Injectivity results for coarse homology theories

We show injectivity results for assembly maps using equivariant coarse homology theories with transfers. Our method is based on the descent principle and applies to a large class of linear groups or, more generally, groups with finite decomposition complexity.


Introduction
For a group G, we consider a functor M : GOrb → C from the orbit category of G to a cocomplete ∞-category C. Often one is interested in the calculation of the object colim GOrb M in C, or equivalently, in the value M ( * ) at the final object * of GOrb. Given a family of subgroups F of G, one can then ask which information about this colimit can be obtained from the restriction of M to the subcategory G F Orb of orbits with stabilizers in F. To this end, one considers the assembly map If M is algebraic or topological K-theory, then such assembly maps appear in the Farrell-Jones or Baum-Connes conjectures; see, for example, Lück and Reich [27] and Bartels [2]. In the present paper, we show split injectivity results about the assembly map by proving a descent principle. This method was first applied by Carlsson and Pederson [15]. For the application of the descent principle, on the one hand, we will use geometric properties of the group G like finite decomposition complexity as introduced by Guentner et al. [17,18]. On the other hand, we use that M extends to an equivariant coarse homology theory with transfers as introduced in [11]. The main theorem of the paper is Theorem 1.11.
We now start by introducing the notation which is necessary to state the theorem and its assumptions in detail. Let G be a group and F be a set of subgroups of G. The ∞-category of spaces will be denoted by Spc. For any small ∞-category C (ordinary categories are considered as ∞-categories using the nerve), we use the notation PSh(C) := Fun(C op , Spc) for the ∞-category of Spc-valued presheaves. Definition 1.3. We denote by E F G the object of the presheaf category PSh(GOrb), which is essentially uniquely determined by In [12,Definition 3.14], we defined the notion of G-equivariant finite decomposition complexity (G-FDC) for a G-coarse space (Definition 3.6). G-FDC is an equivariant version of the notion of finite decomposition complexity FDC which was originally introduced by Guentner et al. [18].
For S in GSet, we let S min denote the G-coarse space with underlying G-set S and the minimal coarse structure (see Example 3.8). In the definition below, ⊗ denotes the Cartesian product in the category GCoarse of G-coarse spaces.
Let F be a family of subgroups of G and X be a G-coarse space.
Definition 1.4. X has G F -equivariant finite decomposition complexity (abbreviated by G F -FDC) if S min ⊗ X has G-FDC for every S in G F Set.
We will consider the following families of subgroups. (iii) FDC denotes the family of subgroups V of G such that V can has V Fin -FDC. (iv) CP denotes the family of subgroups of G generated by those subgroups V such that E Fin V is a compact object of PSh(V Orb).
(v) FDC cp denotes the intersection of FDC and CP.
Remark 1.6. The notation V can in the definition of the family FDC refers to the group V with the canonical coarse structure described in Example 3. 8.
In order to see that FDC is a family of subgroups, we use that the condition that V can has V Fin -FDC is stable under taking subgroups, see Lemma 2.4. An object A of an ∞-category D is called compact if the functor Map(A, −): D → Spc commutes with filtered colimits. The word compact in the definition of CP is understood in this sense.
The family of subgroups of G generated by a set of subgroups of G is the smallest family containing this subset. The condition that E Fin V is compact is not stable under taking subgroups. Hence the family CP may also contain subgroups V with noncompact E Fin V . If F = Fin and F = All, then we omit the symbol All and call Asmbl Fin,M simply the assembly map.
In order to capture the large-scale geometry of metric spaces like G (with its word metric), we introduced the category of G-bornological coarse spaces GBornCoarse in [10,13]. We further defined the notion of an equivariant coarse homology theory. All this will be recalled in detail in section 3.
We can embed the orbit category GOrb into GBornCoarse by a functor i : GOrb → GBornCoarse, which sends a G-orbit S to the G-bornological coarse space S min, max ; see Example 3.8. Note that the convention is that the first index specifies the coarse structure while the second index specifies the bornology. We say that a functor M : GOrb → C can be extended to an equivariant coarse homology theory if there exists an equivariant coarse homology theory F : GBornCoarse → C such that M F • i. We will need various additional properties or structures for an equivariant coarse homology theory.
(1) The property of continuity of an equivariant coarse homology theory was defined in [13,Definition 5.15], see Lemma 3.19.
(2) The property of strong additivity of an equivariant coarse homology theory was defined in [13,Definition 3.12], see Remark 5.13. (3) The additional structure of transfers for an equivariant coarse homology theory is encoded in the notion of a coarse homology theory with transfers which was defined in [11], see Definition 5.5.
Let G can,min denote the G-bornological coarse space consisting of G with the canonical coarse and the minimal bornological structures; see Example 3.8. We furthermore consider a stable ∞-category C and an equivariant coarse homology theory (see Definition 3.13) E : GBornCoarse → C.
To E and G can,min we associate a new equivariant coarse homology theory E Gcan,min : GBornCoarse → C, X → E(G can,min ⊗ X) called the twist of E by G can,min ; see Definition 3.16.
We can now introduce the following assumption on a functor M : GOrb → C. Definition 1.8. We call M a CP-functor if it satisfies the following assumptions.
(i) C is stable, complete, cocomplete, and compactly generated. (ii) There exists an equivariant coarse homology theory E satisfying: (a) M is equivalent to E Gcan,min • i; (b) E is strongly additive; (c) E is continuous; (d) E extends to a coarse homology theory with transfers.
Remark 1.9. We call M a CP-functor since the above assumptions will allow us to apply methods similar to those from Carlsson and Pedersen [15].
(ii) For a group G, let P be the total space of a principal G-bundle and let A denote the functor of nonconnective A-theory (taking values in the ∞-category of spectra). Then P gives rise to a GOrb-spectrum A P sending a transitive G-set S to the spectrum A(P × G S). By [14,Theorem 5.17], A P is a CP-functor.
(iii) More generally, every right-exact ∞-category with G-action C gives rise to a functor KC G : GOrb → Sp. Taking C = Ch b (A) or C = Sp, this recovers KA G and A EG , but one may also consider categories of perfect modules over an arbitrary ring spectrum. Also in this generality, KC G is a CP-functor. See [9] for details and proofs.
We can now state the main theorem of this paper. Let G be a group and M : GOrb → C be a functor. Let F be a family of subgroups.
Theorem 1.11. Assume that M is a CP-functor (Definition 1.8). Furthermore, assume that one of the following conditions holds.
(i) F is a subfamily of FDC cp such that Fin ⊆ F. (ii) F is a subfamily of FDC such that Fin ⊆ F and G admits a finite-dimensional model for E top Fin G. Then the relative assembly map Asmbl F Fin,M admits a left inverse.
Remark 1.12. By Elmendorf's theorem, the homotopy theory of G-spaces is modeled by the presheaf category PSh(GOrb). More precisely, we have a functor Fix : GTop → PSh(GOrb), (1.2) which sends a G-topological space X to the Spc-valued presheaf which associates to S in GOrb the mapping space (Map GTop (S disc , X)). Here S disc is S considered as a discrete G-topological space, Map GTop (S disc , X) in Top is the topological space of equivariant maps from S disc to X, and : For algebraic K-theory (Example 1.10), Corollary 1.13 was first proven by Bartels [1]. Let G be a group and M : GOrb → C be a functor.
Corollary 1.14. Assume that: Then the assembly map Asmbl Fin,M admits a left inverse.
For algebraic K-theory (Example 1.10), this was first proven in [21].
Proof. The corollary follows from Case (ii) of Theorem 1.11.
As an application of Theorem 1.11, we also obtain the following new injectivity result for algebraic K-theory. Proof. By assumption, there exists n in N and an n-dimensional G-simplicial complex X modeling E top F G. Choose a set of representatives S for the G-orbits of vertices in X. Again by assumption, there exists for some k in N and every s in S an at most Now apply the construction of [33, Definition 2.2] to obtain a G-simplicial complex X[Y, υ 0 ] whose dimension is bounded by nk + n + k. After observing that this construction is compatible with taking fixed points in the sense that X[Y, Remark 1.17. Most of the groups for which the Farrell-Jones conjecture is known by now also have finite asymptotic dimension. But, for example, for CAT(0)-groups, which satisfy the Farrell-Jones conjecture [32], this is an open problem. Hence taking some P i to be CAT(0)groups that are not known to have FDC and some P i to be groups that have FDC but for which the Farrell-Jones conjecture is not known, we obtain examples of groups for which Theorem 1.15 applies and the split-injectivity was not known before.

Injectivity results for linear groups
In general it is not an easy task to verify the assumptions on the group G and the family F appearing in Theorem 1.11 and its corollaries. In this section, we provide various cases where the required properties can be shown. Furthermore, we show how Theorem 1.11 can be applied to linear groups.
For a family F of subgroups of G, we consider the G-coarse space S F ,min consisting of the G-set S F := H∈F G/H with the minimal coarse structure. Let X be a G-coarse space. The condition that X has G F -FDC is equivalent to the condition that S F ,min ⊗ X has G-FDC.
The space (G/H) min ⊗ X has G-FDC if and only if the space X has H-FDC. This can be seen by taking an H-equivariant decomposition of X and extending it G-equivariantly to (G/H) min ⊗ X. Hence morally, S F ,min ⊗ X has G-FDC if and only if X has H-equivariant FDC for every group H in the family F in a uniform way. More precisely, the condition that S F ,min ⊗ X has G-FDC is equivalent to the condition, formulated in [23], that the family {(X, H)} H∈F has FDC.
Applying this equivalence of conditions we can transfer the results from [23]. We consider the case X = G can and F = Fin. Then we see that Assumption (iii) of Corollary 1.14 is equivalent to the condition that the family {(G, H)} H∈Fin has FDC. In [23]  Combining Corollary 1.14 with Example 2.1 and Example 2.2, we obtain injectivity results for linear groups over commutative rings with unit and trivial nilradical and for subgroups of virtually connected Lie groups with a uniform upper bound on the cardinality of their finite subgroups. We will now extend these to recover the injectivity results from [22,23] for algebraic K-theory; see Corollary 2.11.
Before we start, we show that the family FDC is closed under subgroups. Let G be a group and let H be a subgroup of G. Let F be a family of subgroups of G.
(ii) The functor A P from Example 1.10 is also a hereditary CP-functor by [14,Theorem 5.17].
We will need the following well-known facts about the Hirsch length, for a proof see [ For g in G, we set F g := g −1 F g. For every h in H, we have the equalities that is, the map F \F gH → F g \F g H, F gh → F g h is an isometry. Hence we can use the covers for the spaces F g ∩ H\H as g varies to obtain for every F \G a cover U 0 ∪ . . . ∪ U n , such that for every i in {0, . . . , n} the subset U i is an R-disjoint union of subspaces of diameter at most S. This shows that {F \G } F ∈Fin(G ) has asymptotic dimension at most n. For g in G and F a finite subgroup of G , F \F gG is isometric to F g \G as above. Since G is normal in G, the group F g is again a finite subgroup of G . Therefore, every element of {F \G} F ∈Fin(G ) is a union of at most k subspaces isometric to elements of {F \G } F ∈Fin(G ) .
Hence also {F \G} F ∈Fin(G ) has asymptotic dimension at most n by the Finite Union Theorem of [7].
Every finite subgroup F of G has a normal subgroup F of index at most k contained in G . Then F \F acts isometrically on F \G with quotient F \G. Hence we can again apply [24,Corollary 1.2] to see that {F \G} F ∈Fin has asymptotic dimension at most n.
be an extension of countable groups and let S be a subgroup of S that is normal in G.
Then h(S/S ) h(S) = n and also for every finite subgroup F of Q, we have be an extension of groups. Denote by Fin(Q) the family of finite subgroups of Q. By φ −1 (Fin(Q)), we denote the family of subgroups of G whose image under φ belongs to Fin(Q). Let M : GOrb → C be a functor.
Theorem 2.9. Assume: Proof. We argue by induction on the derived length k of S. If k = 1, then S is virtually abelian and every group in φ −1 (Fin(Q)) is virtually abelian of Hirsch length at most n, too. Hence the statement follows from case (ii) of Theorem 1.11 since its assumptions are verified by Lemma 2.7 and Lemma 2.8 applied with S the trivial group. Now suppose that the statement holds for k and assume S has derived length k + 1. Note that [S, S] is normal in G and has derived length k. We set G := G/[S, S]. Then there is a finite-dimensional model for E top Fin(G ) G by Lemma 2.8. We consider the factorization of φ as of families of subgroups of G induce a factorization  For convenience, we repeat the arguments from [22,23] to obtain split-injectivity for finitely generated subgroups of linear groups and of virtually connected Lie groups with a finitedimensional classifying space.
Let M : GOrb → C be a functor. Then the assembly map Asmbl Fin,G is split injective.
Proof. Let G be a finitely generated subgroup of a virtually connected Lie group. The adjoint representation induces an extension with abelian kernel and quotient a finite index supergroup Q of a finitely generated subgroup of GL n (C). The group Q has Q Fin -FDC by Example 2.1. Since G admits a finite-dimensional model for E top Fin G, so does Q using the characterization from Example 2.2. By Corollary 1.14, the assembly map Asmbl Fin(Q),M •Res G Q is split-injective. This assembly map is equivalent to Asmbl p −1 (Fin(Q)),M , where p : G → Q is the projection. Because the kernel of p is abelian, the assembly map Asmbl is split-injective by Theorem 2.9. Now let G be a finitely generated subgroup of a linear group over a commutative ring R with unit. Let n be the nilradical of R. Then we have an extension where Q is a finitely generated subgroup of GL n (R/n). Arguing as above, the assembly map Asmbl p −1 (Fin(Q)),M is split-injective by Example 2.1 since R/n has trivial nilradical. Since the group (1 + M n (n)) is nilpotent, the assembly map Asmbl is split-injective by Theorem 2.9.

G-bornological coarse spaces and coarse homology theories
In this section, we recall the definition of the category GBornCoarse of G-bornological coarse spaces and provide basic examples. We further recall the notion of an equivariant coarse homology theory, in particular its universal version Yo s with values in the stable ∞-category GSpX of equivariant coarse motivic spectra. Most of this material has been developed in [13] (see also [10] for the nonequivariant case).
In the definitions below, we will use the following notation.
(1) For a set Z, we let P(Z) denote the power set of Z.
(2) If a group G acts on a set X, then it acts diagonally on X × X and therefore on P(X × X). For U in P(X × X), we set GU := g∈G gU.
(3) For U in P(X × X) and B in P(X), we define the U -thickening U [B] by (4) For U in P(X × X), we define the inverse by Let G be a group and let X be a G-set.
Definition 3.1. A G-coarse structure C on X is a subset of P(X × X) with the following properties.
(i) C is closed under composition, inversion, and forming finite unions or subsets.
(ii) C contains the diagonal diag(X) of X.
(iii) For every U in C, the set GU is also in C.
The pair (X, C) is called a G-coarse space, and the members of C are called (coarse) entourages of X.
Let (X, C) and (X , C ) be G-coarse spaces and let f : X → X be an equivariant map between the underlying sets.
We obtain a category GCoarse of G-coarse spaces and controlled equivariant maps.
The pair (X, B) is called a G-bornological space, and the members of B are called bounded subsets of X.
Let (X, B) and (X , B ) be G-bornological spaces and let f : X → X be an equivariant map between the underlying sets.
We obtain a category GBorn of G-bornological spaces and proper equivariant maps. Let X be a G-set with a G-coarse structure C and a G-bornology B.   We obtain a category GBornCoarse of G-bornological coarse spaces and morphisms. If the structures are clear from the context, we will use the notation X instead of (X, C, B) in order to denote G-bornological coarse spaces.
Let X be a G-set.
We can define the following G-coarse structures on X (i) The minimal coarse structure on X is the G-coarse structure generated by the empty family. It consists of all subsets of diag(X). We denote the corresponding G-coarse space by X min .
(ii) The canonical coarse structure on X is the G-coarse structure generated by the entourages B × B for all finite subsets B of X. We denote the corresponding G-coarse space by X can .
(iii) P(X × X) is the maximal coarse structure on X. We denote the corresponding G-coarse space by X max .
(iv) If X comes equipped with a quasi-metric † d, then the metric coarse structure on X is generated by the subsets {(x, y) | d(x, y) r} of X × X for all r in [0, ∞). We denote the corresponding coarse space by X d . If the quasi-metric d is G-invariant, then we obtain a G-coarse structure and X d is a G-coarse space.
If A is a subset of P(X), then the G-bornology generated by A is the minimal G-bornology containing A, that is, it is the bornology B {gB | g ∈ G, B ∈ A} generated by the set of all G-translates of elements of A.
We can define the following G-bornologies on X.
(i) The minimal G-bornological structure consists of the finite subsets. We denote the corresponding G-bornological space by X min .
(ii) The maximal G-bornological structure consists of all subsets. We denote the corresponding G-bornological space by X max .
(iii) If X comes equipped with a quasi-metric d, the metric bornology on X is generated by the sets {y | d(x, y) r} for all x in X and r in [0, ∞). We denote the corresponding bornological space by X d . If d is G-invariant, then we obtain a G-bornology and X d is a G-bornological space.
Taking any pair of compatible coarse and bornological structures as above, we can form a G-bornological coarse space. These will be denoted by two subscripts, where the first subscript refers to the coarse structure and the second subscript to the bornology. Examples include X can,min , X can,max , X min,min , X min,max , X max,max and, if X comes equipped with an invariant metric, X d,d .
Let X be a G-coarse space with coarse structure C. Then is an invariant equivalence relation on X.
Definition 3.9. We let π 0 (X) denote the G-set of equivalence classes with respect to R C . The elements of π 0 (X) are called the coarse components of X.
We now introduce the notion of an equivariant coarse homology theory; see [13,Section 3] for details.
Let X be a G-bornological coarse space.
Let g, f : X → X be two morphisms in GBornCoarse. Then we say that f is close to g if (f × g)(diag(X)) is a coarse entourage of X . This notion will be used in Condition (i) of the definition below.
Let X be a G-bornological coarse space.
We say that flasqueness of X is implemented by f .
The category GBornCoarse has a symmetric monoidal structure ⊗; see [13,Example 2.17]. If X and Y are G-bornological coarse spaces, then X ⊗ Y has the following description.
(1) The underlying G-coarse space of X ⊗ Y is the Cartesian product in GCoarse of the underlying G-coarse spaces of X and Y . More explicitly, the underlying G-set of X ⊗ Y is X × Y with the diagonal G-action, and the coarse structure is generated by the entourages U × V for all coarse entourages U of X and V of Y .
(2) The bornology on X ⊗ Y is generated by the products A × B for all bounded subsets A of X and B of Y .
Note that X ⊗ Y in general differs from the Cartesian product X × Y in GBornCoarse.
Let C be a cocomplete stable ∞-category and let In this formula, we consider the subsets Y i as G-bornological coarse spaces with the structures induced from X. If Z is another invariant subset, then we use the notation Z ∩ Y := (Z ∩ Y i ) i∈I . Let C be a cocomplete stable ∞-category and consider a functor Definition 3.13. A G-equivariant C-valued coarse homology theory is a functor with the following properties.
(i) (Coarse invariance) For all X in GBornCoarse, the functor E sends the projection {0, 1} max,max ⊗ X → X to an equivalence.
(ii) (Excision) E(∅) 0 and for every equivariant complementary pair (Z, Y) on a G-bornological coarse space X, the square is a push-out.
(iv) (u-Continuity) For every G-bornological coarse space X, the natural map is an equivalence. Here X U denotes the G-bornological coarse space X with the coarse structure replaced by the one generated by U , and C G (X) is the poset of G-invariant coarse entourages of X.
If the group G is clear from the context, then we will often just speak of an equivariant coarse homology theory.
We have a universal equivariant coarse homology theory Yo s : GBornCoarse → GSpX (see [13,Definition 4.9]), where GSpX is a stable presentable ∞-category called the category of coarse motivic spectra. More precisely, we have the following. The symmetric monoidal structure ⊗ descends to GSpX such that Yo s becomes a symmetric monoidal functor [13,Lemma 4.17]. Similarly, all members of Y are flasque. Since Yo s vanishes on flasques, we get Yo s (Z) 0 and Yo s (Y) 0. The inclusion X ∼ = {0} × X → R × X induces an equivalence of X with every member of Z ∩ Y. Consequently, we have a canonical equivalence Yo s (X) Yo s (Z ∩ Y). Therefore, the push-out square in (3.4) is equivalent to a push-out square This square provides an equivalence Σ Yo s (X) Yo s (R ⊗ X). (3.5) Let E : GBornCoarse → C be a functor and let X be a G-bornological coarse space.
Lemma 3.17. If E is an equivariant coarse homology theory, then the twist E X is an equivariant coarse homology theory, too.
Let (X, B) be a G-bornological space.
Continuity is an additional property of an equivariant coarse homology theory E. We refer to [13,Definition 5.15] for the precise definition. For our purposes, it suffices to know the following.
Let X be a G-bornological coarse space and let L(X) denote the poset of all G-invariant locally finite subsets of the underlying bornological space of X. We consider F in L(X) with the G-bornological coarse structure induced from X.
is an equivalence.
In order to capture continuity of equivariant coarse homology theories motivically, we introduce the universal continuous equivariant coarse homology theory whose target GSpX c is the stable presentable ∞-category of continuous equivariant motivic coarse spectra (see [13,Definition 5.21]). We have a canonical colimit-preserving functor such that Yo s c C s • Yo s (see [13, (5.6)]).
Definition 3.21. A morphism in GSpX or GBornCoarse is a continuous equivalence if it becomes an equivalence after application of C s or Yo s c , respectively. Two morphisms in GSpX or GBornCoarse are continuously equivalent if they become equivalent after application of C s or Yo s c , respectively.

Cones and the forget-control map
In this section, we recall the cone construction and the cone sequence. We further introduce the forget-control map and show its compatibility with induction and twisting. We start with discussing G-uniform bornological coarse spaces and the cone construction. Let X be a G-set.
Definition 4.1. A G-uniform structure on X is a subset U of P(X × X) with the following properties.
(i) Every element of U contains the diagonal. (ii) U is closed under inversion, composition, finite intersections, and supersets.
The first three conditions define the notion of a uniform structure, and the last condition reflects the compatibility with the action of G. A G-uniform space is a pair (X, U ) of a G-set X and a G-uniform structure U .
Let (X, U ) and (X , U ) be G-uniform spaces and f : X → X be an equivariant map between the underlying sets.
Let X be a G-set with a G-uniform structure U and a G-coarse structure C. Definition 4.3. We say that U and C are compatible if U ∩ C is not empty.
We obtain the category GUBC of G-uniform bornological coarse spaces. We have the forgetful functor which forgets the uniform structure.
Example 4.6. Let X be a G-set with a quasi-metric d. Then we get a uniform structure on X generated by the subsets {(x, y) ∈ X × X | d(x, y) < r} for all r in (0, ∞). We let X d denote the corresponding uniform space. If d is invariant, then we obtain a G-uniform structure and X d is a G-uniform space.
Expanding the notation for G-bornological coarse spaces, we use triple subscripts to indicate G-uniform bornological coarse spaces, where the first subscript indicates the G-uniform structure, the second subscript indicates the G-coarse structure, and the third subscript indicates the G-bornology.
In particular, if X is a G-set with an invariant quasi-metric d, then we obtain the G-uniform bornological coarse spaces X d,d,d and X d,max,max .
Example 4.7. Let S be a G-set. Then the G-bornological coarse space S min,min equipped with the uniform structure containing all supersets of the diagonal is a G-uniform bornological coarse space which we denote by S disc,min,min .
Let X be a G-uniform bornological coarse space and let Y = (Y i ) i∈I be an equivariant big family. Let C and U denote the coarse and uniform structures of X.
Definition 4.8 [13,Definition 9.15]. An order-preserving function The hybrid structure C h on X is the G-coarse structure generated by the entourages U ∩ U ψ for all U in C G and all U -admissible functions ψ.
We let X h denote the bornological coarse space obtained from X by forgetting the uniform structure and replacing the coarse structure by the hybrid coarse structure.
Definition 4.9. We have the functor If which is called the cone sequence. The first map of the cone sequence is induced by the inclusion X → [0, ∞) × X given by including the point 0 into [0, ∞). The second map is induced by the inclusion O(X) → O ∞ geom (X). Finally, the cone boundary ∂ is given by where the first map is induced by the identity of the underlying sets, and the equivalence is the equivalence (3.5) explained in Example 3.15. We use [13,Proposition 9.31] in order to see that this description of the sequence is equivalent to the original definition from [13,Corollary 9.30].
In various constructions, we form a colimit over the poset of invariant entourages C G (X) of a G-bornological coarse space X. In order to suppress these colimits in an appropriate language, we use the following procedure. We let GBornCoarse C denote the category of pairs ( We have a forgetful functor be a functor to a cocomplete target C and let E be the left Kan extension of F along (4.4). The evaluation of E on a G-bornological coarse space X is then given as follows.
Lemma 4.13. We have an equivalence Proof. By the pointwise formula for the left Kan extension, we have an equivalence in GBornCoarse C /X. This easily implies that the full subcategory of objects of the form Construction 4.14. Let X be a G-bornological coarse space and let U be an invariant entourage of X. Then we can form the G-simplicial complex P U (X) of finitely supported U -bounded probability measures on X (see [13,Definition 11.1] and the subsequent text). We equip P U (X) with the path quasi-metric in which every simplex has the spherical metric. The path quasi-metric determines the uniform and the coarse structure on P U (X). We equip P U (X) with the bornology generated by all subcomplexes P U (B) of measures supported on B for a bounded subset B of X. The resulting G-uniform bornological coarse space will be denoted by P U (X) d,d,b . We denote by P U (X) d,b the underlying bornological coarse space. Note that the bornology in general differs from the metric bornology which would be indicated by a subscript d in the last slot.
Let f : X → X be a morphism of G-bornological coarse spaces and U be an invariant entourage of X such that (f × f )(U ) ⊆ U . Then the push-forward of measures induces a morphism in a functorial way. We have thus constructed a functor If we compose the functor P with the fiber sequence (4.2), then we obtain a fiber sequence of functors GBornCoarse C → GSpX which sends (X, U ) to Definition 4.15. We define the fiber sequence of functors GBornCoarse → GSpX by left Kan extension of (4.5) along the forgetful functor (4.4).
In order to justify this definition, note that a colimit of a diagram of fiber sequences in a stable ∞-category is again a fiber sequence. Since a fiber sequence of functors can be detected objectwise, it is a consequence of the pointwise formula for the Kan extension that a Kan extension of a fiber sequence of functors with values in a stable ∞-category is again a fiber sequence.
If S is a G-set, then we have a twist functor commutes. Note that T Mot S Yo s (S min,min ) ⊗ −, and this functor is equivalent to the left Kanextension of Yo s •T S along Yo s , so in particular it commutes with colimits.
We can extend the twist functor to a functor The twist functor (4.6) further extends to a twist functor Proof. For (X, U ) in GBornCoarse C we construct an isomorphism of G-simplicial complexes which induces the desired isomorphism of G-uniform bornological coarse spaces. Let (s, μ) be a point in S × P U (X). Then there is some n in N, a collection of points x 0 , . . . , x n in X and The map (4.8) sends the point (s, μ) to the point Therefore, the inverse of the isomorphism (4.8) sends ν to the point (s 0 , It is straightforward to check that the isomorphism is G-equivariant, natural in (X, U ), and compatible with the bornologies. Proof. We first discuss the isomorphism in the case of O ∞ geom . For X in GUBC the desired isomorphism We need to verify that the coarse structures agree.
For an admissible function ψ : for all admissible functions ψ, so the bijection f induces a controlled map. Conversely, let p : R × S × X → R × X be the projection map. If φ : N → P((R × S × X) 2 ) G is an admissible function, then the function φ : N → P((R × X) 2 ) G sending n to (p × p)(φ(n)) is also admissible. Moreover, we have   Proof. In a first step we postcompose the diagram from Lemma 4.17 with Yo s and precompose it with the functor P : GBornCoarse C → GUBC. Then we get a corresponding diagram of functors GBornCoarse C → GSpX . We apply the left Kan extension along the forgetful functor GBornCoarse C → GBornCoarse and get the commuting diagram (4.10) Using (4.7) and the fact that T Mot S preserves colimits, the upper line of (4.10) is equivalent to the upper line of the diagram (4.9). It remains to identify the lower line.
We use Lemma 4.16 to identify the lower line of (4.10) with We show that this transformation is an equivalence. To this end, we use the pointwise formula from Lemma 4.13. We therefore must show that the natural morphism is an equivalence. This is clear since U → diag(S) × U is an isomorphism of posets from C G diag (X) to C G diag (S min,min ⊗ X). We therefore get the desired identification of the lower line of the diagram (4.10) with the lower line in (4.9).
If H is a subgroup of G, then we have an induction functor The elements of G × H X will be written in the form [g, x] for g in G and x in X, and we have the equality [gh, . (4.13) This induction functor refines to an induction functor Ind G H : HBornCoarse → GBornCoarse (4.14) for bornological coarse spaces. If X is some H-bornological coarse space, then Ind G H (X) becomes a G-bornological coarse space with the following structures.
(1) The bornological structure on Ind G H (X) is generated by the images under (4.13) of the subsets {g} × B of G × X for all g in G and bounded subsets B of X.
(2) The coarse structure is generated by the entourages Ind G H (U ), which are the images of the entourages diag(G) × U of G × X under the projection (4.13), for all coarse entourages U of X. We can extend the induction functor to a functor ). Then we have a commuting diagram The induction functor (4.14) further extends to an induction functor for uniform bornological coarse spaces. If X is an H-uniform bornological coarse space, then the uniform structure on Ind G,U H (X) is generated by the images of the entourages diag(G) × U of G × X for all uniform entourages U of X under the projection (4.13).
In the following lemma, P G and P H are the versions of the functor P from Construction 4.14 for the groups G and H, respectively.
Proof. For (X, U ) in HBornCoarse C we construct an isomorphism of G-simplicial complexes The isomorphism sends the point [g, μ] to the point n i=0 λ i δ [g,xi] in P Ind G H (U ) (Ind G H (X)). In order to see that this map is invertible, Therefore, the inverse of the isomorphism sends ν to the point It is straightforward to check that the isomorphism is G-equivariant, natural in (X, U ), and compatible with the bornologies. From the explicit description of the coarse and uniform structure on the induction, it follows that In the following statement, we again added subscripts G or H in order to indicate on which categories the respective versions of the functors F, O and O ∞ geom act.

Lemma 4.21. We have a commuting diagram of functors HUBC → GBornCoarse
Proof. We first discuss the isomorphism in the case of the functor O ∞ geom . For X in HUBC the isomorphism is induced by the natural bijection of G-sets which is obviously an isomorphism of G-bornological spaces. We need to show that the hybrid coarse structures agree under f .
denote the projection maps. For an admissible function ψ : N → P((R × X) 2 ) G , define as the function sending n to the image of (p × p)(diag(G) × ψ(n)) under the identification induced by f . Then we have for all admissible functions ψ, so the bijection f induces a controlled map.

A descent result
The main result of the present section is Proposition 5.16. Morally it is a descent result stating that a certain natural transformation from fixed points to homotopy fixed points is an equivalence. The proof is based on the interplay between the covariant and contravariant functoriality of coarse homology theories encoded in their extensions to the ∞-category GBornCoarse tr of G-bornological coarse spaces with transfers. This ∞-category was introduced in [11]. It extends the category GBornCoarse, which only captures the covariant behavior of coarse homology theories.
We start by briefly recalling the construction of the category GBornCoarse tr . Let X be a G-bornological coarse space. Then we let C(X) and B(X) denote the coarse and bornological structures of X. For a subset B of X, we let [B] denote the coarse closure of B, that is, the closure of B with respect to the equivalence relation R C(X) ; see (3.2).
Let now X and Y be G-bornological coarse spaces and f : X → Y be an equivariant map between the underlying G-sets.
Definition 5.1 [11,Definition 2.14]. The map f is called a bounded covering if: (i) f is a morphism between the underlying G-coarse spaces; (ii) the coarse structure C(X) is generated by the sets is an isomorphism of coarse spaces between coarse components; (iv) f is bornological, that is, for every B in B(X) we have f (B) ∈ B(Y ); (v) for every B in B(X) there exists a finite bound (which may depend on B) on the cardinality of the sets (the coarse components of X over V which intersect B nontrivially) for all V in π 0 (Y ).
Note that a bounded covering is not a morphism of bornological coarse spaces in general, since it may not be proper. The composition of two bounded coverings is again a bounded covering; see [12,Lemma 2.18].   (as a simplicial set, this is the edgewise subdivision). We denote by GBornCoarse the category whose objects are G-bornological coarse spaces and whose morphisms are morphisms of the underlying G-coarse spaces.
Then GBornCoarse tr is a certain sub-simplicial set of the simplicial set Let E : GBornCoarse tr → C be a functor.
Definition 5.6. We define the functor Assume now that E is a coarse homology theory with transfers. For every G-set T , we have an equivalence of functors GBornCoarse → C; see Definition 3.16 for notation. The right-hand side is a twist of an equivariant coarse homology theory and therefore again an equivariant coarse homology theory by Lemma 3.17. By Proposition 3.14, we can extend E along Yo s to a functor (denoted by the same symbol for simplicity) which preserves colimits in its second argument.
From now on until the end of this section, we assume that the ∞-category C is stable, cocomplete and complete, and that E is a C-valued equivariant coarse homology theory with transfers. From now on we consider E as a contravariant functor in its first argument.

E(yo(A), X).
Note that the order of the limit and the colimit matters in general.
Let A be in PSh(GSet) and let E be a C-valued equivariant coarse homology theory with transfers. Remark 5.10. By Elmendorf's theorem, the homotopy theory of G-spaces is modeled by the presheaf category PSh(GOrb); see Remark 1.12. This category is equivalent to the category of sheaves Sh(GSet) with respect to the Grothendieck topology on GSet given by disjoint decompositions into invariant subsets. We prefer to identify the sheafification morphism PSh(GSet) → Sh(GSet) with the restriction morphism along the inclusion r : GOrb → GSet since in our special situation it has an additional left adjoint r ! which is not part of general sheaf theory. Proof. We start with the morphism R∈G\S yo(r(R)) → yo(S) induced by the collection of inclusions (r(R) → S) R∈G\S . We claim that it becomes an equivalence after application of r * . Indeed, for T in GOrb we have a commuting square The lower horizontal map is an equivalence since the functor Map GSet (r(T ), −) commutes with coproducts since r(T ) is a transitive G-set.
Since the counit of an adjunction is a natural transformation, we get the following commuting diagram It remains to show that the left vertical arrow is an equivalence. To this end we consider the diagram The left square commutes by the usual relation between the unit and the counit of an adjunction. Since r ! commutes with colimits and r ! yo(R) yo(r(R)) by adjointness, the horizontal morphisms on the right are equivalences. Since r is fully faithful, the unit appearing at the left is an equivalence. Hence, the counit on the right is an equivalence as claimed.
In order to simplify the notation in the arguments below we introduce now the following abbreviation. Let pt denote the one-point G-bornological coarse space. We consider E pt as a contravariant functor from PSh(GSet) to C which sends colimits to limits.
The counit (5.6) induces a transformation Remark 5.13. Recall from [11, Definition 2.61] that we call a coarse homology theory with transfers strongly additive if its sends free unions (see [13,Example 2.16]) of families of G-bornological coarse spaces to products. Note further that for S in GSet the G-bornological coarse space S min,min is the free union of the family (R min,min ) R∈G\S . This is used to see that the morphism (5.9) below is an equivalence.
Lemma 5.14. If E is strongly additive, then the transformation (5.8) is an equivalence.
Proof. Let S be in GSet. Using Lemma 5.11 and the fact that E pt sends colimits to limits, the specialization u S of (5.8) to S is given by the map is an equivalence. Therefore, u S is an equivalence.
The following lemma is the crucial technical ingredient in the proof of the main result of the present section (Proposition 5.16). It allows us to move G-sets from one argument of the functor E to the other.
We consider a G-set S. since both functors send colimits to limits and coincide on representables. Inserting (5.11) and (5.12) into (5.10), we obtain the desired equivalence.
We now state the main result of the present section. Recall that C is a complete and cocomplete, stable ∞-category. Furthermore, E is an equivariant C-valued coarse homology theory with transfers. We let E be defined as in Definition 5.7. We consider an object A in PSh(GSet) and a transitive G-set R in G F Orb. Let Proof. We consider the following commutative diagram in C: Here s is the natural equivalence from Lemma 5.15, and the morphism u from (5.8) is a natural equivalence by Lemma 5.14.
We further use the canonical equivalence r * yo(r(R)) yo(R) for the lower left vertical equivalence, and in addition the fact that r * preserves products for the lower right vertical equivalence. The lower horizontal morphism is an equivalence since where the first equivalence holds true by Assumption (ii) and the second equivalence follows from the fact that R has stabilizers in F, also by assumption.
Remark 5.17. As explained in Remark 1.12, the ∞-category PSh(GOrb) is a model for the homotopy theory of G-spaces. Compactness of E F G as a presheaf on GOrb will play a crucial role in our arguments. This condition is closely related to the existence of a G-compact Identifying presheaves on GOrb with sheaves on GSet, we can consider E F G as an object of PSh(GSet) which satisfies the sheaf condition. But compactness of E F G as an object of PSh(GSet) is a too strong condition. For this reason we consider compact objects A in PSh(GSet) which after sheafification, that is, after application of r * , become equivalent to E F G. The existence of such an object is an important assumption in the following. In Lemma 10.4, we will show that the existence of a finite-dimensional model for E top F G (a much weaker condition than G-compactness) implies the existence of such a compact presheaf A.
A G-simplicial complex is a simplicial complex on which G acts by morphisms of simplicial complexes. We denote by GSimpl the category of G-simplicial complexes and G-equivariant simplicial maps. Let K be a G-simplicial complex.
Definition 5.18. K is G-finite if it consists of finitely many G-orbits of simplices.
We let G F Simpl fin denote the full subcategory of GSimpl of G-finite G-simplicial complexes with stabilizers in F.
We have a canonical functor which equips a G-simplicial complex with the structures induced by the spherical quasi-metric. Hence we have a functor where O ∞ denotes the cone-at-infinity functor from Definition 4.10. Let A be in PSh(GSet).
Proposition 5.19. Assume: Then the natural transformation of functors from G F Simpl fin to C induced by A → * is a natural equivalence.
The functor k sends equivariant decompositions of G-finite G-simplicial complexes to equivariant uniform decompositions of G-uniform bornological coarse spaces by [13,Lemma 10.9]. The functor O ∞ is excisive for those decompositions by [13,Corollary 9.36 and Remark 9.37]. Furthermore, it is homotopy invariant by [13,Corollary 9.38].
Since A natural transformation between two such functors which is an equivalence on G-orbits with stabilizers in F is an equivalence on G-finite G-simplicial complexes with stabilizers in F: by induction on the number of equivariant cells, this follows from application of the Five-Lemma to the Mayer-Vietoris sequences arising from the pushout squares describing simplex attachments. This implies the assertion.

Duality of G-bornological spaces
In this section, we develop a notion of duality for G-bornological spaces that we will use later to compare certain assembly and forget-control maps.
The category GBorn (see Definitions 3.3 and 3.4) of G-bornological spaces and proper equivariant maps has a symmetric monoidal structure ⊗. If Y and X are G-bornological spaces, then Y ⊗ X is the G-bornological space with underlying G-set Y × X (with diagonal action) and the bornology generated by the subsets A × B for bounded subsets A of Y and B of X. Note that this tensor product is not the Cartesian product in GBorn.
Recall that a subset L of a G-bornological space X is called locally finite if L ∩ B is finite for every bounded subset B of X; see Definition 3.18.
For a set A, we let |A| in N ∪ {∞} denote the number of elements of A. For a subset L of X × G, we consider as a subset of X in the natural way. Let X be a G-bornological space and L be a G-invariant subset of X × G.
This implies the assertion.
Let X be a G-bornological space and L be a G-invariant subset of X × G. Proof. The subset L of X ⊗ G min is locally finite if and only L ∩ (B × {g}) is finite for every bounded subset B of X and g in G. Since L is G-invariant, we have bijections This implies the assertion.
Let X and X be two G-bornological spaces with the same underlying G-set. Definition 6.3. We say that X is dual to X if the sets of G-invariant locally finite subsets of X ⊗ G max and X ⊗ G min coincide.
If X and X are two G-bornological coarse spaces, then we say that X is dual to X if the underlying G-bornological space of X is dual to the one of X .
Remark 6.4. Note that duality is not an equivalence relation. In particular, the order is relevant.
Example 6.5. Let S be a G-set with finite stabilizers.
(i) S min is dual to S lax , where S lax (lax stands for locally max) is S with the bornology generated by the G-orbits.
(ii) S lin is dual to S max , where S lin (lin stands for locally min) is S with the bornology given by subsets which have at most finite intersections with each G-orbit.
Let X be a G-bornological space. Definition 6.6. X is called G-bounded if there exists a bounded subset B of X such that GB = X.
If X is a G-bornological space, then we let X max denote the G-bornological space with the same underlying G-set and the maximal bornology.
Let X be a G-bornological space and Y be a bornological space (which we consider as a G-bornological space with the trivial G-action).
Lemma 6.8. Assume: In view of Lemma 6.1 and Lemma 6.2, local finiteness of L in Y ⊗ X ⊗ G max or Y ⊗ X max ⊗ G min is characterized by conditions on the subset L 1 of Y × X; see (6.1) for notation.
We must check that the following conditions on L 1 are equivalent.
We assume that L 1 satisfies Condition (i). Let B be a bounded subset of X and A be a bounded subset of Y . Since X is G-proper, the family (gB) g∈G has finite multiplicity, say bounded by m in N. We get Consequently, L 1 satisfies Condition (ii).
We now assume that L 1 satisfies Condition (ii). Let A be a bounded subset of Y . Since X is G-bounded we can choose a bounded subset B of X such that GB = X. Then Hence L 1 satisfies Condition (i).
The following lemma explains why the notion of duality is relevant. Assume that X and X are G-bornological coarse spaces with the same underlying G-coarse space. Recall the notation Yo s c for the universal continuous equivariant coarse homology theory, see (3.6).
Lemma 6.9. If X is dual to X , then we have a canonical equivalence in GSpX c Proof. This lemma is a special case of the following Lemma 6.10 for the case I = * .
We will need a functorial variant of Lemma 6.9. We consider a small category I and a functor X 0 : I → GCoarse. Assume further that we are given two lifts X, X of X 0 to functors from I to GBornCoarse along the forgetful functor GBornCoarse → GCoarse as depicted in the following diagram: Extending the notion of continuous equivalence (Definition 3.21), we call two functors I → GBornCoarse continuously equivalent if they become equivalent after application of Yo s c .
Lemma 6.10. If X(i) is dual to X (i) for every i in I, then X ⊗ G can,max and X ⊗ G can,min are continuously equivalent.
Proof. For i in I, let L X(i) and L X (i) be the posets of invariant locally finite subsets of X(i) ⊗ G can,max and X (i) ⊗ G can,min equipped with their induced structures, respectively. We first show that the assumption of the lemma implies an equality of posets L X(i) = L X (i) . Indeed, the assumption says that the collections of underlying sets of the elements of L X(i) and L X (i) are equal. In addition, for L in L X(i) its coarse structure coincides with the one induced from X (i) ⊗ G can,min . Finally, in view of the definition of the notion of local finiteness, the induced bornological structures from X(i) ⊗ G can,max and X (i) ⊗ G can,min are the minimal one in both cases.
We have a functor I → Poset which sends i in I to the poset L X(i) and i → i to the map L X(i) → L X(i ) induced by the proper map X(i) → X(i ). We let I X be the Grothendieck construction for this functor.
We have a functor from I X to spans in GBornCoarse which evaluates on the object (i, L) of I X with L ∈ L X(i) to Here L is the set L considered as an element of L X (i) .
We now apply Yo s c and form the left Kan extension of the resulting diagram along the forgetful functor I X → I. Then we get a functor from I to the category of spans in GSpX c which evaluates at i in I to By continuity of Yo s c , see Lemma 3.19, the left and the right morphisms are equivalences as indicated. Therefore, this diagram provides the equivalence claimed in the lemma.

Continuous equivalence of coarse structures
In general, the value of an equivariant coarse homology theory on G-bornological coarse spaces depends nontrivially on the coarse structure. In this section, we show that in the case of a continuous equivariant coarse homology theory, one can change the coarse structure to some extent without changing the value of the homology theory. This is formalized in the notion of a continuous equivalence; see Definition 3.21.
Let X be a G-bornological space with two compatible G-coarse structures C and C such that C ⊆ C . We write X C and X C for the associated G-bornological coarse spaces.
Lemma 7.1. Assume that for every locally finite subset L of X the coarse structures on L induced by C and C coincide. Then id X : X C → X C is a continuous equivalence.
Proof. Let L denote the poset of locally finite subsets of X. Then the claim follows from the commutative square The horizontal maps are equivalences by continuity; see Lemma 3.19. The left vertical map is an equivalence since L XC = L X C for every L in L by assumption, where L XC indicates that we equip L with the coarse structure induced from X C .
The identity on the underlying sets induces a morphism G can,max → G max,max (7.1) of G-bornological coarse spaces. If X is a G-bornological coarse space, then we get an induced morphism Lemma 7.2. If X is G-bounded, then the morphism (7.2) is a continuous equivalence.
Proof. Let L be a G-invariant locally finite subset of the underlying bornological space of X ⊗ G can,max . By Lemma 7.1, it suffices to show that the coarse structure induced on L from X ⊗ G max,max is contained in the coarse structure induced from X ⊗ G can,max (since the other containment is obvious).
Since X is G-bounded (see Definition 6.6) by assumption, there exists a bounded subset A of X such that GA = X. Let U be an invariant entourage of X containing the diagonal. It will suffice to show that (U × (G × G)) ∩ (L × L) is an element of the coarse structure induced on L by X ⊗ G can,max . Note that there is an implicit reordering of the factors in the product to make sense of the intersection. Note Indeed, the condition that This implies that g −1 h ∈ W and g −1 h ∈ W , and hence (h, h ) ∈ G(W × W ).
Hence we conclude that the restriction of U × (G × G) to L is contained in the entourage Corollary 7.5. If X belongs to GSpX bd , then (7.3) induces a continuous equivalence Proof. This follows directly from Lemma 7.2 since the symmetric monoidal structure ⊗ on GSpX commutes with colimits in each variable separately; see [13,Lemma 4.17].
Recall Definition 3.9 of the G-set of coarse components π 0 (X) of a G-coarse space X. Let X be a G-set with two G-coarse structures C and C such that C ⊆ C . We write X C,max and X C ,max for the associated G-bornological coarse spaces with the maximal bornology.
Lemma 7.6. If the canonical map π 0 (X C ) → π 0 (X C ) is an isomorphism, then the morphism X C,max ⊗ G can,min → X C ,max ⊗ G can,min is a continuous equivalence.
Proof. Let L be a locally finite subset of the underlying G-bornological space of X C,max ⊗ G can,min . By Lemma 7.1, it suffices to show that every entourage of the coarse structure induced on L by X C ,max ⊗ G can,min is contained in an entourage of the coarse structure induced from X C,max ⊗ G can,min .
Let W : = G(B × B) be an entourage of G can,min for some bounded subset B of G can,min . We can assume that B contains the neutral element and is closed under inverses since this will only enlarge the entourage W . Furthermore, let V be in C . It suffices to show that (V × W ) ∩ (L × L) is contained in an entourage of the form (U × W 2 ) ∩ (L × L) for some entourage U in C, where W 2 := W • W denotes the composition of W with itself, see (3.1). Note that we are implicitly permuting the factors of the products to make sense of the intersection.
The subset B := W [B] of G is finite. Note that L 1 , see (6.1), and hence also B L 1 are finite. Since π 0 (X C ) ∼ = π 0 (X C ), there exists an invariant entourage U of X such that V ∩ (L 1 × B L 1 ) ⊆ U . We show that this implies Indeed, for l, l in L 1 the condition ((gl, g), (g l , g )) ∈ V × W implies (g, g ) ∈ W . Hence there exists h in G such that hg and hg are contained in B. Then (hg , 1) and thus (g −1 g , and g is in gW [B] = gB . We write g = gb for b in B . Then ((l, 1), (bl , b)) ∈ U × W 2 and hence also ((gl, g), (g l, g )) ∈ U × W 2 by G-invariance of U and W 2 .

Assembly and forget-control maps
Morally, an assembly map is the map induced in an equivariant homology theory by the projection W → * for some G-topological space W with certain relations with classifying spaces. In the present section, W will be the Rips complex associated to a G-bornological coarse space X.
On the other side, the prototype for a forget-control map is the map F ∞ (X) → ΣF 0 (X) induced by the cone boundary.
These two maps will be twisted by G-bornological coarse spaces derived from the G-set G equipped with suitable coarse and bornological structures. The notation for the assembly map associated to a G-bornological coarse space will be α X , and the forget-control map will be denoted by β X .
In this section, we compare the assembly map α X and the forget-control map β X . The main results are Corollary 8.25 and Corollary 8.31.
The comparison argument will go through intermediate versions of the forget-control map denoted by β π0 X and β π weak 0 X . The structure of the comparison argument is as follows.
(1) β X and β π0 X are compared in Lemma 8.12. The combination of these results yields one of the main results (Corollary 8.25). Before we consider the forget-control maps themselves, we investigate preliminary versions of them defined on G-simplicial complexes. Let GSimpl denote the category of G-simplicial complexes. A G-simplicial complex K comes with the invariant spherical path quasi-metric which induces a G-uniform bornological coarse structure on K. We refer to Example 3.8 and Example 4.6 for the corresponding notation. We thus have the following functors d,d , k d,d,max , k d, and Recall Definition 5.18 of the notion of G-finiteness of a G-simplicial complex K.
We let GSimpl conn,prop,fin denote the full subcategory of GSimpl of connected, G-proper, and G-finite G-simplicial complexes.
Extending the notion of continuous equivalence (Definition 3.21), we call two transformations between GSpX -valued functors continuously equivalent, if they become equivalent after application of C s ; see (3.7). Proof. Let K be an object of GSimpl conn,prop,fin . Then we have a commuting square which is natural in K. The left vertical map is an equivalence since K d,d,max → K d,max,max is a coarsening and O ∞ sends coarsenings to equivalences [13,Proposition 9.33]. The right vertical map is a continuous equivalence by Lemma 7.6 because both K d,max and K max,max are coarsely connected. Note that this is the only place where we use that K is connected.
We now claim that we can apply Lemma 6.10 in order to conclude that the map is canonically continuously equivalent to the map Recall from Definition 4.8 the hybrid coarse structure X h associated to a G-uniform bornological coarse space X. Moreover, recall from (4.3) that the cone boundary is given by the map where the second map is induced by the identity of the underlying sets, and the third equivalence follows from excision. We apply Lemma 6.10 to the index category I := GSimpl conn,prop,fin × Δ 1 and the functor X 0 : I → GCoarse given on objects by where the notation [. . .] C indicates that we take the underlying G-coarse spaces. While the action of this functor on the morphisms in I coming from morphisms K → K in GSimpl conn,prop,fin is clear, it sends the morphism (K, 0) → (K, 1) coming from 0 → 1 in Δ 1 to the map given by the identity on the underlying sets. The lifts X and X of this functor to GBornCoarse are given on objects by for X, and by for X , while the lifts on the level of morphisms are clear. We claim that for every ( If the G-simplicial complex K is not connected, then the proof of Proposition 8.2 establishes a modified assertion. For its formulation we first introduce some notation. Let X be a G-coarse space and let U π0 be the entourage from (5.1).
Definition 8.3. We let X π0 denote the G-set X with the G-coarse structure C π0 generated by U π0 . Note the following.
Similar to the transformation β max from (8.2), we define a natural transformation of functors GSimpl → GSpX The following definition is adapted from [31,Definition 3.24]. Let X be a bornological coarse space.
Definition 8.5. X is uniformly discrete if the bornology is the minimal bornology (see Example 3.8) and for every entourage U of X there is a uniform bound for the cardinalities of the sets U [x] for all points x in X.
Remark 8.6. In [10] we called this property strongly bounded geometry. It is not invariant under coarse equivalences. The adjective strongly distinguishes this notion from the notion of bounded geometry which is invariant under coarse equivalences.
Example 8.7. The G-bornological coarse space G can,min is uniformly discrete.
Remark 8.8. Let X be a G-bornological coarse space and U be an invariant entourage of X. The condition that X is uniformly discrete has the following consequences.
(i) P U (X) is a finite-dimensional, locally finite simplicial complex. Furthermore, for X = G can,min the G-simplicial complex P U (G can,min ) is G-finite, that is, it belongs to G Fin(G) Simpl fin ; see Definition 5.18.
(ii) Since X carries the minimal bornology and P U (X) is locally finite, the bornology on P U (X) b (which by definition is generated by the subsets P U (B) for all bounded subsets B of X) coincides with the bornology P U (X) d induced from the spherical path quasi-metric.
Let X be a G-bornological coarse space and let U be an invariant entourage of X. Definition 8.9. We let C π weak 0 denote the coarse structure on P U (X) generated by the entourage We have obvious inclusions of G-coarse structures on P U (X). The coarse structure C π0 was introduced in Definition 8.3 and depends on the coarse structure of P U (X) d given by the path quasi-metric. In contrast, the coarse structure C π weak 0 is given by Definition 8.9 using the coarse structure of X. In analogy to Construction 4.14, we have functors In view of the first inclusion in (8.5), we have a natural transformation The following construction is analogous to Definition 4.15. If we precompose the fibre sequence (4.2) with one of (8.6) or (8.7), then we obtain fiber sequences of functors GBornCoarse C → GSpX which send (X, U ) to and to ,max ), (8.10) respectively. The transformation (8.8) induces a natural transformation of fiber sequence from (8.9) to (8.10).
Again we have a natural transformation of fiber sequences from (8.11) to (8.12). Let X be a G-bornological coarse space. The morphisms in the following definition are induced by the natural transformation denoted by ∂ in Definition 4.15 or Definition 8.10.
The maps Let X be a G-bornological coarse space.
Proof. In view of Definition 4.15 and Lemma 4.13, the morphism β X is given as a colimit of the diagram of morphisms (8.16) indexed by the poset C G (X) (obtained by precomposing (8.3) with the functor P − (X): C G (X) → GSimpl). Similarly, the morphism β π0 X is given as a colimit of the diagram of morphisms indexed by C G (X) (again obtained by precomposing (8.4) with the functor P − (X)). Since X is uniformly discrete and G-finite, for every U in C G (X) the G-simplicial complex P U (X) is G-finite. In addition, the bornology induced from the metric coincides with the bornology induced from X; see Remark 8.8. Finally, since X is G-proper, the G-simplicial complex P U (X) is also G-proper. Hence P U (X) belongs to GSimpl prop,fin .
We can now apply Proposition 8.4 and conclude that the diagrams (parametrized by U in C G (X)) of morphisms (8.16) and (8.17) are canonically equivalent. Therefore, their colimits β X and β π0 X are canonically equivalent, too.
Let X be a G-bornological coarse space.
Proof. We consider an invariant entourage U of X and form the commutative square in GSpX , where the vertical morphisms are induced by (8.8). In view of Lemma 4.13, after taking colimits over U in the poset C G (X), the horizontal maps become equivalent to β π0 X and β π weak 0 X , respectively. The left vertical morphism is an equivalence since it is obtained by applying O ∞ to a coarsening and O ∞ sends coarsenings to equivalences by [13,Proposition 9.34].
It remains to show that the right vertical map becomes a continuous equivalence after taking the colimit over C G (X). We let F(U ) denote the poset of invariant locally finite subsets of the G-bornological space P U (X) max ⊗ G min . We then consider the following commutative diagram where the subscript indicates from which space the bornological coarse structure on L is induced. In view of Lemma 3.19, continuity of Yo s c implies that the vertical maps are equivalences.
For L in F(U ) we know that L 1 := L ∩ (P U (X) × {1}) is finite. There exists an invariant entourage U of X such that U ⊆ U and such that the condition on a subset F of L 1 • F is contained in P U (W ) for some W in π 0 (X) implies the condition • F is contained in a single simplex of P U (X).
Then the coarse structures induced on L from P U (X) π0,max ⊗ G can,min and P U (X) π weak 0 ,max ⊗ G can,min coincide. By a cofinality consideration the upper horizontal map is hence an equivalence. It follows that the lower horizontal map is an equivalence as desired.
Recall from Remark 1.12 that we have functors Let C be a cocomplete stable ∞-category and H : GTop → C be a functor.
Definition 8.14. The functor H is an equivariant homology theory if it is equivalent to the restriction along (8.18) of a colimit-preserving functor PSh(GOrb) → C. Remark 8.15. Note that in [13, Definition 10.3] we use the term strong equivariant homology theory for the objects defined in Definition 8.14 in order to distinguish it from the classical notion of an equivariant homology theory as defined [13,Definition 10.4]. For the purpose of the present paper, we will employ the more natural definition above and drop the word strong.
In view of the universal property of presheaves, the ∞-category Fun colim (PSh(GOrb), C) of colimit-preserving functors is equivalent to the ∞-category Fun(GOrb, C). Therefore, in order to specify an equivariant homology theory or such a colimit preserving functor essentially uniquely, it suffices to specify the corresponding functor in Fun(GOrb, C) In analogy to Construction 4.14, we consider the functor GTop is the underlying G-topological space of the G-uniform space P U (X) d and is the localization as in (8.18). If X is a G-bornological coarse space, then by Lemma 4.13 we have: Corollary 8. 19. The Rips complex of X is given by Remark 8.20. Note that the present definition of the Rips complex differs from the definition given in [13,Definition 11.2]. In the reference, we defined the Rips complex of X as the G-topological space colim U ∈C G (X) P U (X). This definition fits well with the version of O ∞ hlg used there; see Remark 8.17. In contrast, in the present paper we replace the colimit by the homotopy colimit.
For a G-bornological coarse space X, we consider π 0 (X) as a discrete G-topological space. For every U in C G (X), we have a projection P U (X) → π 0 (X) of G-topological spaces. Applying and forming the colimit over C G (X), we obtain a canonical projection morphism Rips(X) → (π 0 (X)) (8.20) in Fix( (P U (G can,min )))(S) for every transitive G-set S. By definition of Fix, we have Fix( (P U (G can,min )))(S) (Map GTop (S disc , P U (G can,min ))).
Since all stabilizers of points in P U (G can,min ) are finite, we see that if S has infinite stabilizers. If S has finite stabilizers, then the argument given in the proof of [13,Lemma 11.4] shows that colim U ∈C G (Gcan,min) π n (Map GTop (S disc , P U (G can,min ))) is trivial for all n in N. This implies colim U ∈C G (Gcan,min) (Map GTop (S disc , P U (G can,min ))) * .
Definition 8.22. The assembly map α X is the map induced by the projection (8.20).
Note that on the target of this map we used (8.19) in order to suppress the symbol . Let GSimpl fin denote the category of G-finite G-simplicial complexes. Recall the functor k d,max,max defined in (8.1).
Lemma 8.23. We have a canonical equivalence of functors GSimpl fin → GSpX for S in GOrb. This implies the desired equivalence.
Let X be a G-bornological coarse space.
Proof. The assumptions on X imply that P U (X) is G-finite for every invariant coarse entourage U of X. Therefore, by Lemma 8.23 we have a canonical equivalence Similarly, we have a canonical equivalence O ∞ hlg (π 0 (X)) O ∞ (π 0 (X) disc,max,max ).
These equivalences yield the lower square in the following diagram. The upper square is induced by a coarsening. Therefore the vertical maps are equivalences by [ Hence the lower horizontal map in (8.22) is equivalent to α X as indicated.
The upper horizontal arrow from (8.22) fits into the commutative square Here the right vertical map is an equivalence by [13,Proposition 9.35]. We now show that the lower horizontal map is an equivalence. The argument is similar to [13,Lemma 10.7]. By choosing a representative in P U (X) for every element of π 0 (X), we obtain a map π 0 (X) × {1} → P U (X) × {1}. This map has a unique extension to a G-equivariant map π 0 (X) × G → P U (X) × G. We now observe that this map is a morphism of G-bornological coarse spaces s : π 0 (X) min,max ⊗ G can,min → P U (X) π weak 0 ,max ⊗ G can,min . It is a right inverse of the projection p : P U (X) π weak 0 ,max ⊗ G can,min → π 0 (X) min,max ⊗ G can,min , and the composition s • p is close to the identity by construction. It follows that p is a coarse equivalence and this implies that lower horizontal map is an equivalence.
It follows that the upper horizontal map in ( Then the assembly map α X and the forget-control map β X from (8.21) and (8.13) are canonically continuously equivalent.
In the following, we derive a version of Corollary 8.25 without the assumption of G-finiteness. To this end, we must modify the definition of the forget-control map.
Let GBornCoarse fin denote the full subcategory of GBornCoarse consisting of G-finite G-bornological coarse spaces. Let E : GBornCoarse fin → C be some functor to a cocomplete target C. Since the image of a G-finite subspace under a morphism of G-bornological coarse spaces is again G-finite the subcategory K(X) is cofinal in GBornCoarse fin /X. This implies the assertion.
Recall Definition 8.5 of the notion of uniform discreteness. In the following, we consider the transformation (8.23) for the functor E := O ∞ hlg • Rips : GBornCoarse → GSpX . Let GBornCoarse udisc be the full subcategory of GBornCoarse of uniformly discrete G-bornological coarse spaces.
Lemma 8.28. The restriction of the transformation to GBornCoarse udisc is an equivalence.
Proof. If X is uniformly discrete, then for every U in C G (X) the complex P U (X) is a locally finite G-simplicial complex. Consequently, P U (X) as a G-topological space is a filtered colimit over its G-compact subsets. In fact, this filtered colimit is a homotopy colimit, so it is preserved by the functor . The subsets P U (L) of P U (X) for invariant G-finite subsets L of X are cofinal in the G-compact subsets of P U (X). All this is used below to justify the equivalence marked by !. At O ∞ hlg (Rips(X)).
We now consider the functor O ∞ hlg • π 0 : GBornCoarse → GSpX . A similar argument as for Lemma 8.28 shows: We do not need to restrict to uniformly discrete spaces here since a discrete G-topological space is always a filtered (homotopy) colimit of its G-finite subspaces.
In the following, we use the abbreviations F x fin for (F x ) fin for x ∈ {∅, 0, ∞}, and we write β X,fin for the image of β X under the (−) fin -construction.
Let X be a G-bornological coarse space.
Then the assembly map α X : O ∞ hlg (Rips(X)) ⊗ G can,min → O ∞ hlg (π 0 (X)) ⊗ G can,min is canonically continuously equivalent to the forget-control map β X,fin : F ∞ fin (X) ⊗ G max,max → ΣF 0 fin (X) ⊗ G max,max . Proof. Since every invariant subspace of X is again uniformly discrete and G-proper, the proposition follows immediately from Corollary 8.25, Lemma 8.27, Lemma 8.28, and Lemma 8.29.
Let X be a G-bornological coarse space and let S be a G-set.
G-bornological coarse space Y , we denote by Y max−B the same coarse space equipped with the maximal bornology. In the lemma below, the group H acts on G × X by h(g, x) := (gh −1 , hx).
Lemma 9.2. The following is a coequalizer in GBornCoarse: (G min,min ⊗ H min,min ⊗ X) max−B ⇒ B H (G min,min ⊗ X) → Ind G H (X), where the first two maps are given by (g, h, x) → (gh, x) and (g, h, x) → (g, hx), respectively.
Let Y be an H-invariant subset of a G-coarse space X. We consider Y as an H-coarse space with the structures induced from X. For every coarse entourage U of X, we define the coarse entourage U Y := (Y × Y ) ∩ U of Y .
In particular, such an A exists if one can represent E top F G by a finite-dimensional G-CW complex.
We then note that r * • Fix Fix Fix • .
Since r * and Fix preserve colimits, By definition we have an identification Fix(S disc ) yo(S). It follows that if we define A := colim I yo(S), then A is a compact object of PSh(GSet) with r * A E F G.
The last assertion of the lemma follows from the more general claim that for every finitedimensional G-CW-complex X with stabilizers in F there exists a finite diagram S X : I X → G F Set such that (X) colim IX (S X,disc ).
Given such a G-CW-complex X, there exists a finite-dimensional G-simplicial complex K with stabilizers in F which is equivariantly homotopy equivalent to X (this works as in the non-equivariant case which can for example be found in [19,Thm. 2C.5]). After one barycentric subdivision, we may assume that K is locally ordered. Then we may regard K as a diagram S : Δ dim(K) inj → G F Set, that is as a finite-dimensional semi-simplicial G-set with stabilizers in F. The homotopy colimit over this finite diagram is equivalent to the barycentric subdivision of K; this can be verified explicitly using the Bousfield-Kan formula for the homotopy colimit amounts to pulling back certain bornologies, and the isomorphism in Lemma 9.3 is compatible with this operation on bornologies.
For the rest of the section, we fix a CP-functor M : GOrb → C. According to Definition 1.8, there is a C-valued strongly additive and continuous equivariant coarse homology theory E with transfers (see Definition 5.5)  We will consider E H as a contravariant functor in its first argument sending colimits to limits.
The following lemma clarifies the relation between E H and E.
that R ∈ H Fin(H) Orb. Since R was an arbitrary H-orbit in S(i) this implies that S(i) ∈ Let A be in PSh(GSet). Recall the notation (−) fin from Definition 8.26. We define the following functors from GOrb to C: Proof. By definition of E G (see (10.4) and Definition 5.7), the map L → M * is equivalent to the map Eι(F ∞ fin ((−) min,min ⊗ G can,min ) ⊗ G max,max ) → ΣEι(F 0 fin ((−) min,min ⊗ G can,min ) ⊗ G max,max ). By the Corollary 8.31 and the assumption that E • ι is continuous (note that G can,min is G-proper, uniformly discrete and coarsely connected), this map is equivalent to the map Eι(O ∞ hlg ( (−) disc × Rips(G can,min ))) ⊗ G can,min ) → Eι(O ∞ hlg ((−) disc ) ⊗ G can,min ) induced by the projection Rips(G can,min ) → * . Since twisting by G can,min commutes with precomposition by ι, this is the map in the statement of the proposition.
Let A be in PSh(GSet). Let F be a family of subgroups of G such that Fin ⊆ F. Recall Definition 1.7 of the relative assembly map.  Let H be a subgroup of G.