\) and, analogously, a natural transformation \(\Phi _{{{\,\mathrm{\mathcal {X}HC}\,}}^G_k}\) from equivariant coarse cyclic homology. By abuse of notation we will denote by \({{\,\mathrm{\mathcal {X}HH}\,}}^G\), \({{\,\mathrm{\mathcal {X}HC}\,}}_{}^G\) and \({{\,\mathrm{\mathcal {X}H}\,}}^G\) both the chain and spectra valued coarse homology theories.
The transformation \(\Phi _{{{\,\mathrm{\mathcal {X}HH}\,}}^G_k}\) is constructed in the following steps:

For every Gbornological coarse space X, we consider its associated klinear category \(V_k^G(X)\) of controlled objects, hence the associated additive cyclic nerve \({{\,\mathrm{CN}\,}}_{}(V_k^G(X))\). For every tensor element \(A_0\otimes _{}\dots \otimes A_n\) in the additive cyclic nerve of \(V_k^G(X)\) and every \(n+1\) points \(x_0,\dots ,x_n\) of X, we define a tracelike map, which gives an element of k (see Notation 4.2);

by letting \(x_0,\dots ,x_n\) vary, this yields a Gequivariant locally finite controlled chain on X, i.e., an element of \(\mathcal {X}C_n^G(X)\) (see Definition 4.3 and Lemma 4.4); by letting \(A_0\otimes _{}\dots \otimes A_n\) vary we get a map \(\varphi :{{\,\mathrm{CN}\,}}_{*}(V_k^G(X))\rightarrow \mathcal {X}C_*^G(X)\) that is a chain map with respect to the differential \(d=\sum (1)^i d_i\) of \({{\,\mathrm{CN}\,}}_{}(V_k^G(X))\) (see Proposition 4.6);

the additive cyclic nerve \({{\,\mathrm{CN}\,}}_{}(V_k^G(X))\) yields a mixed complex with the differentials b and B as in Remark 2.14; the chain map \(\varphi \) extends to a map of mixed complexes \({\tilde{\varphi }}\) (see Lemma 4.7) and yields a natural transformation of equivariant coarse homology theories \(\Phi _{{{\,\mathrm{\mathcal {X}HH}\,}}^G_k}:{{\,\mathrm{\mathcal {X}HH}\,}}^G_k\longrightarrow {{\,\mathrm{\mathcal {X}H}\,}}^G\) (see Theorem 4.8).
We now proceed with the precise construction.
Let \(V_k^G(X)\) be the klinear category of Xcontrolled finitedimensional kvector spaces of Definition 1.10. The additive cyclic nerve associated to \(V_{k}^G(X)\) (see Definition 2.13) is described, in degree n, by
where the index i runs cyclically in the set \(\{0,\dots , n\}\) and the sum ranges over all the \((n+1)\)tuples \(((M_0,\rho _0),\dots ,(M_n,\rho _n))\) of objects of \(V_k^G(X)\).
Remark 4.1
For every controlled morphism \(A_{i}:(M_{i+1},\rho _{i+1})\rightarrow (M_i,\rho _i)\) (see Definition 1.12) in \({{\,\mathrm{Hom}\,}}((M_{i+1},\rho _{i+1}),(M_i,\rho _i))\) and for every pair of points x and y of X, there is a welldefined klinear map
induced by \(A_{i}\).
We use the following notation:
Notation 4.2
Let \(A_0\otimes \cdots \otimes A_n\) be an element of \(\bigotimes _{i=0}^n{{\,\mathrm{Hom}\,}}((M_{i+1},\rho _{i+1}),(M_i,\rho _i))\) and let \(((M_0,\rho _0),\dots ,(M_n,\rho _n))\) be an \((n+1)\)tuple of objects of \(V_k^G(X)\). Let \((x_0,\dots ,x_n)\) be a point of \(X^{n+1}\). The symbol
denotes the linear operator \((A_0\circ \dots \circ A_n)(x_0,\dots ,x_n) :M_{0}(x_{n})\rightarrow M_{0}(x_{n})\) defined as the composition
of the induced operators \(A_i^{x_i,x_{i+1}}:M_i(x_i)\rightarrow M_{i+1}(x_{i+1})\). It is an endomorphism of \(M_0(x_n)\), which is a finitedimensional kvector space.
Let X be a Gbornological coarse space and let \(\mathcal {X}C_n(X)\) be the kvector space generated by the locally finite controlled nchains on X (see Definition 1.6).
Definition 4.3
We let \(\varphi _n:\mathrm {CN}_n(V_k^G(X))\rightarrow \mathcal {X}C_n(X)\) be the map defined on elementary tensors as
and extended linearly.
Lemma 4.4
The nchain \( \varphi _n(A_0\otimes \cdots \otimes A_n)\) is locally finite and controlled.
Proof
In order to prove that \(\varphi _n(A_{0}\otimes \cdots \otimes A_{n})\) is locally finite and controlled we show that its support \({{\,\mathrm{supp}\,}}(\varphi _n(A_{0}\otimes \cdots \otimes A_{n}))\) defined in (1.2) is locally finite and that there exists an entourage \(U\) of X such that every \(x=(x_0,\dots ,x_n)\) in \({{\,\mathrm{supp}\,}}(\varphi _n(A_{0}\otimes \cdots \otimes A_{n}))\) is \(U\)controlled.
We first observe that the operators \(A_i:(M_{i+1},\rho _{i+1})\rightarrow (M_{i},\rho _i)\) 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 \(M_{i+1}\rightarrow M_{i}\circ U_{i}[]\) satisfying an equivariance condition. For every point x in X, \(A_{i}\) restricts to a morphism
where the direct sum has only finitely many nonzero summands.
Let K be a bounded set of X. The set of points \(x_n\in K_{}\) for which \(M_0(x_n)\) is nonzero is finite (as a consequence of Definition 1.13). For such a fixed \(x_n\), there are only finitely many points \(x_{n1}\in U_{n}[K_{}]\) such that the corresponding map \(A_n^{x_{n},x_{n1}}:M_0(x_n)\rightarrow M_n(x_{n1})\) is nonzero. The set \(U_{n}[K_{}]\) is a bounded set of X, the morphism \(A_{n1}:M_{n}\rightarrow M_{n1}\) is \(U_{n1}\)controlled and we can repeat the same argument for \(A_{n1}\), hence for each \(A_{i}\). This implies that the nchain is locally finite because, for the given bounded set K, we have found only finitely many tuples \((x_0,\dots ,x_n)\) in the support of \(\varphi _n(A_0\otimes _{}\cdots \otimes A_n)\) that meet K.
The chain is also Ucontrolled, where U is the entourage \(U:=U_0\circ \dots \circ U_n\) of X. \(\square \)
Remark 4.5
Let X be a Gbornological coarse space. Let \((M,\rho )\) be a \(G\)equivariant Xcontrolled finitedimensional kvector space and let g be an element of the group \(G\). Then, \(\rho (g)\) (Definition 1.10) is a natural isomorphism between the functors M and gM. The diagram
is commutative, for \(A_0\otimes \cdots \otimes A_n\) in \(\mathrm {CN}_n(V_k^G(X))\) with \(A_{i}:(M_{i+1},\rho _{i+1})\rightarrow (M_i,\rho _i)\), where the isomorphisms are induced by \(\rho _i(g)\). Hence, the image of \(\varphi _{n}\) is a \(G\)invariant locally finite controlled nchain on X.
Let \(\partial _{i}:\mathcal {X}C_n^G(X)\rightarrow \mathcal {X}C_{n1}^G(X)\) be the ith differential of the chain complex \(\mathcal {X}C^G(X)\) and let \(d_{i}:{{\,\mathrm{CN}\,}}_n(V_k^G(X))\rightarrow {{\,\mathrm{CN}\,}}_{n1}(V_k^G(X))\) be the ith face map of \({{\,\mathrm{CN}\,}}(V_k^G(X))\). In the next proposition we consider the chain complex \((\mathrm {CN}_{}(V_k^G(X)),d)\) underlying the additive cyclic nerve \({{\,\mathrm{CN}\,}}(V_k^G(X))\).
Proposition 4.6
[8, Proposition 4.3.6] The maps \(\varphi _n:\mathrm {CN}_n(V_k^G(X))\rightarrow \mathcal {X}C_n^G(X)\) of Definition 4.3 extend to a chain map \( \varphi :(\mathrm {CN}_{}(V_k^G(X)),d)\rightarrow (\mathcal {X}C_{}^G(X),\partial ). \)
Proof
The result is a consequence of the additivity of the trace map and of its invariance under cyclic permutations. \(\square \)
If M is a cyclic module, as in Remark 2.14, we get a mixed complex. The chain complex \(\mathcal {X}C^G(X)\) is also a mixed complex with the differential \(B=0\).
Lemma 4.7
The chain map \(\varphi :\mathrm {CN}(V_{k}^G(X))\rightarrow \mathcal {X}C^G(X)\) of Definition 4.3 extends to a map \(\tilde{\varphi }:\mathrm {Mix}(V_{k}^G(X))\rightarrow \mathcal {X}C^G(X)\) that is a morphism of mixed complexes.
Proof
The proof is a simple computation and uses the definition of the operator B (2.7) and that the trace is invariant under cyclic permutations. See also [8, Lemma 4.3.7]. \(\square \)
We can now construct the natural transformation \(\Phi _{{{\,\mathrm{\mathcal {X}HH}\,}}_{k}^G}:{{\,\mathrm{\mathcal {X}HH}\,}}_{k}^G\rightarrow \mathcal {X}C^G_{}\):
Theorem 4.8
The map \(\varphi \) extends to natural transformations
and
of Gequivariant \(\mathbf {Ch}_{\infty }\)valued coarse homology theories.
Proof
The map \(\varphi :(\mathrm {CN}_{}(V_k^G(X)),d)\rightarrow (\mathcal {X}C_{}^G(X),\partial )\) of Definition 4.3 is a chain map by Proposition 4.6. Let \(f:X\rightarrow Y\) be a morphism of Gequivariant bornological coarse spaces. Consider the induced chain map \(\mathcal {X}C^G(f):\mathcal {X}C^G_{}(X)\rightarrow \mathcal {X}C^G_{}(Y)\) and the induced functor \(f_{*}=V^G_{k}(f):V^G_k(X) \rightarrow V^G_k(X')\). By functoriality of the additive cyclic nerve, \(f_*\) induces a morphism \({{\,\mathrm{CN}\,}}_{}(f_{*}):{{\,\mathrm{CN}\,}}_{*}(V^G_k(X))\rightarrow {{\,\mathrm{CN}\,}}_{*}(V^G_k(Y))\) of cyclic modules (hence, a chain map between the underlying chain complexes as well).
The diagram
is commutative. The map \(\varphi \) 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 \({{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}_{k}^G\), Definition 3.10), the map \(\varphi \) yields a natural transformation of equivariant coarse homology theories \(\Phi _{{{\,\mathrm{\mathcal {X}HH}\,}}_{k}^G}:{{\,\mathrm{\mathcal {X}HH}\,}}_{k}^G\rightarrow {{\,\mathrm{\mathcal {X}H}\,}}^G_{}.\)
To every mixed complex C, we associate the chain complex \({{\,\mathrm{Tot}\,}}(\mathcal {B}C)\) (2.3) defined by \({{\,\mathrm{Tot}\,}}_n (\mathcal {B}C)=\bigoplus _{i\ge 0} C_{n2i} \) with differential \(d(c_n,c_{n2},\dots )=(bc_n + Bc_{n2},\dots )\). By Lemma 4.7, we conclude that the map \(\varphi \) extends to a chain map on the total complex as well, and to a natural transformation of coarse homology theories
Here the sum is indexed by the natural numbers because the (mixed complex associated to) the additive cyclic nerve of \(V_{k}^G(X)\) is positively graded. \(\square \)
The following result implies that the transformation \(\Phi _{{{\,\mathrm{\mathcal {X}HH}\,}}_{k}}:{{\,\mathrm{\mathcal {X}HH}\,}}_{k}\rightarrow {{\,\mathrm{\mathcal {X}H}\,}}_{}\) is nontrivial:
Proposition 4.9
If X is the onepoint space \(\{*\}\), then the transformation
induces an equivalence of chain complexes.
Proof
Let \(c:\{*\}^{n+1}\rightarrow k\) be an nchain in \(\mathcal {X}C_n(*)\); we identify this chain with the element \(c\in k\) that is its image. Let \(\iota _n:\mathcal {X}C_n(*)\rightarrow {{\,\mathrm{CN}\,}}_n(V_k(*)))\) be the map sending c to the element \((\cdot c)\otimes (\cdot 1_{k})\otimes \cdots \otimes (\cdot 1_{k})\). This extends to a chain map that gives a section of the trace map, i.e., \(\varphi \circ \iota =\mathrm {id}\).
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
By using these equivalences and the section \(\varphi \circ \iota =\mathrm {id}\), we obtain that, when X is the onepoint space, the transformation \(\Phi _{{{\,\mathrm{\mathcal {X}HH}\,}}_{k}}\) induces an equivalence of chain complexes. \(\square \)
By applying the Eilenberg–MacLane correspondence \(\mathcal {EM}\) (1.3), we now assume that equivariant coarse Hochschild and cyclic homology take values in the \(\infty \)category \(\mathbf {Sp}\) of spectra. The classical trace map constructed by McCarthy [26, Sect. 4.4] extends to a transformation from equivariant coarse algebraic Khomology to equivariant coarse Hochschild homology:
Proposition 4.10
[8, Proposition 4.4.1] There are natural transformations
induced by the Dennis trace maps from algebraic Ktheory to Hochschild homology.
In particular, when X is the Gbornological coarse space \(G_{\mathrm {can,min}}\), the induced map
is the classical Dennis trace map \(K(k[G])\rightarrow {{\,\mathrm{HH}\,}}(k[G])\) by McCarthy’s agreement result [26, Sect. 4.5], by [4, Proposition 8.24] and by Proposition 3.15.
Composing the transformations of Proposition 4.10 and of Theorem 4.8 we get the natural transformation
from equivariant coarse algebraic Khomology to equivariant coarse ordinary homology. When X is the Gbornological coarse space \(G_{\mathrm {can,min}}\), we get a map \(K(k[G])\rightarrow H(G;k)\) from the algebraic Ktheory of the group ring k[G] to the ordinary homology of G with kcoefficients. We believe that further investigations of this transformation can be useful to detect coarse Ktheory classes.
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Acknowledgements
Open Access funding provided by Projekt DEAL. This work formed part of the author’s PhD thesis at Regensburg University. It is a pleasure to again acknowledge Ulrich Bunke, this work would not exist without him. The author also thanks Clara Löh, DenisCharles Cisinski and Alexander Engel for helpful discussions, and the anonymous referees for constructive comments and recommendations. The author has been supported by the DFG Research Training Group GRK 1692 “Curvature, Cycles, and Cohomology” and by the DFG SFB 1085 “Higher Invariants”.
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Caputi, L. Cyclic homology for bornological coarse spaces. J. Homotopy Relat. Struct. 15, 463–493 (2020). https://doi.org/10.1007/s40062020002633
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DOI: https://doi.org/10.1007/s40062020002633
Keywords
 Ktheory and homology
 Algebraic Topology
 Coarse Geometry