$k_{h}\coloneqq\frac{1}{\sqrt{|\tilde{s}^{-1}(\\{h\\})|}}\sum_{\tilde{\ell}\in\tilde{s}^{-1}(\\{h\\})}e_{\tilde{\ell}}\
.$

Note that the fibres of $\tilde{s}$ are non-empty so that the prefactor is
well-defined. We first observe that

$p_{h}\coloneqq\frac{1}{|\tilde{s}^{-1}(\\{h\\})|}\sum_{\tilde{\ell},\tilde{\ell}^{\prime}\in\tilde{s}^{-1}(\\{h\\})}e_{\tilde{\ell}}e_{\tilde{\ell}^{\prime}}^{*}$

is a projection for every $h$ in $H$, and that the family $(p_{h})_{h\in H}$
is mutually orthogonal. We calculate for every finite subset $H^{\prime}$ of
$H$ that

$\displaystyle\sum_{h\in H^{\prime}}k_{h}k_{h}^{*}$ $\displaystyle=$
$\displaystyle\sum_{h\in
H^{\prime}}\frac{1}{|\tilde{s}^{-1}(\\{h\\})|}\sum_{\tilde{\ell},\tilde{\ell}^{\prime}\in\tilde{s}^{-1}(\\{h\\})}e_{\tilde{\ell}}e_{\tilde{\ell}^{\prime}}^{*}$
$\displaystyle=$ $\displaystyle\sum_{h\in H^{\prime}}p_{h}\,,$

which implies that

$\|\sum_{h\in H^{\prime}}k_{h}k_{h}^{*}\|\leq 1$

for every finite subset $H^{\prime}$ of $H$. This verifies the assumption of
[BEa, Cor. 4.1]. A similar calculation shows that $U^{*}U={\mathtt{id}}_{C}$.
Using the explicit formulas one furthermore checks that for every $g$ in $G$
we have $gU\rho_{g}=\rho^{\prime}_{g}U$. By construction, the morphism $U$ is
$V$-controlled, where $V$ is the entourage from (6.2).

The isometry $U$ exhibits $(C,\rho,\mu)$ as a subobject of
$f_{*}(C^{\prime},\rho^{\prime},\mu^{\prime})$. ∎

######  Example 6.10.

The map $G_{max,max}\to*$ is a weak coarse equivalence. If $G$ is infinite we
have $\mathrm{LF}(G_{max,max})=\emptyset$, but $\mathrm{LF}(*)$ is not empty.
Then in this case
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}^{G}_{c}(G_{max,max})\simeq 0$, but in
general
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}^{G}_{c}(*)\simeq{\mathrm{Hg}}({\mathbf{K}}^{u,G})\not\simeq
0$, where ${\mathbf{K}}^{u,G}$ denotes that category of $G$-objects in the
category of unital objects of $\tilde{\mathbf{K}}$. This shows the relevance
of Condition 6.9.2. ∎

In order to check the Condition 6.9.2 the following lemma is sometimes
helpful.

Let $f\colon X\to Y$ be a morphism in $G\mathbf{BornCoarse}$.

######  Lemma 6.11.

Assume:

  1. 1.

$f$ is a weak coarse equivalence.

  2. 2.

There exists an entourage $U$ of $Y$ such that for every point $y$ in $Y$
there is $x$ in $X$ with:

    1. (a)

$f(x)\in U[\\{y\\}]$.

    2. (b)

$G_{x}\subseteq G_{y}$, where $G_{x}$ and $G_{y}$ denote the stabilizers of
$x$ and $y$.

    3. (c)

$|G_{y}/G_{x}|<\infty$.

Then $\mathrm{LF}(f)$ is almost surjective.

###### Proof.

After enlarging $U$ if necessary can assume that $U$ is $G$-invariant. Let $H$
be in $\mathrm{LF}(Y)$. We fix a representative $h$ for every $[h]$ in $H/G$.
Then we can find $\ell_{h}$ in $X$ such that $(f(\ell_{h}),h)\in U$,
$G_{\ell_{h}}\subseteq G_{h}$ and $|G_{h}/G_{\ell_{h}}|<\infty$. We set
$L\coloneqq\bigcup_{[h]\in H/G}G\ell_{h}$ and define the $G$-invariant subset
$s\coloneqq\bigcup_{[h]\in H/G}G(\ell_{h},h)$ of $L\times H$. Then $s$ is a
$G$-equivariant surjective multivalued map and
$(f\times{\mathtt{id}}_{Y})(s)\subseteq U$.

We have $\tilde{L}\cong\bigsqcup_{[h]\in H/G}G\ell_{h}$. For every $[h]$ in
$H/G$ we have $|\tilde{s}^{-1}(h)|=|G_{h}/G_{\ell_{h}}|<\infty$. Thus by
$G$-invariance all fibres of $\tilde{s}$ are finite.

Let $\ell$ be in $L$. If $g\ell_{h}\in q^{-1}(\ell)$, then $gh\in
U^{-1}[f(\\{\ell\\})]$. Since $H$ is locally finite and
$U^{-1}[f(\\{\ell\\})]$ is a bounded subset of $Y$ the intersection $H\cap
U^{-1}[f(\\{\ell\\})]$ is finite. Since
$q^{-1}(\ell)\subseteq\tilde{s}^{-1}(H\cap U^{-1}[f(\\{\ell\\})])$ and the
fibres of $\tilde{s}$ are finite we conclude that the fibres of $q$ are
finite, too. Hence $s$ is proper.

It remains to show that $L$ is locally finite. Let $B$ be a bounded subset of
$X$. Since $f$ is a coarse equivalence, it is bornological and $U^{-1}[f(B)]$
is bounded in $Y$. Since $H$ is locally finite, $H\cap U^{-1}[f(B)]$ is
finite. Since $\tilde{s}$ has finite fibres and $\tilde{s}(q^{-1}(L\cap
B))\subseteq H\cap U^{-1}[f(B)]$, we conclude that $L\cap B$ is finite. ∎

######  Example 6.12.

Let $X$ be in $G\mathbf{BornCoarse}$, and let $U$ be an invariant coarse
entourage. Then $P_{U}(X)$ is defined as the $G$-simplicial set of finitely
supported probability measures on $X$ with $U$-bounded support. We consider
$P_{U}(X)$ as a $G$-bornological coarse space with the coarse structure
induced by the path quasi-metric coming from the spherical path metric on the
simplices, and with the bornology generated by the subcomplexes $P_{U}(B)$ for
the bounded subsets $B$ of $X$.

We consider the canonical inclusion $f\colon X\to P_{U}(X)$ which sends $x$ in
$X$ to the $\delta$-measure at $x$. This inclusion satisfies the assumptions
of Lemma 6.11.

Let us make Proposition 6.9 explicit in this case for the functor
$\mathrm{Hg}=\mathrm{K^{C^{*}Cat}}$. Note that $\mathrm{K^{C^{*}Cat}}$ is a
Morita invariant, finitary homological functor by [BEa, Thms. 12.4 & 14.14].
We assume that $\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}}$ is in
${\mathbf{Fun}}(BG,\mathbf{P}(C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\mathrm{Idem},\infty\mathrm{add}})$.
Then

$\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}(X)\to\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}(P_{U}(X))$
(6.4)

is an equivalence for every invariant entourage $U$ of $X$.

This can be applied in order to define the equivariant coarse assembly map for
$X$ by the naive equivariant generalization of [BEb, Defn. 9.7] by

$\displaystyle\mu_{X}\colon\operatorname*{\mathtt{colim}}_{U\in{\mathcal{C}}^{G}}\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}({\mathcal{O}}^{\infty}(P_{U}(X)))$
$\displaystyle\xrightarrow{\partial}\operatorname*{\mathtt{colim}}_{U\in{\mathcal{C}}^{G}}{\Sigma}\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}(P_{U}(X))$
$\displaystyle\stackrel{{\scriptstyle!}}{{\simeq}}\Sigma\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}(X)\,,$

where the equivalences (6.4) are used in order to get the marked equivalence,
and $\partial$ is the boundary map of the cone sequence [BEKW20b, Cor. 9.30].

By Proposition 6.4 the equivariant coarse homology theory
$\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}$ is strong. Using
[BEKW20b, Sec. 11.3] we see that

$X\mapsto\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}{\mathcal{O}}^{\infty}\mathbf{P}(X)\coloneqq\operatorname*{\mathtt{colim}}_{U\in{\mathcal{C}}^{G}}\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}({\mathcal{O}}^{\infty}(P_{U}(X)))$

gives rise to an equivariant coarse homology theory, and that the assembly map
refines to a natural transformation between equivariant coarse homology
theories

$\mu\colon\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}{\mathcal{O}}^{\infty}\mathbf{P}\to\Sigma\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}\,.$

In particular we get the analogue of [BEKW20b, Cor. 11.26] without having to
twist by a bornological coarse space with a free $G$-action (denoted by $Q$ in
the reference). ∎

######  Example 6.13.

In [BE20, Sec. 8.6] the coarse $K$-homology
$K\\!\mathcal{X}\colon\mathbf{BornCoarse}\to\mathbf{Sp}$ was defined as the
composition $\mathrm{K^{C^{*}Cat}}\circ{\mathbf{C}}^{*}$, where the functor
${\mathbf{C}}^{*}\colon\mathbf{BornCoarse}\to C^{*}\mathbf{Cat}$ sends $X$ in
$\mathbf{BornCoarse}$ to the Roe category ${\mathbf{C}}^{*}(X)$ as introduced
in [BE20, Defn. 8.74]. In this example we explain how this construction fits
into the framework of the present paper. We work in the case of the trivial
group and therefore in the following drop all the notation referring to the
group action.

We choose the datum ${\mathbf{K}}\subseteq{\mathbf{C}}$ as follows:

  * •

${\mathbf{C}}$ is the $C^{*}$-category of all very small Hilbert spaces and
bounded, linear operators.

  * •

${\mathbf{K}}$ is the ideal of compact operators.

We will show that have an equivalence of coarse homology theories

$K\\!\mathcal{X}\simeq\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}\colon
G\mathbf{BornCoarse}\to\mathbf{Sp}\,.$

Note that ${\mathbf{C}}$ is unital, idempotent complete and admits all very
small sums, and ${\mathbf{K}}^{u}$ is the full subcategory of finite-
dimensional Hilbert spaces. The assertion above follows from the claim that we
have an equality of functors

$\mathbf{\bar{C}}_{\mathrm{lf}}^{\mathrm{ctr}}={\mathbf{C}}^{*}\colon\mathbf{BornCoarse}\to
C^{*}\mathbf{Cat}\,.$

We first compare the objects. In both cases an object $(C,\mu)$ on $X$ in
$\mathbf{BornCoarse}$ consists of a Hilbert space $C$ together with a
projection valued, finitely additive measure $\mu$. For both sides we require
that $C$ is the orthogonal sum of the images of the family
$(\mu(\\{x\\}))_{x\in X}$. For ${\mathbf{C}}^{*}(X)$ we in addition require
that $\mu(B)H$ is finite-dimensional for every bounded subset $B$ of $X$. This
condition is equivalent to the condition that $\mu(B)\in{\mathbf{K}}^{u}$
required for $\mathbf{\bar{C}}_{\mathrm{lf}}^{\mathrm{ctr}}(X)$. For
$\mathbf{\bar{C}}_{\mathrm{lf}}^{\mathrm{ctr}}(X)$ we in addition require that
${\mathtt{supp}}(C,\mu)$ is locally finite. Since $\mu(B)$ is finite-
dimensional we have that ${\mathtt{supp}}(C,\mu)\cap B$ is finite for any
bounded subset $B$ of $X$.

In both cases the morphism spaces are defined as the closures of the spaces of
bounded operators of controlled propagation.

Finally, the functoriality for morphisms in $\mathbf{BornCoarse}$ is defined
the same way for $\mathbf{\bar{C}}_{\mathrm{lf}}^{\mathrm{ctr}}(X)$ and
${\mathbf{C}}^{*}(X)$. ∎

##  7 The coarse homology theories $\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}$
and $\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{c,G}$

In this section we introduce two further coarse homology theories
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}$ and
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{c,G}$. The analogue of
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{c,G}$ for left-exact
$\infty$-categories instead of $C^{*}$-categories was considered in [BCKW,
Cor. 4.18]. It is an important ingredient in the proof of an isomorphism
result for an assembly map [BKWa].

From (4.5) and (4.6) in the case of the trivial group we obtain functors

$\mathbf{BornCoarse}\times\mathbf{P}(C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\mathrm{Idem}}\to
C^{*}\mathbf{Cat}\,,\quad(X,{\mathbf{K}}\subseteq{\mathbf{C}})\mapsto\mathbf{\bar{C}}^{\mathrm{ctr}}_{\mathrm{sm}}(X)$
(7.1)

and

$\mathbf{BornCoarse}\times\mathbf{P}(C^{*}\mathbf{Cat}^{\mathrm{nu}})_{i}^{\mathrm{Idem}}\to
C^{*}\mathbf{Cat}\,,\quad(X,{\mathbf{K}}\subseteq{\mathbf{C}})\mapsto\mathbf{\bar{C}}_{\mathrm{lf}}^{\mathrm{ctr}}(X)$
(7.2)

(see Definition 4.1). Below we will further restrict to $\omega$-additive
$C^{*}$-categories. Note that in the case of the trivial group we omit the
superscript $G$ from the notation.

For a group $G$ we have a functor

$G\mathbf{BornCoarse}\to{\mathbf{Fun}}(BG,\mathbf{BornCoarse})\,,\quad
X\mapsto\tilde{X}$

which sends a $G$-bornological coarse space to the underlying bornological
coarse space with the $G$-action by automorphisms. This functor just forgets
the condition that ${\mathcal{C}}^{G}_{X}$ must be cofinal in
${\mathcal{C}}_{X}$ (see Example 2.5). We now fix
$\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}}$ in
${\mathbf{Fun}}(BG,\mathbf{P}(C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\mathrm{Idem}})$.
By composition we get functors

$\displaystyle\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}\colon
G\mathbf{BornCoarse}$
$\displaystyle\xrightarrow{X\mapsto(\tilde{X},\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}})}{\mathbf{Fun}}(BG,\mathbf{BornCoarse}\times\mathbf{P}(C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\mathrm{Idem}})$
$\displaystyle\xrightarrow[\hphantom{X\mapsto(\tilde{X},\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}})}]{\eqref{adsvoihqjoi3rffsdadfc}}{\mathbf{Fun}}(BG,C^{*}\mathbf{Cat})$
(7.3)

and

$\displaystyle\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}\colon
G\mathbf{BornCoarse}$
$\displaystyle\xrightarrow{X\mapsto(\tilde{X},\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}})}{\mathbf{Fun}}(BG,\mathbf{BornCoarse}\times\mathbf{P}(C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\mathrm{Idem}}_{i})$
$\displaystyle\xrightarrow[\hphantom{X\mapsto(\tilde{X},\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}})}]{\eqref{adsvoihqjoi3rffsdadfcccc}}{\mathbf{Fun}}(BG,C^{*}\mathbf{Cat})\,.$
(7.4)

We now use the crossed product functor

$-\rtimes G\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})\to
C^{*}\mathbf{Cat}^{\mathrm{nu}}$

from [Bun, Defn. 5.9] and its restriction to unital $C^{*}$-categories.

######  Definition 7.1.

We define the functors

$\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}\coloneqq\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(-)\rtimes
G\colon G\mathbf{BornCoarse}\to C^{*}\mathbf{Cat}$

and

$\mathbf{\bar{C}}_{\mathrm{lf},G}^{\mathrm{ctr}}\coloneqq\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(-)\rtimes
G\colon G\mathbf{BornCoarse}\to C^{*}\mathbf{Cat}\,.$

Note that the crossed product $\rtimes$ is the maximal crossed product. It
should not be confused with the reduced crossed product [BEa, Sec. 10] which
appears in Proposition 8.2.4. We use the maximal crossed product since it has
the necessary exactness properties which are needed to verify the axioms for a
coarse homology theory below.

######  Lemma 7.2.

$\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}$ and
$\mathbf{\bar{C}}_{\mathrm{lf},G}^{\mathrm{ctr}}$ send weak coarse
equivalences (see Definition 2.18.2) to unitary equivalences.

###### Proof.

We give the argument in the case of
$\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}$. The case of
$\mathbf{\bar{C}}_{\mathrm{lf},G}^{\mathrm{ctr}}$ is analoguous.

If $X\to X^{\prime}$ is a weak coarse equivalence, then by the Lemma 5.2 the
morphism
$\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(X)\to\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(X^{\prime})$
in ${\mathbf{Fun}}(BG,C^{*}\mathbf{Cat})$ is a weak equivalence in the sense
of [Bun, Defn. 7.6] (i.e., a unitary equivalence after forgetting the
$G$-action). Furthermore, by [Bun, Prop. 7.9] the functor $-\rtimes G$ sends
weak equivalences to unitary equivalences. Consequently,
$\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}(X)\to\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}(X^{\prime})$
is a unitary equivalence. ∎

######  Lemma 7.3.

The functors $\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}$ and
$\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}$ are $u$-continuous.

###### Proof.

We give the argument in the case of
$\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}$. The case of
$\mathbf{\bar{C}}_{\mathrm{lf},G}^{\mathrm{ctr}}$ is analoguous.

First of all note that colimits in functor categories are calculated
objectwise. Let $X$ be in $G\mathbf{BornCoarse}$. By Lemma 5.3 and since
${\mathcal{C}}_{X}^{G}$ is cofinal in ${\mathcal{C}}_{X}$ (Definition 2.1.4)
we get

$\operatorname*{\mathtt{colim}}_{U\in{\mathcal{C}}_{X}^{G}}\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(X_{U})\cong\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(X)$
(7.5)

in ${\mathbf{Fun}}(BG,C^{*}\mathbf{Cat})$. Applying $-\rtimes G$ gives

$(\operatorname*{\mathtt{colim}}_{U\in{\mathcal{C}}_{X}^{G}}\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(X_{U}))\rtimes
G\cong\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(X)\rtimes G\,.$

It remains to show that the canonical functor

$\operatorname*{\mathtt{colim}}_{U\in{\mathcal{C}}_{X}^{G}}(\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(X_{U})\rtimes
G)\to(\operatorname*{\mathtt{colim}}_{U\in{\mathcal{C}}_{X}^{G}}\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(X_{U}))\rtimes
G$ (7.6)

is an isomorphism.

We use [Bun, Prop. 7.3] which provides an isomorphism of functors

$-\rtimes G\cong\operatorname*{\mathtt{colim}}_{BG}\circ
L\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Cat})\to C^{*}\mathbf{Cat}\,,$

where $L$ is the functor from [Bun, Eq. (7.1)]. The colimit in (7.5) can be
explicitly calculated as described in [Bun, Lem. 4.6]. From this description
and the explicit description of $L$ after [Bun, Eq. (7.1)] it is obvious that

$\operatorname*{\mathtt{colim}}_{U\in{\mathcal{C}}_{X}^{G}}L(\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(X_{U}))\to
L(\operatorname*{\mathtt{colim}}_{U\in{\mathcal{C}}_{X}^{G}}\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(X_{U}))$

is an isomorphism. Since also the order of taking the colimits over $BG$ and
${\mathcal{C}}_{X}^{G}$ can be interchanged we conclude that (7.6) is an
isomorphism. ∎

Let $X$ be in $G\mathbf{BornCoarse}$, and let $({\mathcal{Y}},Z)$ be a
complementary pair.

######  Lemma 7.4.

We have diagrams

$\textstyle{\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}({\mathcal{Y}}\cap
Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(({\mathcal{Y}}\cap
Z)\subseteq Z)\rtimes
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}({\mathcal{Y}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}({\mathcal{Y}}\subseteq
X)\rtimes
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}(X)}$
(7.7)

and

$\textstyle{\mathbf{\bar{C}}_{\mathrm{lf},G}^{\mathrm{ctr}}({\mathcal{Y}}\cap
Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(({\mathcal{Y}}\cap
Z)\subseteq Z)\rtimes
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\bar{C}}_{\mathrm{lf},G}^{\mathrm{ctr}}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\bar{C}}_{\mathrm{lf},G}^{\mathrm{ctr}}({\mathcal{Y}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}({\mathcal{Y}}\subseteq
X)\rtimes
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\bar{C}}_{\mathrm{lf},G}^{\mathrm{ctr}}(X)}$
(7.8)

where the left horizontal maps are filtered colimits of unitary equivalences
and the right squares are excisive.

###### Proof.

We give the argument in the case of
$\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}$. The case of
$\mathbf{\bar{C}}_{\mathrm{lf},G}^{\mathrm{ctr}}$ is analoguous.

Let ${\mathcal{Y}}=(Y_{i})_{i\in I}$. For every $i$ in $I$ we have a diagram

$\textstyle{\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(Y_{i}\cap
Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}((Y_{i}\cap
Z)\subseteq
Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(Y_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(Y_{i}\subseteq
X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(X)}$

in ${\mathbf{Fun}}(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$. The left horizontal
morphisms are weak equivalences by Lemma 5.6. We apply the functor $-\rtimes
G$ and get

$\textstyle{\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(Y_{i}\cap
Z)\rtimes
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}((Y_{i}\cap
Z)\subseteq Z)\rtimes
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(Z)\rtimes
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(Y_{i})\rtimes
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(Y_{i}\subseteq
X)\rtimes
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(X)\rtimes
G}$

in $C^{*}\mathbf{Cat}^{\mathrm{nu}}$. By [Bun, Prop. 7.9] the left horizontal
functors are unitary equivalences. We now form the colimit over $i$ in $I$ and
use the notation $\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}$ at the
outer corners. Note that the evaluation of a functor on a big family is by
definition the colimit of its values on the members. We furthermore use that
$I$ is filtered in order to remove the colimit symbol for the constant
$I$-diagrams in the right part. We obtain the diagram

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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
224.08383pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
243.66063pt\raise-31.5022pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}({\mathcal{Y}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
64.50037pt\raise-42.0022pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
64.50037pt\raise-42.0022pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\operatorname*{\mathtt{colim}}_{i\in
I}\big{(}\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(Y_{i}\subseteq
X)\rtimes
G\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
223.31996pt\raise-42.0022pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
223.31996pt\raise-42.0022pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}(X)}$}}}}}}}\ignorespaces}}}}\ignorespaces\
.$

In the next step we interchange the order of the colimits and the functor
$-\rtimes G$ in the middle column. The argument is similar as in the proof of
the Lemma 7.3. But in this case the colimit is interpreted in
$C^{*}\mathbf{Cat}^{\mathrm{nu}}$ since the structure maps are not unital. We
use [Bun, Prop. 7.3] which provides an isomorphism of functors

$-\rtimes G\cong\operatorname*{\mathtt{colim}}_{BG}\circ
L\colon{\mathbf{Fun}}(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})\to
C^{*}\mathbf{Cat}^{\mathrm{nu}}\,,$

where $L$ is defined below [Bun, Eq. (7.1)]. The colimit
$\operatorname*{\mathtt{colim}}_{i\in
I}\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(Y_{i}\subseteq X)$ can
be calculated as in [Bun, Lem. 4.6] as a subcategory of
$\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(X)$. It is again clear
that the functor $L$ commutes with such a kind of colimit. Furthermore we can
switch the order of colimits over $BG$ and $I$. We thus get the Diagram (7.7).

It remains to show that the right square is excisive. To this end we note that
the right square is obtained from the square

$\textstyle{\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(({\mathcal{Y}}\cap
Z)\subseteq
Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\
\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}({\mathcal{Y}}\subseteq
X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(X)}$
(7.9)

in ${\mathbf{Fun}}(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$ by applying $-\rtimes
G$. The square (7.9) is excisive after forgetting the $G$-action by the Lemma
5.10. We finally use that $-\rtimes G$ preserves excisive squares by [Bun,
Thm. 8.13] whose right corners are unital. ∎

######  Lemma 7.5.

If ${\mathbf{C}}$ is countably additive, then the functors
$\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}$ and
$\mathbf{\bar{C}}_{\mathrm{lf},G}^{\mathrm{ctr}}$ send flasque
$G$-bornological coarse spaces to flasque $C^{*}$-categories.

###### Proof.

We give the argument in the case of
$\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}$. The case of
$\mathbf{\bar{C}}_{\mathrm{lf},G}^{\mathrm{ctr}}$ is analoguous.

We argue as in the proof of Lemma 5.4. Let $X$ be in $G\mathbf{BornCoarse}$
flasque with flasqueness implemented by $f\colon X\to X$. We construct the
endofunctor
$S\colon\mathbf{\bar{C}}^{\mathrm{ctr}}_{\mathrm{sm}}(X)\to\mathbf{\bar{C}}^{\mathrm{ctr}}_{\mathrm{sm}}(X)$
(not equivariant) as in the proof of Lemma 5.4.

Using [BEa, Lem. 9.9] we extend the endofunctor $\bigoplus_{{\rm
Nature}}{\mathtt{id}}_{{\mathbf{C}}}$ to a weakly equivariant functor
$(\bigoplus_{{\rm Nature}}{\mathtt{id}}_{{\mathbf{C}}},\theta)$ (see also
[BEa, Def. 9.6] or [Bun, Def. 7.10]). The same cocycle then provides an
extension $(S,\theta)$ of $S$ to a weakly equivariant endofunctor. By [Bun,
Prop. 7.12] we get an endofunctor

$(S,\theta)\rtimes
G\colon\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}(X)\to\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}(X)\,.$

The Equality (5.2) induces an equality (adopting appropriate choices for the
sum)

${\mathtt{id}}_{\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}(X)}\oplus\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}(f)\circ(S\rtimes
G)=S\rtimes G\,.$ (7.10)

By the Lemma 7.2 and since $f$ is close to the identity we have a unitary
isomorphism $\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}(f)\circ(S\rtimes
G)\simeq S\rtimes G$. Consequently, we have a unitary isomorphism

${\mathtt{id}}_{\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}(X)}\oplus(S\rtimes
G)\cong S\rtimes G$

showing that $\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}(X)$ is flasque.
∎

Let $\mathrm{Hg}\colon C^{*}\mathbf{Cat}^{\mathrm{nu}}\to\mathbf{M}$ be a
functor.

######  Definition 7.6.

We define the functors

$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}\colon
G\mathbf{BornCoarse}\xrightarrow{\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}}C^{*}\mathbf{Cat}\xrightarrow{\mathrm{Hg}}\mathbf{M}$

and

$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{c,G}\colon
G\mathbf{BornCoarse}\xrightarrow{\mathbf{\bar{C}}_{\mathrm{lf},G}^{\mathrm{ctr}}}C^{*}\mathbf{Cat}\xrightarrow{\mathrm{Hg}}\mathbf{M}\,.$

The subscript $G$ should indicate that the functors involve taking
$G$-coinvariants which is here explicitly realized in terms of the crossed
product. If we want to indicate the dependence on the choice of the ideal
$\tilde{\mathbf{K}}$ we will use the notation
$\mathrm{Hg}{}_{{\mathbf{K}}}{\mathbf{C}}{\mathcal{X}}_{G}$ or
$\mathrm{Hg}{}_{{\mathbf{K}}}{\mathbf{C}}{\mathcal{X}}_{c,G}$. The subscript
$c$ indicates that the construction involves the continuous functor
$\mathbf{\bar{C}}_{\mathrm{lf}}^{\mathrm{ctr}}$. But note that the functors
defined in Definition 7.6 are not continuous except in degenerate cases
(Example 7.9).

######  Theorem 7.7.

Assume:

  1. 1.

$\mathrm{Hg}$ is a finitary homological functor [BEa, Def. 11.5 & 11.6]

  2. 2.

$\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}}$ is in
${\mathbf{Fun}}(BG,\mathbf{P}(C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\mathrm{Idem},\omega\mathrm{add}})$

Then $\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}$ and
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{c,G}$ are equivariant coarse homology
theories (Definition 2.37).

###### Proof.

We give the proof for $\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}$. The case of
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{c,G}$ is analoguous.

Note that $\mathbf{M}$ is stable since it is the target of a homological
functor. We now verify the conditions listed in Definition 2.37.

  1. 1.

$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}$ is coarsely invariant: Indeed, by
Lemma 7.2 the functor $\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}$ sends
weak coarse equivalences to unitary equivalences. Further, $\mathrm{Hg}$ sends
unitary equivalences to equivalences. So
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}$ sends (even weak) coarse
equivalences to equivalences.

  2. 2.

$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}$ is $u$-continuous: This follows
from Lemma 7.3 stating that $\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}$
is $u$-continuous and the fact that $\mathrm{Hg}$ preserves filtered colimits,
since we assume that it is finitary.

  3. 3.

$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}$ is excisive: We clearly have
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}(\emptyset)\simeq 0_{\mathbf{M}}$.

Let now $X$ be in $G\mathbf{BornCoarse}$, and let $({\mathcal{Y}},Z)$ be an
invariant complementary pair on $X$. We apply $\mathrm{Hg}$ to the Diagram
(7.7) and get

$\textstyle{\mathrm{Hg}\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}({\mathcal{Y}}\cap
Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Hg}(\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}(({\mathcal{Y}}\cap
Z)\subseteq Z)\rtimes
G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Hg}\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Hg}\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}({\mathcal{Y}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Hg}(\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{sm}}({\mathcal{Y}}\subseteq
X)\rtimes
G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Hg}\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}(X)}$

Since $\mathrm{Hg}$ preserves filtered colimits and sends unitary equivalences
to equivalences, the left horizontal maps are filtered colimits of
equivalences and hence equivalences. The right square is the image under
$\mathrm{Hg}$ of an excisive square and hence a cocartesian square.
Consequently, the outer square is cocartesian.

  4. 4.

$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}$ annihilates flasques: Indeed, if
$X$ is flasque, then $\mathbf{\bar{C}}_{\mathrm{sm},G}^{\mathrm{ctr}}(X)$ is
flasque by Lemma 7.5. Then $\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}(X)\simeq
0$ by [BEa, Prop. 11.12].∎

In Point 1 of the proof of Theorem 7.7 we have actually shown the following
additional property of these two homology theories:

######  Proposition 7.8.

The functors $\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}$ and
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{c,G}$ send weak coarse equivalences
(Definition 2.18.2) to equivalences.

This should be compared with Proposition 6.9 stating that the functor
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}^{G}_{c}$ sends weak coarse equivalences
to equivalences only under additional assumptions.

######  Example 7.9.

Let $X$ be in $G\mathbf{BornCoarse}$. We assume that there exists a maximal
entourage $U$ in ${\mathcal{C}}_{X}$, and that all coarse components of $X$
(Definition 9.4) are bounded, i.e., the equivalence classes for the
equivalence relation $U$ belong to ${\mathcal{B}}_{X}$. Then the canonical
projection $X\to\pi_{0}(X)_{min,min}$ is a weak coarse equivalence, where
$\pi_{0}(X)$ denotes the $G$-set of coarse components of $X$. In this case

$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{c,G}(X)\to\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{c,G}(\pi_{0}(X)_{min,min})$

is an equivalence.

A concrete example of such a space is $G_{max,max}$. We have
$\pi_{0}(G_{max,max})\cong*$. Hence
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{c,G}(G_{max,max})\to\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{c,G}(*)$
is an equivalence. Note that the target of this morphism is given by
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{c,G}(*)\cong\mathrm{Hg}(\tilde{{\mathbf{K}}}^{u}\rtimes
G)$, which is in general non-trivial. If $G$ is infinite, then $G_{max,max}$
does not have any invariant, locally finite subsets. This shows that the
coarse homology theory $\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{c,G}$ is in
general not continuous. See also Example 6.10. ∎

######  Proposition 7.10.

The coarse homology theories $\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}$ and
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{c,G}$ are strong (see Definition 2.40).

###### Proof.

We give the argument in the case of
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}$. The case of
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{c,G}$ is analoguous.

Let $X$ in $G\mathbf{BornCoarse}$ be weakly flasque (Definition 2.39) with
weak flasqueness implemented by $f\colon X\to X$. We apply $\mathrm{Hg}$ to
the Equation (7.10) obtained in the proof of Lemma 7.5. Using the additivity
of $\mathrm{Hg}$ shown in [BEa, Prop. 11.10] we get

${\mathtt{id}}_{\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}(X)}+\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}(X)(f)\circ\mathrm{Hg}(S\rtimes
G)=\mathrm{Hg}(S\rtimes G)$

in ${\mathtt{End}}_{\mathbf{M}}(\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}(X))$.
From Condition 2.39.1 we get
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}(X)(f)\simeq{\mathtt{id}}_{\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}(X)}$,
and the resulting relation
${\mathtt{id}}_{\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}(X)}+\mathrm{Hg}(S\rtimes
G)=\mathrm{Hg}(S\rtimes G)$ implies that
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{G}(X)\simeq 0$. ∎

##  8 Calculations

In this section we calculate the values of some of the functors introduced in
the preceding sections on bornological coarse spaces derived from $G$-sets.
The calculations for $\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}^{G}_{c}$ are of
particular relevance for the results stated in detail in Section 12.

We fix $\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}}$ in
${\mathbf{Fun}}(BG,\mathbf{P}(C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\mathrm{Idem}})$,
see Definition 4.1 for notation. Applying the functor from (7.4) to $X$ in
$G\mathbf{BornCoarse}$ we obtain the object
$\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X)$ in
${\mathbf{Fun}}(BG,C^{*}\mathbf{Cat})$. Instead of the reduced crossed product
as in Definition 7.1 we can form, as a start, the algebraic crossed product
$\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X)\rtimes^{\mathrm{alg}}G$
in $\mathbf{{}^{*}{\mathbf{Cat}}_{{\mathbb{C}}}}$, see [Bun, Defn. 5.1]. On
the other hand we have the uncompleted Roe category
$\mathbf{C}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X\otimes G_{can,min})$ in
$\mathbf{{}^{*}{\mathbf{Cat}}_{{\mathbb{C}}}}$, introduced in Definition
(4.4), where $G_{can,min}$ is defined in Example 2.13 and the symmetric
monoidal structure $\otimes$ is explained in Definition 2.15.

######  Proposition 8.1.

Assume:

  1. 1.

Every locally finite subset of $X$ has a maximal entourage.

  2. 2.

${\mathbf{C}}$ is $|G|$-additive.

Then there exists a unitary equivalence

$\phi\colon\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X)\rtimes^{\mathrm{alg}}G\to\mathbf{C}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X\otimes
G_{can,min})$

in $\mathbf{{}^{*}{\mathbf{Cat}}_{{\mathbb{C}}}}$ which is natural in $X$.

###### Proof.

We fix for every object $C$ in ${\mathbf{C}}$ an orthogonal sum
$(\bigoplus_{g\in G}gC,(e^{C}_{g})_{g\in G})$. For any $X$ in
$G\mathbf{BornCoarse}$ satisfying Condition 1 we will then construct a functor

$\phi\colon\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X)\rtimes^{\mathrm{alg}}G\to\mathbf{C}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X\otimes
G_{can,min})$

in $\mathbf{{}^{*}{\mathbf{Cat}}_{{\mathbb{C}}}}$ which turns out to be
natural in $X$. To show that it is a unitary equivalence we will furthermore
construct a functor

$\psi\colon\mathbf{C}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X\otimes
G_{can,min})\to\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X)\rtimes^{\mathrm{alg}}G$

in $\mathbf{{}^{*}{\mathbf{Cat}}_{{\mathbb{C}}}}$ (with no naturality
requirements) and unitary isomorphisms

$u\colon\psi\circ\phi\xrightarrow{\cong}{\mathtt{id}}_{\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X)\rtimes^{\mathrm{alg}}G}\,,\quad
v\colon\phi\circ\psi\xrightarrow{\cong}{\mathtt{id}}_{\mathbf{C}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X\otimes
G_{can,min})}\,.$

Condition 1 implies that every morphism in
$\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X)$ is controlled.
Indeed, if $(C,\rho,\mu)$ and $(C^{\prime},\rho^{\prime},\mu^{\prime})$ are
objects of $\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X)$, then by
Definition 4.7.2

$L\coloneqq{\mathtt{supp}}(C,\rho,\mu)\cup{\mathtt{supp}}(C^{\prime},\rho^{\prime},\mu^{\prime})$

is a locally finite subset of $X$. By Condition 1 we can find an entourage $U$
in ${\mathcal{C}}_{X}$ such that $U\cap(L\times L)$ is maximal in the coarse
structure on $L$ induced from $X$. Then every morphism
$(C,\rho,\mu)\to(C^{\prime},\rho^{\prime},\mu^{\prime})$ in
$\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X)$ is $U$-controlled
(see Example 4.17). In contrast, a morphism in
$\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X)$ for general $X$ in
$G\mathbf{BornCoarse}$ is only in the norm closure of the controlled
morphisms.

We start with the construction of $\phi$.

  1. 1.

objects: Note that the objects of
$\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X)\rtimes^{\mathrm{alg}}G$
are the objects $(C,\mu)$ of
$\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X)$. Using the choice
of the sums adopted at the beginning of this proof we define

$\phi(C,\mu)\coloneqq\big{(}\bigoplus_{g\in G}gC,\rho,\tilde{\mu}\big{)}\,.$

Here the projection-valued measure $\tilde{\mu}$ is determined by

$\tilde{\mu}(\\{(x,g)\\})\coloneqq
e^{C}_{g}(g\mu)(g^{-1}(\\{x\\}))e_{g}^{C,*}$

for all $(x,g)$ in $X\times G$. Furthermore, the unitary cocycle
$\rho\coloneqq(\rho_{h})_{h\in G}$ is defined by
$\rho_{h}\colon\bigoplus_{g\in G}gC\to h\bigoplus_{g\in G}gC$ is given by

$\rho_{h}\coloneqq\sum_{g\in G}he^{C}_{g}e_{hg}^{C,*}\,.$

One checks that $\rho$ satisfies the cocycle conditions, and that
$\tilde{\mu}$ is equivariant (see Definitions 3.1.1 and 3.4.3). One
furthermore checks that $\phi(C,\mu)$ is locally finite.

  2. 2.

morphisms: We will use the pair notation for morphisms as in [Bun, Defn. 5.1].
We consider a morphism $(A,g)\colon(C,\mu)\to(C^{\prime},\mu^{\prime})$ in
$\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X)\rtimes^{\mathrm{alg}}G$
with $A\colon(C,\mu)\to g^{-1}(C^{\prime},\mu^{\prime})$. We then define

$\phi(A,g)\coloneqq\sum_{h\in G}e^{C^{\prime}}_{hg^{-1}}hAe_{h}^{C,*}\,.$

One checks in a straightforward manner that $\phi(A,g)$ is equivariant (see
Definition 3.1.2).

We must check that $\phi(A,g)$ is controlled. We claim that if $A$ is
$U$-controlled for $U$ in ${\mathcal{C}}_{X}$, then $\phi((A,g))$ is
$G(U\times\\{(g^{-1},1)\\})$-controlled. Let $x,x^{\prime}$ be in $X$. Then
$(g^{-1}\mu^{\prime})(g(\\{x^{\prime}\\}))A\mu(\\{x\\})\not=0$ implies that
$(x^{\prime},x)\in U$. Let now $l,l^{\prime}$ be in $G$ and assume that
$0\not=\tilde{\mu}^{\prime}(\\{(x^{\prime},l^{\prime})\\})\phi(A,g)\tilde{\mu}(\\{(x,l)\\})$.
Then we have

$\displaystyle 0$ $\displaystyle\not=$
$\displaystyle\tilde{\mu}^{\prime}(\\{(x^{\prime},l^{\prime})\\})\phi(A,g)\tilde{\mu}(\\{(x,l)\\})$
$\displaystyle=$ $\displaystyle
e^{C^{\prime}}_{l^{\prime}}(l^{\prime}\mu^{\prime})(l^{\prime,-1}\\{x^{\prime}\\})e^{C^{\prime},*}_{l^{\prime}}\big{(}\sum_{h\in
G}e^{C^{\prime}}_{hg^{-1}}hAe^{C,*}_{h}\big{)}e^{C}_{l}(l\mu)(l^{-1}(\\{x\\})e^{C,*}_{l}$
$\displaystyle=$ $\displaystyle
e^{C^{\prime}}_{l^{\prime}}(l^{\prime}\mu^{\prime})(l^{\prime,-1}\\{x^{\prime}\\})(l^{\prime}g)Ae^{C,*}_{l^{\prime}g}e^{C}_{l}(l\mu)(l^{-1}(\\{x\\})e^{C,*}_{l}\,.$

We conclude that $l^{\prime}g=l$ and hence

$0\not=e^{C^{\prime}}_{lg^{-1}}(lg^{-1}\mu^{\prime})((lg^{-1})^{-1}\\{x^{\prime}\\})lA(l\mu)(l^{-1}(\\{x\\})e^{C,*}_{l}\,,$

and therefore also $(l^{-1}x^{\prime},l^{-1}x)\in U$. We finally conclude that

$((x^{\prime},l^{\prime}),(x,l))\in G(U\times\\{(g^{-1},e)\\})\,.$

This finishes the verification of the claim.

We now observe that $G\\{(g^{-1},e)\\}\in{\mathcal{C}}_{G_{can}}$, and
$GU\in{\mathcal{C}}_{X}$ since $X$ is a $G$-bornological coarse space. Since

$G(U\times\\{(g^{-1},e)\\})\subseteq GU\times
G\\{(g^{-1},e)\\}\in{\mathcal{C}}_{X\otimes G_{can,min}}$

we see that $\phi(A,g)$ is controlled.

By an inspection one checks that $\phi$ is a functor in
$\mathbf{{}^{*}{\mathbf{Cat}}_{{\mathbb{C}}}}$ which is natural $X$.

Next we construct $\psi$.

  1. 1.

objects: Let $(C,\rho,\tilde{\mu})$ be an object of
$\mathbf{C}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X\otimes G_{can,min})$. We choose
an image $(\tilde{C},\tilde{u})$ of the projection
${\tilde{\mu}}(X\times\\{e\\})$. Then we define
$\mu\colon\mathcal{P}_{X}\to\mathrm{Proj}(\tilde{C})$ to be the restriction of
${\tilde{u}}^{*}{\tilde{\mu}}{\tilde{u}}$ to $X\times\\{e\\}$. We now set

$\psi(C,\rho,\tilde{\mu})\coloneqq(\tilde{C},\mu)\,.$ (8.1)

One checks that $(\tilde{C},\mu)$ is locally finite.

  2. 2.

morphisms: Let
$A\colon(C,\rho,\tilde{\mu})\to(C^{\prime},\rho^{\prime},{\tilde{\mu}}^{\prime})$
be a morphism in $\mathbf{C}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X\otimes
G_{can,min})$. For every $g$ in $G$ we set

$A_{g}\coloneqq(g^{-1}{\tilde{u}}^{\prime})^{*}\rho^{\prime}_{g^{-1}}A{\tilde{u}}\colon\tilde{C}\to
g^{-1}\tilde{C}^{\prime}\,.$ (8.2)

If $A$ is $U$-controlled for $U$ in ${\mathcal{C}}_{X\otimes G_{can,min}}$,
then $A_{g}$ is $U\cap((X\times X)\times{(e,e)})$-controlled for all $g$ in
$G$. Furthermore, if $A_{g}\not=0$, then $g^{-1}\in\mathrm{pr}_{G\times
G}(U)[\\{e\\}]$. In view of the definition of the coarse structure on
$X\otimes G_{can,min}$ we have $\mathrm{pr}_{G\times
G}(U)\in{\mathcal{C}}_{G_{can,min}}$ and therefore $\mathrm{pr}_{G\times
G}(U)[\\{e\\}]\in{\mathcal{B}}_{G_{can,min}}$, i.e., this set is finite. We
now define

$\psi(A)\coloneqq\sum_{g\in G}(A_{g},g)\,.$ (8.3)

We just have seen that this sum has only finitely many non-trivial terms and
therefore defines a morphism from $(\tilde{C},\mu)$ to
$(\tilde{C}^{\prime},\mu^{\prime})$ in
$\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X)\rtimes^{\mathrm{alg}}G$.
One checks that this construction is compatible with the composition and the
involution.

We now construct the unitary isomorphisms $u$ and $v$.

  1. 1.

$u\colon\psi\circ\phi\xrightarrow{\cong}{\mathtt{id}}$: Let $(C,\mu)$ be an
object in $\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X)$. Then
$\phi(C,\mu)=(\bigoplus_{g\in G}gC,{\rho,\tilde{\mu}})$. In the construction
of $\psi$ we have chosen the image $(\widetilde{\bigoplus_{g\in
G}gC},{\tilde{u}})$ of $\tilde{\mu}(X\times\\{e\\})$ such that

$u_{(C,\mu)}\coloneqq e^{C,*}_{e}{\tilde{u}}\colon\widetilde{\bigoplus_{g\in
G}gC}\to C$

is a unitary isomorphism. The family
$u\coloneqq(u_{(C,\mu)})_{(C,\mu)\in\mathrm{Ob}(\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X))}$
provides the desired unitary isomorphism.

  2. 2.

$v\colon\phi\circ\psi\xrightarrow{\cong}{\mathtt{id}}$: Let
$(C,\rho,\tilde{\mu})$ be an object of
$\mathbf{C}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X\otimes G_{can,min})$. In the
construction of $\psi$ we have fixed an image $(\tilde{C},\tilde{u})$ of
${\tilde{\mu}}(X\times\\{e\\})$ such that
$\psi(C,\rho,\tilde{\mu})=(\tilde{C},\mu)$, see (8.1). Then
$\phi(\tilde{C},\mu)=(\bigoplus_{g\in G}g\tilde{C},\rho,\tilde{\tilde{\mu}})$.

Using the invariance condition for $\tilde{\mu}$ stated in Definition 3.4.3,
one checks that $(g\tilde{C},\rho_{g}^{-1}g(u))$ represents an image of
${\tilde{\mu}}(X\times\\{g\\})$. Since $(C,\rho,\mu)$ is determined on points,
by Lemma 3.13 the object $C$ is the orthogonal sum of the images of the family
of projections $\tilde{\mu}(X\times\\{g\\})$. Therefore, setting

$u_{g}\coloneqq\rho_{g}^{-1}g(u)\,,$ (8.4)

we get a unitary isomorphism

$v_{(C,\rho,\mu)}\colon\bigoplus_{g\in G}g\tilde{C}\xrightarrow{\sum_{g\in
G}u_{g}e^{\tilde{C},*}_{g}}C\,.$ (8.5)

The family
$(v_{(C,\rho,\mu)})_{(C,\rho,\mu)\in\mathrm{Ob}(\mathbf{C}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X\otimes
G_{can,min}))}$ is the desired unitary isomorphism $v$.∎

Let $\tilde{\mathbf{C}}$ in ${\mathbf{Fun}}(BG,C^{*}\mathbf{Cat})$. In [BEa,
Def. 17.7] we have introduced a functor

${\mathbf{C}}[-]\colon
G{\mathbf{Set}}\to{{\mathbf{Fun}}(BG,C^{*}\mathbf{Cat})}\,.$

If ${\mathbf{C}}$ is $|G|$-additive, then ${\mathbf{C}}[\tilde{X}]$ is
$G$-additive for every $\tilde{X}$ in $G{\mathbf{Set}}$, and we can define the
functor

${\mathbf{C}}[-]\rtimes_{r}G\colon G{\mathbf{Set}}\to C^{*}\mathbf{Cat}$

[BEa, (17.10)]. We now consider
$\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}}$ in
${\mathbf{Fun}}(BG,\mathbf{P}(C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\mathrm{Idem}})$.
Since $\tilde{{\mathbf{K}}}$ is an invariant ideal in $\tilde{{\mathbf{C}}}$
we know from [BEa, Rem. 17.11] that ${\mathbf{K}}^{u}$ is hereditarily
additive [BEa, Def. 17.10]. In this situation, in [BEa, Def. 17.12] we defined
the subfunctor

${{}_{{\mathbf{K}}}{\mathbf{C}}_{\mathrm{lf}}[-]}\colon
G{\mathbf{Set}}\to{{\mathbf{Fun}}(BG,C^{*}\mathbf{Cat})}$

of ${\mathbf{C}}[-]$. Furthermore, in [BEa, Def. 17.15] we defined the
subfunctor

${{}_{{\mathbf{K}}}{\mathbf{C}}_{\mathrm{lf}}[-]\rtimes_{r}G}\colon
G{\mathbf{Set}}\to C^{*}\mathbf{Cat}$ (8.6)

of ${\mathbf{C}}[-]\rtimes_{r}G$ such that
${}_{{\mathbf{K}}}{\mathbf{C}}_{\mathrm{lf}}[\tilde{X}]\rtimes_{r}G$ is the
full subcategory of ${\mathbf{C}}[\tilde{X}]\rtimes_{r}G$ on the objects of
${}_{{\mathbf{K}}}{\mathbf{C}}_{\mathrm{lf}}[\tilde{X}]$.

In the following proposition it is useful to add the choice of the ideal to
the notation of the Roe categories.

######  Proposition 8.2.

We assume that ${\mathbf{C}}$ is $|G|$-additive. For $\tilde{X}$ in
$G{\mathbf{Set}}$ we have the following natural isomorphisms (or unitary
equivalence, respectively):

  1. 1.

${}_{{\mathbf{C}}}\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X_{min,min})\cong\mathbf{C}[\tilde{X}]$
in ${\mathbf{Fun}}(BG,C^{*}\mathbf{Cat})$.

  2. 2.

${}_{{\mathbf{K}}}\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X_{min,max})\cong{{}_{{\mathbf{K}}}\mathbf{C}_{\mathrm{lf}}[\tilde{X}]}$
in ${\mathbf{Fun}}(BG,C^{*}\mathbf{Cat})$.

  3. 3.

${}_{{\mathbf{K}}}\mathbf{\bar{C}}_{\mathrm{lf},G}^{\mathrm{ctr}}(X_{min,max})\cong{}_{{\mathbf{K}}}{\mathbf{C}}_{\mathrm{lf}}[\tilde{X}]\rtimes
G$ in $C^{*}\mathbf{Cat}$.

  4. 4.

${}_{{\mathbf{K}}}\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{min,max}\otimes
G_{can,min})\simeq{}_{{\mathbf{K}}}{\mathbf{C}}_{\mathrm{lf}}[\tilde{X}]\rtimes_{r}G$
in $C^{*}\mathbf{Cat}$.

###### Proof.

Assertion 1: This isomorphism follows from a comparison of [BEa, Defn. 17.7]
with the definition of
${}_{{\mathbf{C}}}\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}$, as
we explain in the following.

Recall that an object of
${}_{{\mathbf{C}}}\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X_{min,min})$
is a pair $(C,\mu)$ of an object $C$ of ${\mathbf{C}}$ and a function
$\mu\colon\mathcal{P}_{X}\to\mathrm{Proj}(C)$. The asserted isomorphism
identifies this object with the object $(C,(\mu(\\{x\\}))_{x\in X})$ of
${\mathbf{C}}[\tilde{X}]$. The requirement that $(C,\mu)$ is determined on
points corresponds to the requirement that $C$ is the orthogonal sum of the
images of the family of projections $(\mu(\\{x\\}))_{x\in X}$. Since
$X_{min,min}$ has the minimal bornology the set $X$ itself is locally finite,
and hence local finiteness of the support of $(C,\mu)$ is no restriction.
Finally, since we work with the ideal ${\mathbf{C}}$ in ${\mathbf{C}}$ there
is no restriction on the projections $\mu(\\{x\\})$.

Since $X_{min,min}$ has the minimal coarse structure the condition that a
morphism $A\colon(C,\mu)\to(C^{\prime},\mu^{\prime})$ is controlled
corresponds to the condition $\mu^{\prime}(\\{x^{\prime}\\})A\mu(\\{x\\})=0$
for all $x,x^{\prime}$ in $X$ with $x\not=x$ as required for a morphism in
$\mathbf{C}[{\tilde{X}}]$.

Assertion 2: The morphisms $X_{min,max}\to X_{min,min}$ in
$G\mathbf{BornCoarse}$ and
$(\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}})\to(\tilde{\mathbf{C}}\to\tilde{\mathbf{C}})$
in
${\mathbf{Fun}}(BG,\mathbf{P}(C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\mathrm{Idem}}_{i})$
by (4.6) induce a morphism

${}_{{\mathbf{K}}}\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X_{min,max})\to{}_{{\mathbf{C}}}\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X_{min,min})\,.$
(8.7)

It is clear from the definitions that it is an inclusion of a full
subcategory. Similarly, by [BEa, Def. 17.9] we have an inclusion of a full
subcategory
${}_{{\mathbf{K}}}{\mathbf{C}}_{\mathrm{lf}}[\tilde{X}]\to{\mathbf{C}}[\tilde{X}]$.
We obtain the desired isomorphism by observing that the isomorphism above
identifies these subcategories, as will be explained in the following.

Since locally finite subsets in $X_{min,max}$ are finite the objects $(C,\mu)$
in
${}_{{\mathbf{K}}}\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X_{min,max})$
are supported on finite subsets. This corresponds exactly to the condition
that the family $(\mu(\\{x\\}))_{x\in X}$ has only finitely many non-zero
members. Furthermore, the requirement that $\mu(\\{x\\})$ is a morphism in
${\mathbf{K}}$ is equivalent to the condition that every object representing
an image of this projection is a unital object of the ideal ${\mathbf{K}}$
(see [BEa, Def. 2.10]).

Assertion 3: This isomorphism follows from the isomorphism in Assertion 2 and
Definition 7.1.

Assertion 4: We apply Proposition 8.1 to $X_{min,min}$ and
$\tilde{\mathbf{C}}\subseteq\tilde{\mathbf{C}}$ in order to get the first
equivalence in the chain

${}_{{\mathbf{C}}}\mathbf{C}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{min,min}\otimes
G_{can,min})\xrightarrow{\psi,\simeq}{}_{{\mathbf{C}}}\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X_{min,min})\rtimes^{\mathrm{alg}}G\xrightarrow{\kappa}\mathbf{L}^{2}(G,{}_{{\mathbf{C}}}\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X_{min,min}))\,.$

The second functor $\kappa$ is the functor [BEa, (10.6)] appearing in the
definition of the reduced crossed product [BEa, Def. 10.14].

We claim that $\kappa\circ\psi$ induces on the morphism spaces of
${}_{{\mathbf{C}}}\mathbf{C}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{min,min}\otimes
G_{can,{min}})$ the same norm as the embedding

${}_{{\mathbf{C}}}\mathbf{C}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{min,min}\otimes
G_{can,min})\to{}_{{\mathbf{C}}}\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{min,min}\otimes
G_{can,min})\,.$

Let $(C,\rho,\mu)$ be an object of
${}_{{\mathbf{C}}}\mathbf{C}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{min,min}\otimes
G_{can,{min}})$ and choose an image $(\tilde{C},u)$ of the projection
$\mu(X\times\\{e\\})$. From the proof of Proposition 8.1 we recall the unitary
isomorphism (8.5). Let now
$A\colon(C,\rho,\mu)\to(C^{\prime},\rho^{\prime},\mu^{\prime})$ be a morphism
in
${}_{{\mathbf{C}}}\mathbf{C}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{min,min}\otimes
G_{can,{min}})$. We then observe by a straightforward calculation that

$v^{*}_{(C^{\prime},\rho^{\prime},\mu^{\prime})}Av_{(C,\rho,\mu)}=\kappa(\psi(A))\,.$

Since $v^{*}_{(C^{\prime},\rho^{\prime},\mu^{\prime})}$ and $v_{(C,\rho,\mu)}$
are unitary this shows the claim.

In view of the definition [BEa, Def. 10.8] of the reduced crossed product the
claim immediately implies that $\psi$ extends by continuity to a unitary
equivalence

${}_{{\mathbf{C}}}\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{min,min}\otimes
G_{can,min})\xrightarrow{\bar{\psi},\simeq}{}_{{\mathbf{C}}}\mathbf{\tilde{\bar{C}}}^{\mathrm{ctr}}_{\mathrm{lf}}(X_{min,min})\rtimes_{r}G\,.$

Consequently, its inverse $\phi$ from Proposition 8.1 also extends to a
unitary equivalence $\bar{\phi}$. We consider the bold part of the following
commuting diagram

whose maps are defined as follows:

  * •

The left and right upper vertical morphisms are the canonical embeddings into
the respective completions.

  * •

The lower left vertical morphism in the diagram is induced by the morphisms
$(\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}})\to(\tilde{\mathbf{C}}\to\tilde{\mathbf{C}})$
in
${\mathbf{Fun}}(BG,\mathbf{P}(C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\mathrm{Idem}}_{i})$
and $X_{min,max}\to X_{min,min}$ in $G\mathbf{BornCoarse}$, see (4.6). By an
inspection of the definitions one observes that it is an embedding of a full
subcategory.

  * •

The middle vertical morphism in the diagram is also induced by the morphisms
$(\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}})\to(\tilde{\mathbf{C}}\to\tilde{\mathbf{C}})$
in
${\mathbf{Fun}}(BG,\mathbf{P}(C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\mathrm{Idem}}_{i})$
and $X_{min,max}\to X_{min,min}$ in $G\mathbf{BornCoarse}$, see (8.7),
together with passing to the corresponding completion of the crossed product.

  * •

The functor $\phi^{\prime}$ is another instance of the unitary equivalence
constructed in Proposition 8.1.

  * •

The left pentagon commutes by the naturality asserted in Proposition 8.1.

  * •

The lower right vertical morphism is an embedding of a full subcategory by
definition, see (8.6).

  * •

The right pentagon commutes by the naturality asserted in the previous cases
of this Proposition 8.2.

Since in the diagram the lower left and right morphisms are embeddings of full
subcategories and the upper left and right morphisms are the canonical
embeddings into the respective completions, commutativity of the diagram
implies that we can define the dotted arrow by continuous extension of the
composition of the upper horizontal maps. Hence we can conclude that the
dotted arrow is the desired unitary equivalence. ∎

######  Remark 8.3.

In general the maximal and the reduced crossed products do not coincide. We
have a canonical morphism from the maximal crossed product to the reduced one.
Combining this with the Proposition 8.2(3 & 4) we get a functor

${}_{{\mathbf{K}}}\mathbf{\bar{C}}_{\mathrm{lf},G}^{\mathrm{ctr}}(X_{min,max})\to{}_{{\mathbf{K}}}\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{min,max}\otimes
G_{can,min})$ (8.8)

which in general is not an equivalence. This stays in contrast to [BCKW, Cor.
5.4] which asserts that the analogue of this morphism in the case of
controlled objects in left-exact $\infty$-categories is always an equivalence.

If $G$ is amenable, then as a consequence of [BEa, Thm. 10.19] the functor in
(8.8) is an isomorphism. ∎

##  9 Transfers

In [BEKW20a, Defn. 2.27] we introduced an extension

$\iota\colon G\mathbf{BornCoarse}\to G\mathbf{BornCoarse}_{{\mathtt{tr}}}$
(9.1)

of the category of $G$-bornological coarse spaces to an $\infty$-category
$G\mathbf{BornCoarse}_{{\mathtt{tr}}}$. It has the same objects as
$G\mathbf{BornCoarse}$. Its $1$-morphisms are spans

$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w}$$\scriptstyle{f}$$\textstyle{X}$$\textstyle{Y}$

where $f$ is a morphism in $G\mathbf{BornCoarse}$ which is in addition
bornological, see Definition 2.7(2), and where $w$ is a bounded covering (see
Definition 9.5 below). The higher morphisms are diagrams which exhibit
compositions of spans. We refer to [BEKW20a, Defn. 2.27] for more details.
They will not be needed in the present paper.

We consider a functor $E\colon G\mathbf{BornCoarse}\to\mathbf{M}$.

######  Definition 9.1.

$E$ admits transfers if there exists a functor $E_{{\mathtt{tr}}}\colon
G\mathbf{BornCoarse}_{{\mathtt{tr}}}\to\mathbf{M}$ and an equivalence
$E_{{\mathtt{tr}}}\circ\iota\simeq E$.

In the following it is useful to model $\infty$-categories by quasi-
categories, and to distinguish the notation for categories ${\mathcal{C}}$
from the notation ${\mathtt{N}}({\mathcal{C}})$ for the corresponding quasi-
categories. Basic references for this model are [Lur09, Lur].

The category $C^{*}\mathbf{Cat}$ is the underlying $1$-category of a strict
$(2,1)$-category $C^{*}\mathbf{Cat}_{2,1}$ whose $2$-isomorphisms are unitary
isomorphisms of functors. We have a $2$-nerve functor ${\mathtt{N}}_{2}$ from
$(2,1)$-categories to quasi-categories. This functor sends a $(2,1)$-category
to the coherent nerve of the fibrant simplicial category obtained by applying
the usual nerve functor to the morphism groupoids. We have a canonical
embedding

$\ell_{2}\colon{\mathtt{N}}(C^{*}\mathbf{Cat})\to{\mathtt{N}}_{2}(C^{*}\mathbf{Cat}_{2,1})\,.$

Assume that $\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}}$ is in
${\mathbf{Fun}}(BG,\mathbf{P}(C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\mathrm{Idem}})$
and consider the functor

$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}\colon G\mathbf{BornCoarse}\to
C^{*}\mathbf{Cat}$

defined in Definition 4.21.2. We can consider the functor

${\mathtt{N}}(G\mathbf{BornCoarse})\xrightarrow{{\mathtt{N}}(\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}})}{\mathtt{N}}(C^{*}\mathbf{Cat})\xrightarrow{\ell_{2}}{\mathtt{N}}_{2}(C^{*}\mathbf{Cat}_{2,1})\,.$

We want to assert that this functor admits transfers. The more precise
formulation of Definition 9.1 requires that there is a functor
$\mathbf{\bar{C}}_{\mathrm{lf},\mathrm{tr}}^{G,\mathrm{ctr}}\colon
G\mathbf{BornCoarse}_{{\mathtt{tr}}}\to{\mathtt{N}}_{2}(C^{*}\mathbf{Cat}_{2,1})$
such that

$\ell_{2}\circ{\mathtt{N}}(\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}})\simeq\mathbf{\bar{C}}_{\mathrm{lf},\mathrm{tr}}^{G,\mathrm{ctr}}\circ\iota\,,$
(9.2)

where $\iota\colon{\mathtt{N}}(G\mathbf{BornCoarse})\to
G\mathbf{BornCoarse}_{{\mathtt{tr}}}$ is the canonical embedding [BEKW20d,
Def. 2.33].

The following is the main theorem of the present section.

######  Theorem 9.2.

If $\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}}$ is in
${\mathbf{Fun}}(BG,\mathbf{P}(C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\mathrm{Idem},\infty\mathrm{add}})$,
then the functor

$\ell_{2}\circ{\mathtt{N}}(\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}})\colon{\mathtt{N}}(G\mathbf{BornCoarse})\to{\mathtt{N}}_{2}(C^{*}\mathbf{Cat}_{2,1})$

admits transfers.

######  Remark 9.3.

We can not expect that the functor
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}$ with values in
$C^{*}\mathbf{Cat}$ admits transfers. The problematic point is that the
contravariant extension of the functor to bounded coverings involves choices
on the level of objects. As a consequence, functoriality is satisfied only up
to canonical $2$-isomorphisms. ∎

Before we start with the proof of Theorem 9.2 we recall some further notions
from coarse geometry which will be used in the arguments.

Let $W$ be in $G\mathbf{Coarse}$. Then we consider the element

$U_{\pi_{0}(W)}\coloneqq\bigcup_{U\in{\mathcal{C}}_{W}}U$

of $\mathcal{P}_{W\times W}$. Using the conditions on ${\mathcal{C}}_{W}$
listed in Definition 2.1 we see that $U_{\pi_{0}(W)}$ is a $G$-invariant
equivalence relation on $W$.

######  Definition 9.4.

The equivalence classes in $W$ for $U_{\pi_{0}(W)}$ are the coarse components
of $W$.

We let $\pi_{0}(W)$ denote the set of coarse components. Since
$U_{\pi_{0}(W)}$ is $G$-invariant, the group $G$ acts on the set $\pi_{0}(W)$.

Let $W,X$ be objects of $G\mathbf{BornCoarse}$ and $w\colon W\to X$ be a
morphism between the underlying $G$-coarse spaces.

######  Definition 9.5.

$w$ is a bounded covering if:

  1. 1.

For every coarse component $V$ of $W$ the restriction $f_{|V}\colon V\to f(V)$
is an isomorphism of coarse spaces from $V$ to a coarse component of $X$.

  2. 2.

The coarse structure of $W$ is generated by the entourages
$U_{\pi_{0}(W)}\cap(w\times w)^{-1}(U)$ for all $U$ in ${\mathcal{C}}_{X}$.

  3. 3.

$w$ is bornological.

  4. 4.

For every $B$ in ${\mathcal{B}}_{W}$ there exists a finite bound (depending on
$B$) on the cardinality of the fibres of $\pi_{0}(B)\to\pi_{0}(X)$.

We will indicate bounded coverings using the double-headed arrow symbol .

We consider objects $W,V,U,Z$ in $G\mathbf{BornCoarse}$ and a square

$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{w}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{Z}$

in $G\mathbf{Coarse}$.

######  Definition 9.6 ([BEKW20a, Def. 2.19]).

The square is admissible if it has the following properties:

  1. 1.

The square is cartesian in $G\mathbf{Coarse}$.

  2. 2.

$f$ and $g$ are proper and bornological.

  3. 3.

$u$ and $w$ are bounded coverings.

###### Proof of Theorem 9.2.

We must construct a functor

$\mathbf{\bar{C}}_{\mathrm{lf},\mathrm{tr}}^{G,\mathrm{ctr}}\colon
G\mathbf{BornCoarse}_{{\mathtt{tr}}}\to{\mathtt{N}}_{2}(C^{*}\mathbf{Cat}_{2,1})$

satisfying (9.2). We can use the method explained in [BEKW20a, Sec. 2.1].
Following this reference, in order to construct
$\mathbf{\bar{C}}_{\mathrm{lf},\mathrm{tr}}^{G,\mathrm{ctr}}$ we must provide
the following data:

  1. 1.

We start with the functor
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}\colon G\mathbf{BornCoarse}\to
C^{*}\mathbf{Cat}$.

  2. 2.

For every bounded covering $w\colon W\to Z$ we must provide a $1$-morphism

$w^{*}:\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(Z)\to\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(W)\,.$

  3. 3.

For every two composable bounded coverings $w\colon W\to Z$ and $v\colon V\to
W$ we must provide a $2$-isomorphism

$a_{v,w}\colon(w\circ v)^{*}\Rightarrow v^{*}\circ w^{*}\,.$

  4. 4.

For every admissible square

$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{w}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{Z}$

we must provide a $2$-isomorphism

$b_{g,u}\colon f_{*}\circ w^{*}\Rightarrow u^{*}\circ g_{*}\,.$

This data must satisfy the following conditions:

  1. 1.

If the bounded covering $w\colon W\to Z$ is an isomorphism, then
$w^{*}=w^{-1}_{*}$.

  2. 2.

If the composable bounded coverings $w\colon W\to Z$ and $v\colon V\to W$ are
both isomorphisms, then $a_{v,w}={\mathtt{id}}$. This equality makes sense,
because in this case $v^{*}w^{*}=(wv)^{*}$ by the strict functoriality of
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}$.

  3. 3.

If $w\colon W\to Z$ and $v\colon V\to W$ and $u\colon U\to V$ are three
composable bounded coverings, then the square of $2$-isomorphisms

$\textstyle{(w\circ v\circ
u)^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{vu,w}}$$\scriptstyle{a_{u,wv}}$$\textstyle{(v\circ
u)^{*}\circ
w^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{u,v}\circ
w^{*}}$$\textstyle{u^{*}\circ(w\circ
v)^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u^{*}\circ
a_{v,w}}$$\textstyle{u^{*}\circ v^{*}\circ w^{*}}$

commutes.

  4. 4.

In the case of an admissible square with morphisms $w,f,g,u$, if $u$ (and
therefore $w$) is an isomorphism, then we have $b_{g,u}={\mathtt{id}}$. This
equality again makes sense since

$f_{*}w^{*}=f_{*}w^{-1}_{*}=u^{-1}_{*}g_{*}=u^{*}g_{*}$

by the strict functoriality of
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}$.

  5. 5.

In the case of an admissible square with morphisms $w,f,g,u$, if $f$ and $g$
are identities and therefore $w=u$, then we have $b_{g,u}={\mathtt{id}}$.

  6. 6.

For every diagram

$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$$\scriptstyle{m}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{n}$$\scriptstyle{s}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\textstyle{Z}$

consisting of two admissible squares we have the relation

$b_{gh,r}=(b_{g,r}\circ h_{*})(n_{*}\circ b_{h,s})\,.$ (9.3)

  7. 7.

For every diagram

$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$$\scriptstyle{m}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\scriptstyle{h}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v}$$\textstyle{Y}$

consisting of two admissible squares we have the relation

$(a_{s,v}\circ f_{*})b_{f,vs}=(s^{*}\circ b_{f,v})(b_{h,s}\circ
u^{*})(m_{*}\circ a_{t,u})\,.$ (9.4)

Let $X$ be in $G\mathbf{BornCoarse}$, and let $(C,\rho,\mu)$ be an object of
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X)$. For a subset $Y$ of $X$
we will use the notation $(C(Y),u^{C}_{Y})$ in order to denote a
representative of the image of the projection $\mu(Y)$.

We let $(\bigoplus_{Y\in\pi_{0}(X)}C(Y),(e_{Y})_{Y\in\pi_{0}(X)})$ denote a
representative of the orthogonal sum of the family $(C(Y))_{Y\in\pi_{0}(X)}$.
By Lemma 3.13 applied to the partition $(Y)_{Y\in\pi_{0}(X)}$ of $X$ we have
an isomorphism

$\sum_{Y\in\pi_{0}(X)}u^{C}_{Y}e_{Y}^{*}\colon\bigoplus_{Y\in\pi_{0}(X)}C(Y)\xrightarrow{\cong}C\,.$

Let $(C^{\prime},\rho^{\prime},\mu^{\prime})$ be a second object,
$(\bigoplus_{Y\in\pi_{0}(X)}C^{\prime}(Y),(e^{\prime}_{Y})_{Y\in\pi_{0}(X)})$
be the corresponding choice of a sum, and let
$A\colon(C,\rho,\mu)\to(C^{\prime},\rho^{\prime},\mu^{\prime})$ be a morphism
in $\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X)$. For $Y,Y^{\prime}$ in
$\pi_{0}(X)$ we set

$A_{Y^{\prime},Y}\coloneqq u_{Y^{\prime}}^{C^{\prime},*}Au^{C}_{Y}\colon
C(Y)\to C^{\prime}(Y^{\prime})\,.$

Since $A$ is $U_{\pi_{0}}(X)$-controlled we actually have $A_{Y^{\prime},Y}=0$
if $Y\not=Y^{\prime}$. In order to simplify the notation we set

$A_{Y}\coloneqq A_{Y,Y}\,.$

The morphism $A$ is uniquely determined by the family
$(A_{Y})_{Y\in\pi_{0}(X)}$. In the following we discuss how one can
reconstruct the morphism $A$ from a family $(A_{Y})_{Y\in\pi_{0}(X)}$. Assume
that a family $(A_{Y}\colon C(Y)\to C^{\prime}(Y))_{Y\in\pi_{0}(X)}$ is given.
We say that $(A_{Y})_{Y\in\pi_{0}(X)}$ is bounded if
$\sup_{Y\in\pi_{0}(X)}\|A_{Y}\|<\infty$. If $(A_{Y})_{Y\in\pi_{0}(X)}$ is
bounded, then we can define a morphism $A\colon C\to C^{\prime}$ uniquely such
that

$\textstyle{\bigoplus_{Y\in\pi_{0}(X)}C(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\oplus_{Y\in\pi_{0}(X)}A_{Y}}$$\scriptstyle{\cong}$$\scriptstyle{\sum_{Y\in\pi_{0}(X)}u^{C}_{Y}e^{*}_{Y}}$$\textstyle{\bigoplus_{Y\in\pi_{0}(X)}C^{\prime}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\sum_{Y\in\pi_{0}(X)}u^{C^{\prime}}_{Y}e^{\prime,*}_{Y}}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A}$$\textstyle{C^{\prime}}$

commutes.

By Definition 3.4.3 we have the relation
$\rho_{g}\circ\mu(Y)=(g\mu)(g^{-1}Y)\circ\rho_{g}$. We therefore get a
canonical unitary

$\rho_{g,Y}\coloneqq(gu^{C}_{g^{-1}Y})^{*}\circ\rho_{g}\circ u^{C}_{Y}\colon
C(Y)\to g(C(g^{-1}Y))\,.$ (9.5)

We say that the family $(A_{Y})_{Y\in\pi_{0}(X)}$ is equivariant if it
satisfies

$gA_{g^{-1}Y}\circ\rho_{g,Y}=\rho^{\prime}_{g,Y}\circ A_{Y}$

for all $g$ in $G$. If $(A_{Y})_{Y\in\pi_{0}(X)}$ is bounded and equivariant,
then it induces a morphism of $G$-objects
$A\colon(C,\rho)\to(C^{\prime},\rho^{\prime})$.

Finally one must ensure that $A$ is controlled. We consider the projection-
valued function

$\mu_{Y}\coloneqq
u^{C,*}_{Y}\mu_{|\mathcal{P}_{Y}}u^{C}_{Y}\colon\mathcal{P}_{Y}\to\mathrm{Proj}(C(Y))\,.$
(9.6)

Then $(C(Y),\mu_{Y})$ is an object of
$\mathbf{\bar{C}}_{\mathrm{lf}}^{\mathrm{ctr}}(Y)$. Assume that
$(A_{Y})_{Y\in\pi_{0}(X)}$ is an equivariant and bounded family. Let $U$ be in
$\mathcal{P}_{X\times X}$. We say that $(A_{Y})_{Y\in\pi_{0}(X)}$ is
$U$-controlled if $A_{Y}$ is $U\cap(Y\times Y)$-controlled for all $Y$ in
$\pi_{0}(X)$. If $(A_{Y})_{Y\in\pi_{0}(X)}$ is $U$-controlled, then the
induced morphism
$A\colon(C,\rho,\mu)\to(C^{\prime},\rho^{\prime},\mu^{\prime})$ is a
$U$-controlled morphism in $\mathbf{C}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X)$.

We again consider an equivariant and bounded family
$(A_{Y})_{Y\in\pi_{0}(X)}$. We now assume that there exists a sequence
$((A_{n,Y})_{Y\in\pi_{0}(X)})_{n\in{\rm Nature}}$ of bounded equivariant
families and a family $(U_{n})_{n\in{\rm Nature}}$ in ${\mathcal{C}}_{X}$ such
that $(A_{n,Y})_{Y\in\pi_{0}(X)}$ is $U_{n}$-controlled for every $n$ in ${\rm
Nature}$, and

$\operatorname*{\mathtt{lim}}_{n\to\infty}\sup_{Y\in\pi_{0}(X)}\|A_{n,Y}-A_{Y}\|=0\,.$

Let $A_{n}\colon(C,\rho,\mu)\to(C^{\prime},\rho^{\prime},\mu^{\prime})$ be the
$U_{n}$-controlled morphism in $\mathbf{C}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X)$
defined by the family $(A_{n,Y})_{Y\in\pi_{0}(X)}$. Then we have
$\operatorname*{\mathtt{lim}}_{n\to\infty}A_{n}=A$ in
${\mathtt{Hom}}_{{\mathbf{C}}}(C,C^{\prime})$, and consequently
$A\colon(C,\rho,\mu)\to(C^{\prime},\rho^{\prime},\mu^{\prime})$ is a morphism
in $\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X)$.

Note that if $A\colon(C,\rho,\mu)\to(C^{\prime},\rho^{\prime},\mu^{\prime})$
is a morphism in $\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X)$, then
the associated family $(A_{Y})_{Y\in\pi_{0}(X)}$ has all the properties needed
for the reconstruction of $A$.

We now start with the construction of the data for
$\mathbf{\bar{C}}_{\mathrm{lf},\mathrm{tr}}^{G,\mathrm{ctr}}$.

  1. 1.

The functor $\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}\colon
G\mathbf{BornCoarse}\to C^{*}\mathbf{Cat}$ is already constructed, see
Definition 4.21.2.

  2. 2.

Let $w\colon W\to Z$ be a bounded covering. We construct the functor

$w^{*}\colon\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(Z)\to\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(W)\,.$

    1. (a)

objects: Let $(C,\rho,\mu)$ be an object in
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(Z)$. We choose an orthonormal
sum $(\bigoplus_{Y\in\pi_{0}(W)}C(w(Y)),(e^{w^{*}}_{Y})_{Y\in\pi_{0}(W)})$. In
particular, if $w$ is an isomorphism, then we choose the representative
$(C,(u^{C}_{w(Y)})_{Y\in\pi_{0}(W)})$. We then set

$w^{*}(C,\rho,\mu)\coloneqq(w^{*}C,w^{*}\rho,w^{*}\mu)\,,$

where the right-hand side has the following description.

      1. i.

We set

$w^{*}C\coloneqq\bigoplus_{Y\in\pi_{0}(W)}C(w(Y))\,.$

      2. ii.

For a subset $B$ of $W$ and $Y$ in $\pi_{0}(W)$ we define the projection

$(w^{*}\mu)(B)_{Y}\coloneqq\mu_{w(Y)}(w(B\cap Y))$

on $C(w(Y))$, see (9.6). We then define $(w^{*}\mu)(B)$ as the projection on
$w^{*}C$ given by

$(w^{*}\mu)(B)\coloneqq\oplus_{Y\in\pi_{0}(W)}(w^{*}\mu)(B)_{Y}\,.$

By Definition 9.5.4, if $B$ is in ${\mathcal{B}}_{W}$, then the sum has
finitely many non-zero summands. One now checks that $w^{*}\mu$ is a finitely
additive, projection-valued measure which is full. Furthermore,
$(w^{*}C,w^{*}\mu)$ is locally finite.

      3. iii.

For $g$ in $G$, using (9.5), we define

$\mathclap{(w^{*}\rho)_{g}\colon\bigoplus_{Y\in\pi_{0}(W)}C(w(Y))\xrightarrow{\oplus_{Y}\rho_{g,Y}}\bigoplus_{Y\in\pi_{0}(W)}g(C(w(g^{-1}Y)))\cong
g\bigoplus_{Y\in\pi_{0}(W)}C(w(Y))\,.}$

The last isomorphism is fixed by the commutativity of the diagram

$\mathclap{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
49.2594pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 73.2594pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{g(C(w(g^{-1}Y)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
52.16545pt\raise-11.13889pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.61111pt\hbox{$\scriptstyle{\tilde{e}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 27.75732pt\raise-30.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
145.32614pt\raise-11.54916pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.0464pt\hbox{$\scriptstyle{ge^{w^{*}}_{g^{-1}Y}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 180.46536pt\raise-30.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
203.14114pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-49.2594pt\raise-40.5pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\bigoplus_{Y\in\pi_{0}(W)}g(C(w(g^{-1}Y)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
97.44557pt\raise-46.65pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\cong}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 164.8349pt\raise-40.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
104.04715pt\raise-40.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
164.8349pt\raise-40.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{g\bigoplus_{Y\in\pi_{0}(W)}C(w(Y))}$}}}}}}}\ignorespaces}}}}\ignorespaces}\,.$

where $\tilde{e}$ is the canonical embedding of the summand with index $Y$.

One now checks in a straightforward manner that

$w^{*}(C,\phi,\rho)=(w^{*}C,w^{*}\phi,w^{*}\rho)\,.$

is a well-defined object in
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(W)$.

    2. (b)

morphisms: Let $A\colon(C,\rho,\mu)\to(C^{\prime},\rho^{\prime},\mu^{\prime})$
be a morphism in $\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(Z)$. Then we
let $w^{*}(A)\colon w^{*}C\to w^{*}C^{\prime}$ be the morphism which is
uniquely determined by the family $(A_{w(Y)})_{Y\in\pi_{0}(W)}$. We now
explain why $w^{*}A$ is well-defined. First of all note that
$(A_{B})_{B\in\pi_{0}(Z)}$ is bounded and equivariant. We conclude that
$(A_{w(Y)})_{Y\in\pi_{0}(W)}$ is also bounded. Since $w$ is equivariant we
furthermore see that $(A_{w(Y)})_{Y\in\pi_{0}(W)}$ is also equivariant. There
exists a family $(A_{n})_{n\in{\rm Nature}}$ of morphisms
$A_{n}\colon(C,\rho,\mu)\to(C^{\prime},\rho^{\prime},\mu^{\prime})$ in
$\mathbf{C}_{\mathrm{lf}}^{G,\mathrm{ctr}}(Z)$ and a family $(U_{n})_{\in{\rm
Nature}}$ in ${\mathcal{C}}_{Z}$ such that $A_{n}$ is $U_{n}$-controlled for
every $n$ in ${\rm Nature}$, and
$\operatorname*{\mathtt{lim}}_{n\to\infty}A_{n}=A$ in
${\mathtt{Hom}}_{{\mathbf{C}}}(C,C^{\prime})$. Then
$(A_{n,w(Y)})_{Y\in\pi_{0}(W)}$ is $U_{\pi_{0}(W)}\cap(w\times
w)^{-1}(U_{n})$-controlled for every $n$ in ${\rm Nature}$. By Definition
9.5.2 we have $U_{\pi_{0}(W)}\cap(w\times w)^{-1}(U_{n})\in{\mathcal{C}}_{W}$,
and furthermore
$\operatorname*{\mathtt{lim}}_{n\to\infty}\sup_{Y\in\pi_{0}(W)}\|A_{n,w(Y)}-A_{w(Y)}\|=0$.
By the discussion above we see that $A$ is controlled. We have the following
formula:

$w^{*}A=\sum_{B\in\pi_{0}(W)}e^{\prime,w^{*}}_{B}u^{C^{\prime},*}_{w(B)}Au_{w(B)}^{C}e^{w^{*},*}_{B}\,.$
(9.7)

One checks in a straightforward manner that $w^{*}$ is compatible with
compositions and the involution.

  3. 3.

We consider two composable bounded coverings
$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w}$$\textstyle{Z}$.
We construct the $2$-isomorphism

$a_{v,w}\colon(wv)^{*}\Rightarrow v^{*}w^{*}$

as follows. Let $(C,\rho,\mu)$ be an object of
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(Z)$. Then we define

$a_{v,w}(C,\rho,\mu)\colon((wv)^{*}C,(wv)^{*}\phi,(wv)^{*}\rho)\to(v^{*}w^{*}C,v^{*}w^{*}\phi,v^{*}w^{*}\rho)$

by

$a_{v,w}(C,\rho,\mu)\coloneqq\sum_{B\in\pi_{0}(V)}e_{B}^{v^{*}}u_{v(B)}^{w^{*}C,*}e_{v(B)}^{w^{*}}e_{B}^{(wv)^{*},*}\,.$
(9.8)

One checks by a calculation that $a_{v,w}(C,\rho,\mu)$ is a unitary
isomorphism. One furthermore checks that

$a_{v,w}\coloneqq(a_{v,w}(C,\rho,\mu))_{(C,\rho,\mu)\in\mathrm{Ob}(\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(Z))}$

is a unitary natural isomorphism.

  4. 4.

We now consider an admissible square

$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{w}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{Z}$

Then must define the transformation

$b_{g,u}\colon f_{*}\circ w^{*}\Rightarrow u^{*}\circ g_{*}\,.$

Let $(C,\rho,\mu)$ be an object of
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(V)$. Then we define

$b_{g,u}(C,\rho,\mu)\colon f_{*}w^{*}(C,\rho,\mu)\to u^{*}g_{*}(C,\rho,\mu)$

by

$b_{g,u}(C,\rho,\mu)\coloneqq\sum_{B\in\pi_{0}(U)}e^{u^{*}}_{B}u^{g_{*}C,*}_{u(B)}\sum_{A\in\pi_{0}(f^{-1}(B))}u^{C}_{w(A)}e_{A}^{w^{*},*}\,.$
(9.9)

One checks by a calculation that $b_{g,u}(C,\rho,\mu)$ is a unitary
isomorphism.

We must verify that the functors and $2$-isomorphisms constructed above
satisfy the seven conditions listed above. All these conditions are verified
by straightforward (but lengthy) calculations. These calculations start with
inserting the explicit definitions of the transformations given above and then
proceed by employing the orthogonality relations for the families of canonical
isometries. Since the details do not really lead to further insights we will
not bore the reader with further details. ∎

Let $\mathrm{Hg}\colon C^{*}\mathbf{Cat}\to\mathbf{M}$ be a functor to an
$\infty$-category $\mathbf{M}$.

######  Theorem 9.7.

If $\mathrm{Hg}$ sends unitary equivalences to equivalences, then the
composition

$\mathrm{Hg}\circ\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}\colon
G\mathbf{BornCoarse}\to\mathbf{M}$

admits transfers.

###### Proof.

The proof is analogous to the proof of [BEKW20c, Prop. 2.63] combined with the
discussion around the diagram in [BEKW20a, (3.8)].

We consider $\mathrm{Hg}$ as a functor
${\mathtt{N}}(\mathrm{Hg})\colon{\mathtt{N}}(C^{*}\mathbf{Cat})\to\mathbf{M}$.

As explained in [Del12] or [Bun19, Thm. 1.4] the category $C^{*}\mathbf{Cat}$
has a simplicial model category structure whose weak equivalences $W$ are the
unitary equivalences. Applying the coherent nerve functor we obtain an
$\infty$-category ${\mathtt{N}}^{\mathrm{coh}}(C^{*}\mathbf{Cat})$. Then we
have a canonical inclusion functor
${\mathtt{N}}(C^{*}\mathbf{Cat})\to{\mathtt{N}}^{\mathrm{coh}}(C^{*}\mathbf{Cat})$.
This inclusion is an explicit model of the Dwyer–Kan localization
$\ell\colon{\mathtt{N}}(C^{*}\mathbf{Cat})\to{\mathtt{N}}(C^{*}\mathbf{Cat})[W^{-1}]$.

We now construct the following commutative diagram

$\textstyle{{\mathtt{N}}(C^{*}\mathbf{Cat})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathtt{N}}(\mathrm{Hg})}$$\scriptstyle{\ell}$$\scriptstyle{(1)}$$\scriptstyle{\ell_{2,1}}$$\textstyle{{\mathtt{N}}(C^{*}\mathbf{Cat})[W^{-1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(2)}$$\scriptstyle{\simeq}$$\scriptstyle{\tilde{\mathrm{Hg}}}$$\textstyle{\mathbf{M}\,.}$$\textstyle{{\mathtt{N}}^{\mathrm{coh}}(C^{*}\mathbf{Cat})}$$\textstyle{{\mathtt{N}}_{2}(C^{*}\mathbf{Cat}_{2,1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(3)}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{Hg}_{2,1}}$
(9.10)

The morphism $(1)$ sends unitary equivalences to equivalences. Consequently we
obtain the morphism $(2)$ and the filler of the corresponding triangle from
the universal property of the Dwyer–Kan localization $\ell$. As explained
above the morphism (2) is an equivalence. The morphism (3) is actually an
isomorphism of simplicial sets.

Since $\mathrm{Hg}$ sends unitary equivalences to equivalences we obtain the
factorization $\tilde{\mathrm{Hg}}$ and the upper triangle from the universal
property of the Dwyer–Kan localization $\ell$. We now define
$\mathrm{Hg}_{2,1}\coloneqq\tilde{\mathrm{Hg}}\circ(2)^{-1}\circ(3)$. The
diagram (9.10) provides an equivalence of functors

${\mathtt{N}}(\mathrm{Hg})\simeq\mathrm{Hg}_{2,1}\circ\ell_{2,1}\colon{\mathtt{N}}({\mathbf{Cat}})\to\mathbf{M}\,.$

This equivalence provides the right triangle in the following diagram:

$\textstyle{{\mathtt{N}}(G\mathbf{BornCoarse})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{{\mathtt{N}}(\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}})}$$\textstyle{{\mathtt{N}}(C^{*}\mathbf{Cat})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ell_{2,1}}$$\scriptstyle{{\mathtt{N}}(\mathrm{Hg})}$$\textstyle{\mathbf{M}\,.}$$\textstyle{G\mathbf{BornCoarse}_{{\mathtt{tr}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{\bar{C}}_{\mathrm{lf},\mathrm{tr}}^{G,\mathrm{ctr}}}$$\textstyle{{\mathtt{N}}_{2}(C^{*}\mathbf{Cat}_{2,1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Hg}_{1,2}}$
(9.11)

The existence of the left square is exactly the statement of Theorem 9.2. The
outer part of the diagram (9.11) exhibits the required extension of
$\mathrm{Hg}\circ\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}$. ∎

##  10 Induction from and restriction to subgroups

Let $G$ be a group and consider a subgroup $H$. We consider a coefficient
category $\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}}$ in
${\mathbf{Fun}}(BG,\mathbf{P}(C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\mathrm{Idem}})$.
Then we can form the functor

$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}\colon G\mathbf{BornCoarse}\to
C^{*}\mathbf{Cat}$

(Definition 4.21.2). Furthermore, we can restrict the $G$-action on
$\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}}$ to an $H$-action (this will
not be written explicitly) and define the functor

$\mathbf{\bar{C}}_{\mathrm{lf}}^{H,\mathrm{ctr}}\colon H\mathbf{BornCoarse}\to
C^{*}\mathbf{Cat}\,.$

We have an induction functor

${\mathtt{Ind}}_{H}^{G}\colon H\mathbf{BornCoarse}\to G\mathbf{BornCoarse}\,,$
(10.1)

see [BEKW20b, Sec. 6] (it will be recalled in the proof of Proposition 10.1).

For the following statement we implicitly compose the functors
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}$ and
$\mathbf{\bar{C}}_{\mathrm{lf}}^{H,\mathrm{ctr}}$ with the functor
$C^{*}\mathbf{Cat}\to C^{*}\mathbf{Cat}_{2,1}$, where
$C^{*}\mathbf{Cat}_{2,1}$ is the $(2,1)$-category of $C^{*}$-categories,
functors and unitary isomorphisms.

######  Proposition 10.1.

If ${\mathbf{C}}$ is $|H\backslash G|$-additive, then we have a natural
equivalence

$\phi\colon\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}\circ{\mathtt{Ind}}_{H}^{G}\xrightarrow{\simeq}\mathbf{\bar{C}}_{\mathrm{lf}}^{H,\mathrm{ctr}}$

of functors from $H\mathbf{BornCoarse}$ to the $(2,1)$-category
$C^{*}\mathbf{Cat}_{2,1}$.

######  Remark 10.2.

The reason for considering the target of the two functors in the
$(2,1)$-category is that the transformation $\phi$ which we will construct is
only natural for $X$ in $H\mathbf{BornCoarse}$ up to unique isomorphisms which
are in general not equalities. This is because in the construction we must
choose objects representing images of projections, and these choices are only
determined up to unique unitary isomorphism [BEa, Lem. 2.14]. See also Remark
10.3. ∎

###### Proof of Proposition 10.1.

We start with recalling the construction of the induction functor (10.1) from
[BEKW20b, Sec. 6]. Let $X$ be in $H\mathbf{BornCoarse}$. We define
${\mathtt{Ind}}_{H}^{G}(X)$ in $G\mathbf{BornCoarse}$ as follows.

  * •

The underlying $G$-set of ${\mathtt{Ind}}_{H}^{G}(X)$ is $G\times_{H}X$ with
the left-action of $G$ on the left factor. Here $G\times_{H}X$ is the quotient
of the set $G\times X$ by $H$ with respect to the action given by
$h(g,x)=(gh^{-1},hx)$.

  * •

The bornology on $G\times_{H}X$ is generated by the images under the natural
projection $G\times X\to G\times_{H}X$ of the subsets $\\{g\\}\times B$ for
all bounded subsets $B$ of $X$.

  * •

The coarse structure is generated by the images of ${\mathtt{diag}}_{G}\times
U$ under this projection for all entourages $U$ of $X$.

The functor (10.1) is now defined as follows:

  * •

objects: The functor ${\mathtt{Ind}}_{H}^{G}$ sends $X$ in
$H\mathbf{BornCoarse}$ to ${\mathtt{Ind}}_{H}^{G}(X)$ in
$G\mathbf{BornCoarse}$.

  * •

morphisms: The functor ${\mathtt{Ind}}_{H}^{G}$ sends a morphism $f\colon X\to
Y$ in $H\mathbf{BornCoarse}$ to the morphism in $G\mathbf{BornCoarse}$ given
by the map $G\times_{H}f\colon G\times_{H}X\to G\times_{H}Y$ of underlying
sets induced from ${\mathtt{id}}_{G}\times f\colon X\times G\to Y\times G$.

For $X$ in $H\mathbf{BornCoarse}$ we now construct a functor

$\phi_{X}\colon\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}({\mathtt{Ind}}_{H}^{G}(X))\to\mathbf{\bar{C}}_{\mathrm{lf}}^{H,\mathrm{ctr}}(X)\,.$

  1. 1.

objects: Let $(C,\rho,\mu)$ be an object of
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}({\mathtt{Ind}}_{H}^{G}(X))$.
We have a $H$-equivariant inclusion $i\colon X\to G\times_{H}X$ given by
$x\mapsto[e,x]$. Here $[g,x]$ denotes the image in $G\times_{H}X$ of $(g,x)$
under the natural projection $G\times X\to G\times_{H}X$. We choose an image
$(\tilde{C},u)$ of the projection $\mu(i(X))$ on $C$. Then we define
$\tilde{\rho}=(\tilde{\rho}_{h})_{h\in H}$ by
$\tilde{\rho}_{h}\coloneqq(hu^{*})\rho_{h}u\colon\tilde{C}\to h\tilde{C}$ and
$\tilde{\mu}\colon\mathcal{P}(X)\to\mathrm{Proj}(\tilde{C})$ by
$\tilde{\mu}(Y)\coloneqq u^{*}\mu(Y)u$. The functor $\phi_{X}$ sends an object
$(C,\rho,\mu)$ to the object $(\tilde{C},\tilde{\rho},\tilde{\mu})$ in
$\mathbf{\bar{C}}_{\mathrm{lf}}^{H,\mathrm{ctr}}(X)$. One checks the
conditions listed in Definitions 3.4 and 4.7 by straightforward calculations
and thus verifies that the object $(\tilde{C},\tilde{\rho},\tilde{\mu})$ is
well-defined.

  2. 2.

morphisms: If $A\colon(C,\rho,\mu)\to(C^{\prime},\rho^{\prime},\mu^{\prime})$
is a morphism in
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}({\mathtt{Ind}}_{H}^{G}(X))$,
then the functor $\phi_{X}$ sends $A$ to
$u^{\prime,*}Au\colon(\tilde{C},\tilde{\rho},\tilde{\mu})\to(\tilde{C}^{\prime},\tilde{\rho}^{\prime},\tilde{\mu}^{\prime})$.
One checks that this is a morphism in
$\mathbf{\bar{C}}_{\mathrm{lf}}^{H,\mathrm{ctr}}(X)$ and that this
construction is compatible with the involution and composition. Here it is
important that $i(X)$ is a union of coarse components of $G\times_{H}X$ and
therefore $A$ does not “propagate” between $i(X)$ and its complement.

We now argue that $\phi_{X}$ is a unitary equivalence. We first see using the
$G$-equivariance of morphisms in
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}({\mathtt{Ind}}_{H}^{G}(X))$
and the fact that the $G$-translates of $i(X)$ form a coarsely disjoint
partition of ${\mathtt{Ind}}_{H}^{G}(X)$ that $\phi_{X}$ is bijective on
morphism spaces. This immediately implies that $\phi_{X}$ is fully faithful.
It remains to argue that $\phi_{X}$ is essentially surjective. Let
$(\tilde{C},\tilde{\rho},\tilde{\mu})$ in
$\mathbf{\bar{C}}_{\mathrm{lf}}^{H,\mathrm{ctr}}(X)$ be given. We choose a
section $s\colon H\backslash G\to G$ of the projection $G\to H\backslash G$
such that $s([e])=e$. Then we choose a sum $(C,(e_{[k]})_{[k]\in H\backslash
G})$ of the family of objects $(s(k)\tilde{C})_{[k]\in H\backslash G}$ in
${\mathbf{C}}$. It is at this point where we use the assumption that
${\mathbf{C}}$ is $|H\backslash G|$-additive. We then observe that there exist
a unique extension of the $H$-cocycle $\tilde{\rho}$ on $\tilde{C}$ to a
$G$-cocycle $\rho$ on $C$ such that
$he^{*}_{[e]}\rho_{h}e_{[e]}=\tilde{\rho}_{h}$ for all $h$ in $H$.
Furthermore, there exists a unique extension of $\tilde{\mu}$ to a
$G$-equivariant $\mathrm{Proj}(C)$-valued measure on $G\times_{H}X$ such that
for any subset $Y$ of $X$ we have
$\tilde{\mu}(Y)=e_{[e]}^{*}\mu(i(Y))e_{[e]}$. Applying $\phi_{X}$ to the
object $(C,\rho,\mu)$ of
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}({\mathtt{Ind}}_{H}^{G}(X))$
we get an object which is unitarily isomorphic to the original object
$(\tilde{C},\tilde{\rho},\tilde{\mu})$.

Let $f\colon X\to X^{\prime}$ be a morphism in $H\mathbf{BornCoarse}$, and let
$(C,\rho,\mu)$ be in
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}({\mathtt{Ind}}_{H}^{G}(X))$.
In the construction of $\phi_{X}$ we have chosen an image $(\tilde{C},u)$ of
$\mu(i(X))$. Similarly, in the construction of $\phi_{X^{\prime}}$ applied to
${\mathtt{Ind}}_{H}^{G}(f)_{*}(C,\rho,\mu)$ we have chosen an image
$(\tilde{C}^{\prime},u^{\prime})$ of the projection
${\mathtt{Ind}}_{H}^{G}(f)_{*}\mu(i^{\prime}(X^{\prime}))$ on $C$. We now note
that $\mu(i(X))={\mathtt{Ind}}_{H}^{G}(f)_{*}\mu(i^{\prime}(X^{\prime}))$.
Conseqently, there exists a unique unitary isomorphism
$v_{f,(C,\rho,\mu)}\colon\tilde{C}\to\tilde{C}^{\prime}$ such that
$v_{f,(C,\rho,\mu)}u=u^{\prime}$. The family
$v_{f}\coloneqq(v_{f,(C,\rho,\mu)})_{(C,\rho,\mu)\in\mathrm{Ob}(\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}({\mathtt{Ind}}_{H}^{G}(X)))}$
is a unitary isomorphism of functors $v_{f}\colon
f_{*}\phi_{X}\to\phi_{X^{\prime}}{\mathtt{Ind}}_{H}^{G}(f)_{*}$ filling the
diagram

$\textstyle{\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}({\mathtt{Ind}}_{H}^{G}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Rightarrow,v(f)}$$\scriptstyle{{\mathtt{Ind}}_{H}^{G}(f)_{*}}$$\scriptstyle{\phi_{X}}$$\textstyle{\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}({\mathtt{Ind}}_{H}^{G}(X^{\prime}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{X^{\prime}}}$$\textstyle{\mathbf{\bar{C}}_{\mathrm{lf}}^{H,\mathrm{ctr}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{*}}$$\textstyle{\mathbf{\bar{C}}_{\mathrm{lf}}^{H,\mathrm{ctr}}(X^{\prime})}$

If $f^{\prime}\colon X^{\prime}\to X^{\prime\prime}$ is a second morphism in
$H\mathbf{BornCoarse}$, then using the uniqueness of the choices of the
unitary isomorphisms $v_{f,(C,\rho,\mu)}$, etc. one checks that

$(v_{f^{\prime}}\circ{\mathtt{Ind}}_{H}^{G}(f)_{*})(f^{\prime}_{*}\circ
v_{f})=v_{f^{\prime}f}\,.\qed$

######  Remark 10.3.

One could explicitly strictify the functor $\phi$ to a natural transformation
of $C^{*}\mathbf{Cat}$-valued functors as follows. In a first step one chooses
for every object $C$ in ${\mathbf{C}}$ and every projection $p$ on $C$ a
representative $(\tilde{C},u)$ of the image of $p$. In the construction of
$\phi_{X}$ we then use this choice for the mage of $\mu(i(X))$.

If we proceed in this way, then the transformations $v_{f}$ for morphisms $f$
in $H\mathbf{BornCoarse}$ all become identities. ∎

Let

$\mathtt{Res}^{G}_{H}\colon G\mathbf{BornCoarse}\to H\mathbf{BornCoarse}$

be the functor which sends a $G$-bornological coarse space to the
$H$-bornological coarse space obtained by restriction of the action from $G$
to $H$. We have a canonical natural transformation

$\psi\colon\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}\to\mathbf{\bar{C}}_{\mathrm{lf}}^{H,\mathrm{ctr}}\circ\mathtt{Res}^{G}_{H}$
(10.2)

of functors from $G\mathbf{BornCoarse}$ to $C^{*}\mathbf{Cat}$. It evaluation
$\psi_{X}$ on $X$ in $G\mathbf{BornCoarse}$ is given as follows:

  1. 1.

objects: The functor $\psi_{X}$ sends the object $(C,\rho,\mu)$ of
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X)$ with
$\rho=(\rho_{g})_{g\in G}$ to the object $(C,\rho_{|H},\mu)$ of
$\mathbf{\bar{C}}_{\mathrm{lf}}^{H,\mathrm{ctr}}(\mathtt{Res}^{G}_{H}(X))$,
where $\rho_{|H}\coloneqq(\rho_{h})_{h\in H}$.

  2. 2.

morphisms: The functor $\psi_{X}$ is the canonical inclusion of $G$-invariant
into $H$-invariant morphisms.

Assume that $X$ is a $G\mathbf{BornCoarse}$. Then we have an isomorphism

$c_{X}\colon{\mathtt{Ind}}_{H}^{G}(\mathtt{Res}^{G}_{H}(X))\to(G/H)_{min,min}\otimes
X$

in $G\mathbf{BornCoarse}$ given by the map of underlying sets

$G\times_{H}X\to G/H\times X\,,\quad[g,x]\mapsto(g,gx)\,.$

The family $c=(c_{X})_{X\in G\mathbf{BornCoarse}}$ is a natural isomorphism of
functors

${\mathtt{Ind}}_{H}^{G}\circ\mathtt{Res}^{G}_{H}\to(G/H)_{min,min}\otimes-\colon
G\mathbf{BornCoarse}\to G\mathbf{BornCoarse}\,.$

We note that the projection $p_{X}\colon(G/H)_{min,min}\otimes X\to X$ is a
bounded covering (Definition 9.5). The family $p\colon(p_{X})_{X\in
G\mathbf{BornCoarse}}$ is a natural transformation of functors

$p\colon(G/H)_{min,min}\otimes\iota\to\iota$

where $\iota\colon G\mathbf{BornCoarse}\to
G\mathbf{BornCoarse}_{{\mathtt{tr}}}$ is as in (9.1). We now assume that
$\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}}$ belongs to
${\mathbf{Fun}}(BG,\mathbf{P}(C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\mathrm{Idem},\infty\mathrm{add}})$.
Then the pull-back $p_{X}^{*}$ is defined by Theorem 9.2. We consider the
composition

$\displaystyle\mathclap{\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X)\xrightarrow{p_{X}^{*}}\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}((G/H)_{min,min}\otimes
X)\stackrel{{\scriptstyle
c_{X,*}^{-1}}}{{\cong}}\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}({\mathtt{Ind}}_{H}^{G}(\mathtt{Res}^{G}_{H}(X)))\stackrel{{\scriptstyle\phi_{\mathtt{Res}(X)}}}{{\simeq}}\mathbf{\bar{C}}_{\mathrm{lf}}^{H,\mathrm{ctr}}(\mathtt{Res}^{G}_{H}(X))}$

which we will denote by $\tilde{\psi}_{X}$. The family
$\tilde{\psi}\coloneqq(\tilde{\psi}_{X})_{X\in G\mathbf{BornCoarse}}$ is a
natural transformation

$\tilde{\psi}\colon\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}\to\mathbf{\bar{C}}_{\mathrm{lf}}^{H,\mathrm{ctr}}\circ\mathtt{Res}^{G}_{H}$
(10.3)

of functors from $G\mathbf{BornCoarse}$ to $C^{*}\mathbf{Cat}_{2,1}$. We must
again consider values in the $(2,1)$-category ${\mathbf{Cat}}_{2,1}$ since the
naturality of $\tilde{\psi}$ is only up to canonical unitary isomorphisms
instead of equalities. Even if we would strictify $\phi$ as in Remark 10.3 we
would still get non-trivial fillers of the naturality squares from the
$2$-categorial nature of $p$.

The following proposition compares $\psi$ in (10.2) with $\tilde{\psi}$ in
(10.3).

######  Proposition 10.4.

There exists a unitary isomorphism $u\colon\psi\to\tilde{\psi}$ of natural
transformations from $\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}$ to
$\mathbf{\bar{C}}_{\mathrm{lf}}^{H,\mathrm{ctr}}\circ\mathtt{Res}^{G}_{H}$.

###### Proof.

For $X$ in $G\mathbf{BornCoarse}$ we must construct a unitary isomorphism

$u_{X}\colon\psi_{X}\to\tilde{\psi}_{X}\,.$

Let $(C,\rho,\mu)$ be in $\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X)$.
Then we must construct an isomorphism

$u_{X,(C,\rho,\mu)}\colon\psi_{X}(C,\rho,\mu)\to\tilde{\psi}_{X}(C,\rho,\mu)\,.$

In order to avoid lengthy formulas we explain the object
$\tilde{\psi}_{X}(C,\rho,\mu)$ in words. In the first step of the construction
of $p_{X}^{*}$ in the proof of Theorem 9.2 for every coarse component $Y$ of
$X$ we have chosen an image $(C(Y),u_{Y})$ of $\mu(Y)$. Then $C$ is
canonically unitarily isomorphic to the orthogonal sum of the family
$(C(Y))_{Y\in\pi_{0}(X)}$. The underlying object of $p_{X}^{*}(C,\rho,\mu)$ is
given by an orthogonal sum over the objects $C(p_{X}(Z))$ indexed by the set
of coarse components $Z$ of $(G/H)_{min,min}\otimes X$. The subsum over the
components in $[e]\times X$ is canonically unitarily isomorphic to $C$. Since
$c_{X}^{-1}([e]\times X)=i(X)$ we conclude using the details of the
construction of $\phi_{X}$ in the proof of Proposition 10.1 that the
underlying object of $\tilde{\psi}_{X}(C,\rho,\mu)$ is canonically unitarily
isomorphic to $C$. We then check that this isomorphism is compatible with the
cocycles and measures and yields the desired unitary isomorphism
$u_{X,(C,\rho,\mu)}$.

In all cases above the canonical unitary isomorphisms are determined uniquely
by their compatibilty with the structure maps of the chosen images of
projections.

One then checks that the family
$u_{X}\coloneqq(u_{X,(C,\rho,\mu)})_{(C,\rho,\mu)\in\mathrm{Ob}(\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}})}$
is a unitary isomorphism of functors.

Let $f\colon X\to X^{\prime}$ be a morphism in $G\mathbf{BornCoarse}$. Then
from the $2$-categorial naturality of $(p_{X}^{*})$ and $\phi$ we get a
unitary isomorphism

$v_{f}\colon\mathtt{Res}^{G}_{H}(f)_{*}\tilde{\psi}_{X}\to\tilde{\psi}_{X^{\prime}}f_{*}\,.$

Then one finally checks, using the uniqueness of the unitary isomorphisms
going into the constructions of $u_{X}$ and $v_{f}$, that the composition

$\textstyle{\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{X}}$$\scriptstyle{\tilde{\phi}_{X}}$$\scriptstyle{f_{*}}$$\textstyle{\Downarrow
u_{X}}$$\textstyle{\mathbf{\bar{C}}_{\mathrm{lf}}^{H,\mathrm{ctr}}(\mathtt{Res}^{G}_{H}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathtt{Res}^{G}_{H}(f)_{*}}$$\textstyle{\Downarrow
v_{f}}$$\textstyle{\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\phi}_{X^{\prime}}}$$\scriptstyle{\phi_{X^{\prime}}}$$\textstyle{\Downarrow
u_{X^{\prime}}^{-1}}$$\textstyle{\mathbf{\bar{C}}_{\mathrm{lf}}^{H,\mathrm{ctr}}(\mathtt{Res}^{G}_{H}(X^{\prime}))}$

of isomorphisms of functors is the identity. ∎

Let $\mathrm{Hg}\colon C^{*}\mathbf{Cat}^{\mathrm{nu}}\to\mathbf{M}$ be a
finitary homological functor. Assume that
$\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}}$ belongs to
${\mathbf{Fun}}(BG,\mathbf{P}(C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\mathrm{Idem},\infty\mathrm{add}})$.

######  Corollary 10.5.

  1. 1.

We have a natural equivalence
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{c}^{G}\circ{\mathtt{Ind}}_{H}^{G}\to\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}^{H}_{c}$
of $H$-equivariant coarse homology theories.

  2. 2.

We have a natural transformation
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}^{G}_{c}\to\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{c}^{H}\circ\mathtt{Res}^{G}_{H}$
of $G$-equivariant coarse homology theories induced by $\psi$ in (10.2).

###### Proof.

We use Theorem 6.3 in order to conclude that
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{c}^{G}$ and
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{c}^{H}$ are $G$\- and $H$-equivariant
coarse homology theories, respectively. The composition
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}_{c}^{G}\circ{\mathtt{Ind}}_{H}^{G}$ is
an $H$-equivariant coarse homology theory by [BEKW20d, Lem. 4.19] (which
settles the universal case). The natural equivalence in Assertion 1 is
obtained by applying the extension ${\mathrm{Hg}}_{1,2}$ (see (9.11)) to the
transformation $\phi$ obtained in Proposition 10.1. ∎

######  Remark 10.6.

Assume that $E^{G}\colon G\mathbf{BornCoarse}\to\mathbf{M}$ is a
$G$-equivariant coarse homology theory. Then by [BEKW20d, Lem. 4.19] we can
define an $H$-equivariant coarse homology theory by
$E^{G}\circ{\mathtt{Ind}}_{H}^{G}:H\mathbf{BornCoarse}\to\mathbf{M}$.

If $E^{G}$ admits transfers, then we can furthermore define a natural
transformation

$E^{G}\to E^{G}\circ{\mathtt{Ind}}_{H}^{G}\circ\mathtt{Res}^{G}_{H}$ (10.4)

as the composition

$E^{G}\xrightarrow{p^{*}\circ
c_{*}^{-1}}E^{G}\circ{\mathtt{Ind}}^{G}_{H}\circ\mathtt{Res}^{G}_{H}\,.$

For $\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}}$ in
${\mathbf{Fun}}(BG,\mathbf{P}(C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\mathrm{Idem},\infty\mathrm{add}})$
we have the two equivariant coarse homology theories
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}^{G}_{c}$ and
$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}^{H}_{c}$ which are defined independently
from each other. By Corollary 10.5.1 we know that the transformation $\phi$
provides an equivalence of $H$-equivariant coarse homology theories

$\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}^{G}_{c}\circ{\mathtt{Ind}}_{H}^{G}\xrightarrow{\mathrm{Hg}(\phi),\simeq}\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}^{H}_{c}\,.$
(10.5)

Consequently, we can consider the natural transformation

$r\colon\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}^{G}_{c}\xrightarrow{\eqref{3rfjoifqwefqwfqwf}}\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}^{G}_{c}\circ{\mathtt{Ind}}^{G}_{H}\circ\mathtt{Res}^{G}_{H}\stackrel{{\scriptstyle\eqref{ewfqwfewfwqfqwef4}}}{{\simeq}}\mathrm{Hg}{\mathbf{C}}{\mathcal{X}}^{H}_{c}\circ\mathtt{Res}^{G}_{H}\,.$
(10.6)

The combination of Proposition 10.4 and Corollary 10.5.2 then says that the
abstract transformation (10.6) is induced by the obvious inclusion of
$C^{*}$-categories $\psi$ in (10.2). This fact will be further employed in
[BL]. ∎

##  11 Strong additivity of $K{\mathbf{C}}{\mathcal{X}}_{c}^{G}$

Recall from Definition 2.36 the notion of a strongly additive functor on
$G\mathbf{BornCoarse}$. Recall that the topological $K$-theory functor (see
[BEa, Def. 12.3])

$\mathrm{K^{C^{*}Cat}}\colon C^{*}\mathbf{Cat}^{\mathrm{nu}}\to\mathbf{Sp}$

is a finitary homological functor [BEa, Thm. 12.4] (see [BEa, Def. 11.5 &
11.6] for definitions). For the purpose of this section its main additional
property is [BEa, Thm. 13.7] stating that $\mathrm{K^{C^{*}Cat}}$ preserves
arbitrary very small products of finitely additive $C^{*}$-categories.

Assume that $\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}}$ is in
${\mathbf{Fun}}(BG,\mathbf{P}(C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\mathrm{Idem},\infty\mathrm{add}})$.
By Theorem 6.3 we have a continuous equivariant coarse homology theory

$\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}_{c}^{G}\colon
G\mathbf{BornCoarse}\to\mathbf{Sp}\,.$

######  Theorem 11.1.

The functor $\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}_{c}^{G}$ is
strongly additive.

The proof of Theorem 11.1 will be given further below. We first start with the
corresponding additivity statement for the functor
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}$.

Let $(X_{i})_{i\in I}$ be a family in $G\mathbf{BornCoarse}$. Recall
Definition 2.35 of the free union $\bigsqcup^{\mathrm{free}}_{i\in I}X_{i}$ of
this family. For $i$ in $I$ let $f_{i}\colon
X_{i}\to\bigsqcup^{\mathrm{free}}_{i\in I}X_{i}$ denote the inclusion. In the
formulation of the following lemma the symbol $0$ stands for a functor which
sends every object to a zero object.

######  Lemma 11.2.

There exists a unitary equivalence

$\phi\colon\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}\big{(}\bigsqcup^{\mathrm{free}}_{i\in
I}X_{i}\big{)}\to\prod_{i\in
I}\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{i})$

in $C^{*}\mathbf{Cat}$ such that for all $i,j$ in $I$ we have a unitary
isomorphism

$\mathrm{pr}_{j}\circ\phi\circ
f_{i,*}\cong\begin{cases}{\mathtt{id}}_{\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{i})}&\text{if
}i=j\,,\\\ 0&\text{if }j\not=i\,.\end{cases}$ (11.1)

###### Proof.

We will construct functors

$\phi\colon\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}\big{(}\bigsqcup^{\mathrm{free}}_{i\in
I}X_{i}\big{)}\to\prod_{i\in
I}\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{i})\,,\quad\psi\colon\prod_{i\in
I}\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{i})\to\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}\big{(}\bigsqcup^{\mathrm{free}}_{i\in
I}X_{i}\big{)}$

and then show that they are inverse to each other up to unitary isomorphism.

We start with the construction of $\phi$.

  1. 1.

objects: Let $(C,\rho,\mu)$ be an object of
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(\bigsqcup^{\mathrm{free}}_{i\in
I}X_{i})$. Using that ${\mathbf{C}}$ is idempotent complete, for every $i$ in
$I$ we choose an image $(C_{i},u_{i})$ of the projection $\mu(X_{i})$. We
further define $\rho_{i}=(\rho_{g,X_{i}})_{g\in G}$ as in (9.5), where
$\rho_{g,X_{i}}\colon C_{i}\to gC_{i}$ (since $X_{i}$ is $G$-invariant), and
$\mu_{i}\coloneqq\mu_{X_{i}}$ by (9.6). In this way we get an object

$\phi(C,\rho,\mu)\coloneqq((C_{i},\rho_{i},\mu_{i}))_{i\in I}$

of $\prod_{i\in I}\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{i})$.

  2. 2.

morphisms: Let $A\colon(C,\rho,\mu)\to(C^{\prime},\rho^{\prime},\mu^{\prime})$
be a morphism in
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(\bigsqcup^{\mathrm{free}}_{i\in
I}X_{i})$. Note that the subsets $X_{i}$ of the free union are mutually
coarsely disjoint. Therefore $A$ does not propagate between different
components. For every $i$ in $I$ we consider the restriction $A_{i}\coloneqq
u_{i}^{*}Au_{i}$ of $A$ to $C_{i}$. Then we have $\sup_{i\in
I}\|A_{i}\|=\|A\|$. We define the morphism $\phi(A)$ in $\prod_{i\in
I}\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{i})$ by

$\phi(A)\coloneqq(A_{i})_{i\in I}\colon((C_{i},\rho_{i},\mu_{i}))_{i\in
I}\to((C^{\prime}_{i},\rho^{\prime}_{i},\mu^{\prime}_{i}))_{i\in I}\,.$

One checks in a straightforward manner that this construction defines a
functor $\phi$ in $C^{*}\mathbf{Cat}$. Note that it depends on the choices
made for the images of the projection $\mu$.

We now define the functor $\psi$.

  1. 1.

objects: Let $((C_{i},\rho_{i},\mu_{i}))_{i\in I}$ be an object of
$\prod_{i\in I}\mathbf{\bar{C}}_{\mathrm{sm}}^{G,\mathrm{ctr}}(X_{i})$. Since
${\mathbf{C}}$ admits all very small sums we can choose a sum
$(\bigoplus_{i\in I}C_{i},(e_{i})_{i\in I})$. Using [BEa, Ex. 4.16] we define

$\psi(((C_{i},\rho_{i},\mu_{i}))_{i\in I})\coloneqq(\bigoplus_{i\in
I}C_{i},\oplus_{i\in I}\rho_{i},\oplus_{i\in I}\mu_{i})\,.$

One checks again in a straightforward manner that this object is well-defined.

  2. 2.

morphisms: Let

$(A_{i})_{i\in I}\colon((C_{i},\rho_{i},\mu_{i}))_{i\in
I}\to((C^{\prime}_{i},\rho^{\prime}_{i},\mu^{\prime}_{i}))_{i\in I}$

be a morphism in $\prod_{i\in
I}\mathbf{\bar{C}}_{\mathrm{sm}}^{G,\mathrm{ctr}}(X_{i})$. Then $\sup_{i\in
I}\|A_{i}\|<\infty$ and we use [BEa, Ex. 4.16] to define a morphism

$\oplus_{i\in I}A_{i}\colon\bigoplus_{i\in I}C_{i}\to\bigoplus_{i\in
I}C^{\prime}_{i}$

in ${\mathbf{C}}$. One checks that it is compatible with the cocycles, i.e.,
$\oplus_{i\in I}A_{i}$ is a morphism in
${\mathbf{C}}^{G}(\bigsqcup^{\mathrm{free}}_{i\in I}X_{i})$. It remains to
show that it is a morphism in the Roe category. Fix $\varepsilon$ in
$(0,\infty)$. Then for every $i$ in $I$ there exists an entourage $U_{i}$ of
$X_{i}$ and a $U_{i}$-controlled operator
$A_{i}^{\prime}\colon(C_{i},\rho_{i},\mu_{i})\to(C^{\prime}_{i},\rho_{i}^{\prime},\mu_{i}^{\prime})$
in $\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{i})$ such that
$\|A_{i}-A_{i}^{\prime}\|\leq\varepsilon$. Then $\oplus_{i\in
I}A_{i}^{\prime}$ is in
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(\bigsqcup^{\mathrm{free}}_{i\in
I}X_{i})$. It is controlled by the entourage $\bigcup_{i\in I}U_{i}$ in the
coarse structure of the free union, and we have $\|\oplus_{i\in
I}A_{i}-\oplus_{i\in I}A_{i}^{\prime}\|\leq\varepsilon$. Since we can choose
$\varepsilon$ arbitrary small we conclude that $\oplus_{i\in I}A_{i}$ is a
morphism in
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(\bigsqcup^{\mathrm{free}}_{i\in
I}X_{i})$.

One checks in a straightforward manner that this construction defines a
functor $\psi$ in $C^{*}\mathbf{Cat}$. It depends on the choices involved in
the sums.

We now show that $\phi$ and $\psi$ are mutually inverse to each other.

  1. 1.

The unitary $u\colon\phi\circ\psi\xrightarrow{\cong}{\mathtt{id}}$: Let
$((C_{i},\rho_{i},\mu_{i}))_{i\in I}$ be an object of $\prod_{i\in
I}\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{i})$. For every $j$ in
$I$ we choose an image $((\bigoplus_{i\in I}C_{i})_{j},u^{\prime}_{j})$ of
$(\oplus_{i\in I}\mu_{i})(X_{j})$ and define

$a_{j}\coloneqq e_{j}^{*}u^{\prime}_{j}\colon(\bigoplus_{i\in I}C_{i})_{j}\to
C_{j}\,.$

We get a unitary isomorphism

$u_{((C_{i},\rho_{i},\mu_{i}))_{i\in I}}\coloneqq(a_{j})_{j\in
I}\colon((\bigoplus_{i\in I}C_{i})_{j})_{j\in I}\to(C_{j})_{j\in I}\,.$

The family $(u_{((C_{i},\rho_{i},\mu_{i}))_{i\in
I}})_{((C_{i},\rho_{i},\mu_{i}))_{i\in I}\in\mathrm{Ob}(\prod_{i\in
I}\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{i}))}$ is the desired
unitary isomorphism of functors.

  2. 2.

The unitary
$v\colon{\mathtt{id}}\stackrel{{\scriptstyle\cong}}{{\to}}\psi\circ\phi$: Let
$(C,\rho,\mu)$ be an object of
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(\bigsqcup^{\mathrm{free}}_{i\in
I}X_{i})$. By using [BEa, Cor. 4.1] we can define a morphism

$v_{(C,\rho,\mu)}\coloneqq\sum_{i\in I}u_{i}e_{i}^{*}\colon\bigoplus_{i\in
I}C_{i}\to C\,.$

We now use Lemma 3.13 in order to conclude that $\oplus_{i\in I}v_{i}$ is an
unitary isomorphism. Then the family
$(v_{(C,\rho,\mu)})_{(C,\rho,\mu)\in\mathrm{Ob}(\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(\bigsqcup^{\mathrm{free}}_{i\in
I}X_{i}))}$ is the desired unitary isomorphism of functors.

We now show that $\phi$ has the asserted property (11.1). Let $i$ be in $I$.
We construct the unitary isomorphism

$w\colon\mathrm{pr}_{i}\circ\phi\circ
f_{i,*}\xrightarrow{\cong}{\mathtt{id}}_{\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{i})}$

as follows. Let $(C,\rho,\mu)$ be an object in
$\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{i})$. Then
$u_{i}\colon(f_{*}C)_{i}\to C$ is an isomorphism, where $(f_{*}C)_{i}$ denotes
the choice of the image of $(f_{i,*}\mu)(X_{i})$ involved in the construction
of $\phi$. We set $w_{(C,\rho,\mu)}\coloneqq u_{i}$. The family
$(w_{(C,\rho,\mu)})_{(C,\rho,\mu)\in\mathrm{Ob}(\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{i}))}$
is the desired unitary isomorphism.

If $i,j$ are in $I$ and $i\not=j$, then $(f_{i,*}C)_{j}=0$. Consequently,
$\mathrm{pr}_{j}\circ\phi\circ f_{i,*}\cong 0$. ∎

###### Proof of Theorem 11.1.

Since $\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}$ is excisive, it
is $\pi_{0}$-excisive (Definition 2.33). Let $(X_{i})_{i\in I}$ be a family in
$G\mathbf{BornCoarse}$. For $j$ in $J$ we consider the invariant coarsely
disjoint partition $(X_{j},\bigcup_{i\in I\setminus\\{j\\}}X_{i})$ of
$\bigsqcup^{\mathrm{free}}_{i\in I}X_{i}$. Since
$\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}$ is $\pi_{0}$-excisive
we get a decomposition

$\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}\big{(}\bigsqcup^{\mathrm{free}}_{i\in
I}X_{i}\big{)}\simeq\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}(X_{j})\oplus\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}\big{(}\bigsqcup^{\mathrm{free}}_{i\in
I\setminus\\{j\\}}X_{i}\big{)}\,.$

We let

$p_{j}\colon\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}\big{(}\bigsqcup^{\mathrm{free}}_{i\in
I}X_{i}\big{)}\to\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}(X_{j})$

be the corresponding projection onto the first summand. We must show that

$(p_{j})_{j\in
J}\colon\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}\big{(}\bigsqcup^{\mathrm{free}}_{i\in
I}X_{i}\big{)}\to\prod_{j\in
I}\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}(X_{j})$

is an equivalence. We consider the diagram

$\textstyle{\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}(\bigsqcup^{\mathrm{free}}_{i\in
I}X_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(p_{i})_{i\in
J}}$defn.$\textstyle{\prod_{i\in
I}\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}(X_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$defn.$\textstyle{\mathrm{K^{C^{*}Cat}}(\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(\bigsqcup^{\mathrm{free}}_{i\in
I}X_{i}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\simeq,\mathrm{K^{C^{*}Cat}}(\phi)}$Lemma
11.2 $\textstyle{\prod_{i\in
I}\mathrm{K^{C^{*}Cat}}(\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{i}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{pr}_{k}}$$\textstyle{\mathrm{K^{C^{*}Cat}}(\prod_{j\in
I}\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{j}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\mathrm{K^{C^{*}Cat}}(\mathrm{pr}_{k})}$$\scriptstyle{\simeq}$$\scriptstyle{!}$$\textstyle{\mathrm{K^{C^{*}Cat}}(\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{j}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{K^{C^{*}Cat}}(f_{j,*})}$$\scriptstyle{\delta_{jk}{\mathtt{id}}}$$\textstyle{\mathrm{K^{C^{*}Cat}}(\mathbf{\bar{C}}_{\mathrm{lf}}^{G,\mathrm{ctr}}(X_{k}))}$

where the marked equivalence together with commutativity of the right triangle
for every $k$ in $I$ are stated in [BEa, Thm. 13.7], and commutativity of the
lower left square is asserted by (11.1). The big outer square commutes for
every $j,k$ in $J$ in view of the construction of the map $p_{k}$. This
implies that the middle pentagon commutes as well. This finally implies the
assertion of the theorem. ∎

##  12 CP-functors

Let $G$ be a group and let $G\mathbf{Orb}$ denote its orbit category. In this
section we describe a functor $G\mathbf{Orb}\to\mathbf{Sp}$ associated to a
pair $\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}}$ in
${\mathbf{Fun}}(BG,\mathbf{P}(C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\mathrm{Idem},\infty\mathrm{add}})$.
The main theorem states that this functor is a CP-functor.

Recall the functor

${}_{{\mathbf{K}}}{\mathbf{C}}_{\mathrm{lf}}[-]\rtimes_{r}G\colon
G{\mathbf{Set}}\to C^{*}\mathbf{Cat}$

from (8.6). We can consider the functor

$\mathrm{K^{C^{*}Cat}}({}_{{\mathbf{K}}}{\mathbf{C}}_{\mathrm{lf}}[-]\rtimes_{r}G)\colon
G{\mathbf{Set}}\to\mathbf{Sp}\,.$ (12.1)

######  Definition 12.1.

We define $\mathrm{K^{C^{*}Cat}}{\mathbf{C}}_{G,r}\colon
G\mathbf{Orb}\to\mathbf{Sp}$ to be the restriction of (12.1) to the orbit
category.

Before we can state the main theorem about
$\mathrm{K^{C^{*}Cat}}{\mathbf{C}}_{G,r}$ we must recall the notion of a CP-
functor. We have the functor

$i\colon G\mathbf{Orb}\to G\mathbf{BornCoarse}\,,\quad S\mapsto S_{min,max}$

(see Example 2.12). We consider a functor

$M\colon G\mathbf{Orb}\to\mathbf{M}\,.$

######  Definition 12.2 ([BEKW20d, Defn. 1.8]).

$M$ is a CP-functor if

  1. 1.

$\mathbf{M}$ is stable, admits all very small colimits and limits, and is
compactly generated.

  2. 2.

There exists an equivariant coarse homology theory $E\colon
G\mathbf{BornCoarse}\to\mathbf{M}$ (see Definition 2.37) such that:

    1. (a)

$M$ is equivalent to $E_{G_{can,min}}\circ i$ (see Examples 2.38 & 2.13).

    2. (b)

      1. i.

$E$ is continuous (Definition 2.31).

      2. ii.

$E$ is strongly additive (Definition 2.36).

      3. iii.

$E$ admits transfers (Definition 9.1).

######  Theorem 12.3.

Assume that $\tilde{\mathbf{K}}\subseteq\tilde{\mathbf{C}}$ is in
${\mathbf{Fun}}(BG,\mathbf{P}(C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\mathrm{Idem},\infty\mathrm{add}})$.
Then the functor $\mathrm{K^{C^{*}Cat}}{\mathbf{C}}_{G,r}$ is a CP-functor.

###### Proof.

The target category of the functor $\mathrm{K^{C^{*}Cat}}{\mathbf{C}}_{G,r}$
is the large category $\mathbf{Sp}$ of small spectra. This category is stable,
compactly generated and admits all small colimits and limits. This verifies
Condition 12.2.1.

For $E$ in Definition 12.2 we take the equivariant coarse homology theory
$\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}$ obtained by
specializing Definition 6.1.2 to the case $\mathrm{Hg}=\mathrm{K^{C^{*}Cat}}$.

By applying $\mathrm{K^{C^{*}Cat}}$ to the equivalence stated in Proposition
8.2.4 we obtain an equivalence

$\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c,G_{can,{min}}}\circ
i\simeq\mathrm{K^{C^{*}Cat}}{\mathbf{C}}_{G,r}\,.$

This verifies Condition 12.2.2a.

The coarse homology theory
$\mathrm{K^{C^{*}Cat}}{\mathbf{C}}{\mathcal{X}}^{G}_{c}$ has the properties
required in 12.2.2b:

  1. 1.

It is continuous by Theorem 6.3.

  2. 2.

It is strongly additive by Theorem 11.1.

  3. 3.

It admits transfers by Theorem 9.7 which is applicable for
$\mathrm{Hg}=\mathrm{K^{C^{*}Cat}}$ since $\mathrm{K^{C^{*}Cat}}$, being a
homological functor, sends unitary equivalences to equivalences.∎

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