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 k-algebras. 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 k-linear 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 k-linear 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 k-linear category to its associated dg-category 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 G-bornological 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 K-homology
[4, Sect. 8] and does not require assumptions on G or k. For every G-set X and additive category with G-action \(\mathbf {A}\), 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 G-bornological 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), u-continuity (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 dg-categories 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
$$\begin{aligned} {{{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}}_{k}^G(\mathrm {id}):{{{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}}_{k}^G(X)\rightarrow {{{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}}_{k}^G(X) \end{aligned}$$
is equivalent to the 0-morphism 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 u-continuous.
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 k-linear categories, hence of dg-categories. The functor \(\mathrm {Mix}\) commutes with filtered colimits, and we get the equivalence
$$\begin{aligned} {{{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}}_{k}^G(X)\simeq {{\,\mathrm{colim}\,}}_{U\in \mathcal {C}^G}{{{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}}_{k}^G(X_U) \end{aligned}$$
in \(\mathbf {Mix}_{\infty }\), which shows that the functor \({{{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}}_{k}^G\) is u-continuous. \(\square \)
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 Karoubi-filtration 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 Karoubi-filtration 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:
$$\begin{aligned} {{\,\mathrm{Hom}\,}}_{\mathcal {U}/\mathcal {A}}(U,V):={{\,\mathrm{Hom}\,}}_\mathcal {U}(U,V)/{\sim } \end{aligned}$$
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 G-bornological 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
$$\begin{aligned} a:V_k^G(Z)/V_k^G(Z\cap \mathcal {Y})\rightarrow V_k^G(X)/V_k^G(\mathcal {Y}) \end{aligned}$$
(3.1)
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 k-linear categories:
$$\begin{aligned} V_k^G(Z\cap \mathcal {Y})\rightarrow V_k^G(Z)\rightarrow V_k^G(Z)/V_k^G(Z\cap \mathcal {Y}) \end{aligned}$$
and
$$\begin{aligned} V_k^G(\mathcal {Y})\rightarrow V_k^G(X)\rightarrow V_k^G(X)/V_k^G(X\cap \mathcal {Y}). \end{aligned}$$
By
[31, Example. 1.8, Proposition 2.6] (see also
[8, Remark 3.3.12]), Karoubi filtrations induce short exact sequences of dg-categories. 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 co-Cartesian 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 dg-categories. 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 co-Cartesian 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 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 \(\mathbf {Ch}_{\infty }\) the \(\infty \)-category of chain complexes. The \(G\)-equivariant coarse Hochschild homology \({{\,\mathrm{\mathcal {X}HH}\,}}_k^G\) (with k-coefficients) 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
$$\begin{aligned} {{\,\mathrm{\mathcal {X}HH}\,}}_k^G:G\mathbf {BornCoarse}\rightarrow \mathbf {Ch}_\infty \quad \text { and } \quad {{\,\mathrm{\mathcal {X}HC}\,}}_k^G:G\mathbf {BornCoarse}\rightarrow \mathbf {Ch}_\infty \end{aligned}$$
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, u-continuity, 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:
$$\begin{aligned} {{\,\mathrm{\mathcal {X}HH}\,}}_{k}^{G,\otimes }:{{\,\mathrm{N}\,}}(G\mathbf {BornCoarse}^\otimes )\rightarrow \mathbf {Ch}_{\infty }^\otimes \end{aligned}$$
and
$$\begin{aligned} {{\,\mathrm{\mathcal {X}HC}\,}}_{k}^{G,\otimes }:{{\,\mathrm{N}\,}}(G\mathbf {BornCoarse}^\otimes )\rightarrow \mathbf {Ch}_{\infty }^\otimes , \end{aligned}$$
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
$$\begin{aligned} {{\,\mathrm{\mathcal {X}\mathrm {Mix}}\,}}_{k}^{G,\otimes }:{{\,\mathrm{N}\,}}(G\mathbf {BornCoarse}^\otimes )\rightarrow \mathbf {Mix}_{\infty }^\otimes . \end{aligned}$$
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 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
$$\begin{aligned} C^{{{\,\mathrm{HH}\,}}}_{*}(A;k)\quad \text { and } \quad C^{{{\,\mathrm{HC}\,}}}_{*}(A;k) \end{aligned}$$
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 one-point bornological coarse space, endowed with a trivial G-action.
Proposition 3.14
There are equivalences of chain complexes
$$\begin{aligned} \mathcal {X}\mathrm {HH}_{k}(*)\simeq C_{*}^{\mathrm {HH}}(k;k)\quad \text { and } \quad \mathcal {X}\mathrm {HC}_{k}(*)\simeq C_{*}^{\mathrm {HC}}(k;k) \end{aligned}$$
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 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 \(\mathbf {Vect}_k^{\mathrm {f.d.}}\) of finite-dimensional k-vector 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 G-bornological 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:
$$\begin{aligned} {{\,\mathrm{\mathcal {X}HH}\,}}_k^G(G_{\mathrm {can,min}})\simeq C_{*}^{{{\,\mathrm{HH}\,}}}(k[G];k) \end{aligned}$$
and
$$\begin{aligned} {{\,\mathrm{\mathcal {X}HC}\,}}_k^G(G_{\mathrm {can,min}})\simeq C_{*}^{{{\,\mathrm{HC}\,}}}(k[G];k) \end{aligned}$$
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 finite-dimensional k-vector 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 dg-categories). Therefore, the result follows from the chain of equivalences of mixed complexes
$$\begin{aligned} \mathrm {Mix}(V_k^G(G))\simeq \mathrm {Mix}(\mathbf {Mod}^{\mathrm {fg,free}}(k[G]))\simeq \mathrm {Mix}(\mathbf {Mod}^{\mathrm {fg,proj}}(k[G]))\simeq \mathrm {Mix}(k [G]), \end{aligned}$$
where the last equivalence is again true by Theorem 2.16. \(\square \)
Let X be a G-set and let \(X_{\mathrm {min,max}}\) denote the G-bornological 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:
$$\begin{aligned} {{\,\mathrm{\mathcal {X}HH}\,}}_{k}^G((G/ H)_{\mathrm {min,max}}\otimes G_{\mathrm {can,min}})\simeq C_{*}^{{{\,\mathrm{HH}\,}}}(k[H];k); \end{aligned}$$
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 forget-control 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 forget-control map \(\beta \)
[4, Definition 11.10] and of the coarse assembly map \(\alpha \)
[4, Definition 10.24]. By
[4, Theorem 11.16], the forget-control map for a G-equivariant coarse homology theory E can be compared with the classical assembly map for the associated G-equivariant 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 spectra-valued 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 forget-control 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 G-homology theory \(\mathbf {HH}_k^G\).
Furthermore, the assembly map \(\alpha _{E_{\mathbf {Fin}}G,G_{\mathrm {can,min}}}\) for the G-homology theory \(\mathbf {HH}_k^G\) (hence, the forget-control 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].