2.1 Groupoids

A groupoid is a small category whose morphisms are all invertible. As usual, we identify the groupoid with its set of morphisms, say G, and view its set of objects (also called units) ${G^{(0)}}$ as a subset of G by identifying objects with the corresponding identity morphisms. By definition, our groupoid G comes with range and source maps $\mathrm {r}: \: G \to {G^{(0)}}$ , $\mathrm {s}: \: G \to {G^{(0)}}$ , a multiplication map

and an inversion map $G \to G, \, g \mapsto g^{-1}$ such that $\mathrm {r}(g^{-1}) = \mathrm {s}(g)$ , $\mathrm {s}(g^{-1}) = \mathrm {r}(g)$ , $g g^{-1} = \mathrm {r}(g)$ and $g^{-1} g = \mathrm {s}(g)$ . These structure maps satisfy a list of conditions so that G becomes a small category (see, for instance, [Reference Renault76, Chapter I, Section 1]).

We are interested in the case of topological groupoids, that is, our groupoid G is endowed with a topology such that range, source, multiplication and inversion maps are all continuous. We do not assume that G is Hausdorff, but ${G^{(0)}}$ is always assumed to be Hausdorff in the subspace topology. We call ${G^{(0)}}$ the unit space. We will also always assume that ${G^{(0)}}$ is locally compact. A topological groupoid is called étale if the range map (and hence also the source map) is a local homeomorphism. It follows that ${G^{(0)}}$ is an open subspace of G in that case. An open subspace $U \subseteq G$ is called an open bisection if the restricted range and source maps $\mathrm {r} \vert _U: \: U \to \mathrm {r}(U), \, g \mapsto \mathrm {r}(g)$ , $\mathrm {s} \vert _U: \: U \to \mathrm {s}(U), \, g \mapsto \mathrm {s}(g)$ are bijections (and hence homeomorphisms). If G is étale, then G has a basis for its topology consisting of open bisections. Note that open bisections are always locally compact and Hausdorff because they are homeomorphic to open subspaces of the unit space. A topological groupoid G is called ample if it is étale and its unit space ${G^{(0)}}$ is totally disconnected. An étale groupoid is ample if and only if it has a basis for its topology consisting of compact open bisections.

A topological groupoid G is called minimal if for all $x \in {G^{(0)}}$ , the orbit $G.x \mathrel {:=} \left \{ \mathrm {r}(g) \text {: } g \in \mathrm {s}^{-1}(x) \right \}$ is dense in ${G^{(0)}}$ . Let $M(G)$ be the set of all nonzero Radon measures $\mu $ on ${G^{(0)}}$ which are invariant, that is, $\mu (\mathrm {r}(U)) = \mu (\mathrm {s}(U))$ for all open bisections $U \subseteq G$ . An ample groupoid G is said to have groupoid strict comparison for compact open sets (abbreviated by comparison in the following) if for all nonempty compact open sets $U, V \subseteq {G^{(0)}}$ with $\mu (U) < \mu (V)$ for all $\mu \in M(G)$ , there exists a compact open bisection $\sigma \subseteq G$ with $\mathrm {s}(\sigma ) = U$ and $\mathrm {r}(\sigma ) \subseteq V$ (see for instance [Reference Ma and Wu54, § 6]). Note that we restrict ourselves to nonempty open sets U and V because we want our definition of comparison to cover purely infinite groupoids, where $M(G) = \emptyset $ (see below).

Examples of groupoids with comparison include locally compact $\sigma $ -compact Hausdorff ample groupoids with compact unit spaces which are almost finite in the sense of [Reference Matui56, § 6] (see also [Reference Ma and Wu54, Proposition 7.2]). This covers many examples, for instance, AF groupoids, classes of transformation groupoids and tiling groupoids, as we explain below in § 2.2. Another class of groupoids with comparison is given by purely infinite minimal groupoids, in the following sense: An ample groupoid G is purely infinite minimal if and only if for all compact open subspaces $U, V \subseteq {G^{(0)}}$ with $V \neq \emptyset $ , there exists a compact open bisection $\sigma \subseteq G$ such that $\mathrm {s}(\sigma ) = U$ and $\mathrm {r}(\sigma ) \subseteq V$ (compare [Reference Matui57, § 4.2], but we do not require essential principality or Hausdorffness). Concrete examples are discussed in § 2.2. By definition, it is clear that purely infinite minimal groupoids have comparison.

2.2 Examples of groupoids

Let us discuss several classes of examples of topological groupoids.

2.2.4 Graph groupoids

Groupoids attached to graphs have been constructed in [Reference Renault76, Reference Deaconu15] (see also [Reference Kumjian, Pask, Raeburn and Renault49, Reference Paterson71, Reference Nyland and Ortega67], for instance). We will refer to these as graph groupoids, and remark that they are special cases of Deaconu-Renault groupoids. The reader will find criteria when graph groupoids are purely infinite minimal in [Reference Nyland and Ortega67].

Let us describe groupoids associated with shifts of finite type (abbreviated by SFT groupoids), which are special cases of graph groupoids. Consider a shift of finite type encoded by a finite directed graph with vertices $E^0$ and edges $E^1$ . Let A be the corresponding adjacency matrix, that is, $A = (A(j,i))_{j, i \in E^0}$ , where $A(j,i)$ is the number of edges from i to j. Assume that A is irreducible, in the sense that for all $j, i$ , there exists n such that $A^n(j,i)> 0$ and that A is not a permutation matrix. Let $X_A$ be the infinite path space of our graph, that is, $X_A$ consists of infinite sequences $(x_k)_k$ such that the target of $x_{k+1}$ is the domain of $x_k$ . Equip $X_A$ with the product topology. Define the one-sided shift $\sigma _A: \: X_A \to X_A$ by setting $(\sigma _A(x_k))_k = x_{k+1}$ . The groupoid attached to our shift of finite type is given by

$$ \begin{align*}G_A \mathrel{:=} \left\{ (x,n,y) \in X_A \times \mathbb{Z} \times X_A \text{: } \exists \, l, m \in \mathbb{Z} \text{ with } l, m \geq 0 \text{ such that } n = l-m \text{ and } \sigma_A^l(x) = \sigma_A^m(y) \right\}. \end{align*} $$

The topology of $G_A$ is generated by sets of the form

$$ \begin{align*}\left\{ (x,l-m,y) \in G_A \text{: } x \in U, \, y \in V, \, \sigma_A^l(x) = \sigma_A^m(y) \right\}, \end{align*} $$

where $l, m \in \mathbb {Z}$ with $l, m \geq 0$ , and U, V are open subspaces of $X_A$ such that $\sigma _A^l$ and $\sigma _A^m$ induce homeomorphisms

The unit space of $G_A$ is given by $\left \{ (x,0,x) \in G_A \text {: } x \in X_A \right \}$ , which is canonically homeomorphic to $X_A$ . Source and range maps are given by $\mathrm {s}(x,n,y) = y$ , $\mathrm {r}(x,n,y) = x$ and multiplication is given by $(x,n,y)(y,n',z) = (x,n+n',z)$ . In this setting, our groupoid $G_A$ is always purely infinite minimal, with unit space homeomorphic to the Cantor space. Note that compared to the convention in [Reference Matui57], the direction of our arrows is reversed.

2.3 Groupoid homology

Let us discuss groupoid homology in the general setting of non-Hausdorff groupoids. We refer the reader to [Reference Crainic and Moerdijk14, Reference Matui56] for more information about groupoid homology.

2.3.1 Functions with compact open support and definition of groupoid homology

Let Z be a topological space and $\mathscr {O}$ a family of subspaces $O \subseteq Z$ which are Hausdorff, open, locally compact and totally disconnected in the subspace topology, such that $Z = \bigcup _{O \in \mathscr {O}} O$ . This implies that $\mathscr {O}$ determines the topology of Z because a subset of Z is open if and only if its intersection with every $O \in \mathscr {O}$ is open.

Let $\mathsf {C}$ be a $\mathbb {Z}$ -module, that is, an abelian group. Given $c \in \mathsf {C}$ and a subset $U \subseteq Z$ , let $c_U$ denote the function $Z \to \mathsf {C}$ with $c_U \equiv c$ on U and $c_U \equiv 0$ on $Z \setminus U$ . Define

$$ \begin{align*}\mathscr{C}(Z,\mathsf{C}) \mathrel{:=} \mathrm{span} \left\{ c_U \text{: } U \text{ compact open subspace of some } O \in \mathscr{O}, \, c \in \mathsf{C} \right\}. \end{align*} $$

By construction, $\mathscr {C}(Z,\mathsf {C})$ consists of functions $Z \to \mathsf {C}$ . Clearly, $\mathscr {C}(Z,\mathsf {C})$ is an abelian group. As observed in [Reference Steinberg87, Proposition 4.3], $\mathscr {C}(Z,\mathsf {C})$ is also the linear span of all functions of the form $c_K$ , where c runs through all $c \in \mathsf {C}$ and K runs through all subspaces of Z which are compact, open and Hausdorff.

If Z is Hausdorff, then $\mathscr {C}(Z,\mathsf {C})$ is the set of continuous $\mathsf {C}$ -valued functions on Z with compact (open) support. In that case, disjointification is a key technique in the analysis of $\mathscr {C}(Z,\mathsf {C})$ . In the non-Hausdorff setting, disjointification is not possible in general because intersections of compact sets might not be compact. Instead, the result below (Lemma 2.2) serves as a replacement. We include it because similar proof techniques will appear frequently in the non-Hausdorff setting.

Given $U \subseteq Z$ , let $\mathsf {C}_U \mathrel {:=} \left \{ c_U \text {: } c \in \mathsf {C} \right \}$ . Consider $\bigoplus _U \mathsf {C}_U$ , where the sum runs over all compact open subsets U of some $O \in \mathscr {O}$ , and let $\mathscr {I}$ be the subgroup of $\bigoplus _U \mathsf {C}_U$ generated by elements of the form $c_{U \amalg V} - c_U - c_V$ , where $U, V$ are disjoint compact open subspaces of some $O \in \mathscr {O}$ .

Lemma 2.2. The kernel of the canonical projection map $\pi : \: \bigoplus _U \mathsf {C}_U \to \mathscr {C}(Z,\mathsf {C}), \, c_U \mapsto c_U$ coincides with $\mathscr {I}$ .

Proof. Suppose that $f = \sum _{i \in I} (c_i)_{U_i}$ satisfies $\pi (f) = 0$ , where I is a finite index set. Suppose that $\left \{ O_1, \dotsc , O_n \right \}$ is a finite subset of $\mathscr {O}$ such that for every $i \in I$ there exists $1 \leq m \leq n$ with $U_i \subseteq O_m$ . We proceed inductively on n.

If $n = 1$ , then all $U_i$ are contained in some $O \in \mathscr {O}$ . Then we can disjointify $U_i$ in O, that is, we let $\left \{ V_j \right \}$ be the set of nonempty subspaces of O of the form $\bigcap _{i \in I'} U_i \cap \bigcap _{i' \in I \setminus I'} U_i^c$ , where $I'$ runs through all nonempty subsets of I and $U_i^c = Z \setminus U_i$ . By construction, $\left \{ V_j \right \}$ is a family of pairwise disjoint subsets, and because O is Hausdorff, every $V_j$ is compact open. Moreover, every $U_i$ is a disjoint union of $V_j$ because $U_i = \bigcup _{i \in I'} (\bigcap _{i \in I'} U_i \cap \bigcap _{i' \in I \setminus I'} U_i^c)$ . So we can write $U_i = \coprod _{j_i} V_{j_i}$ . Hence, it follows that $(c_i)_{U_i} \equiv \sum _{j_i} (c_i)_{V_{j_i}} \ \mathrm {mod} \ \mathscr {I}$ . Hence, we obtain $f \equiv \sum _j (\tilde {c}_j)_{V_j} \ \mathrm {mod} \ \mathscr {I}$ . But now $\pi (f) = 0$ implies that every $\tilde {c}_j$ must be zero because the $V_j$ are pairwise disjoint. Hence, $f \equiv \sum _j (\tilde {c}_j)_{V_j} = 0 \ \mathrm {mod} \ \mathscr {I}$ , that is, $f \in \mathscr {I}$ .

Now, suppose that $n> 1$ . By disjointifying, we may assume that all the $U_i$ which are contained in a single $O_m$ are pairwise disjoint. Now, fix $i \in I$ with $U_i \subseteq O_n$ . If there exists $z \in Z$ with $z \in U_i$ , $z \notin U_{i'}$ for all $i' \neq i$ , then $c_i = 0$ as $0 = \pi (f)(z) = c_i$ . Hence, we may assume $U_i \subseteq \bigcup _{i \neq i' \in I} U_{i'}$ . Actually, we even have $U_i \subseteq \bigcup _{i' \in I'} U_{i'}$ , where $I'$ is a subset of $I \setminus \left \{ i \right \}$ such that for every $i' \in I'$ , we have $U_{i'} \subseteq O_m$ for some $1 \leq m < n$ . Here, we are using that all $U_{i'}$ with $U_{i'} \subseteq O_n$ are pairwise disjoint. Therefore, for every $x \in U_i$ there exists a compact open neighbourhood $V_x$ of x with $V_x \subseteq U_{i'}$ for some $i' \in I'$ . As $U_i$ is compact, we can write $U_i$ as a finite union $U_i = \bigcup _{j \in J} V_j$ , where $V_j = V_{x_j}$ for some $x_j \in U_i$ . As $U_i$ is contained in $O_n$ , all $V_j$ are also contained in $O_n$ so that we can (after disjointifying) assume that the $V_j$ are pairwise disjoint. Thus, $(c_i)_{U_i} \equiv \sum _{j \in J} (c_i)_{V_j} \ \mathrm {mod} \ \mathscr {I}$ and thus $f \equiv \sum _{j \in J} (c_i)_{V_j} + \sum _{i \neq i' \in I} (c_{i'})_{U_{i'}} \ \mathrm {mod} \ \mathscr {I}$ . Now, run this procedure for all $i \in I$ such that $U_i \subseteq O_n$ . In this way, we are able to replace $\left \{ O_1, \dotsc , O_n \right \}$ by $\left \{ O_1, \dotsc , O_{n-1} \right \}$ and then apply induction hypothesis.

We derive the following immediate consequence.

Corollary 2.3. $\mathscr {C}(Z,\mathsf {C}) \cong \mathscr {C}(Z,\mathbb {Z}) \otimes _{\mathbb {Z}} \mathsf {C}$ .

Let G be an ample groupoid with locally compact Hausdorff unit space ${G^{(0)}}$ . Let $\mathsf {C}$ be as above. Define

$$ \begin{align*}\mathscr{C}(G,\mathbb{Z}) \mathrel{:=} \mathrm{span} \left\{ 1_U \text{: } U \subseteq G \text{ compact open bisection} \right\}. \end{align*} $$

Note that since every compact open bisection is automatically Hausdorff, $\mathscr {C}(G,\mathbb {Z})$ coincides with $\mathscr {C}(Z,\mathbb {Z})$ as defined above, for $Z = G$ and $\mathscr {O}$ given by the collection of all open bisections. $\mathscr {C}(G,\mathbb {Z})$ becomes a $\mathbb {Z}$ -algebra with respect to convolution given by $(f_1 f_2) (g) = \sum _{h_1 h_2 = g} f_1(h_1) f_2(h_2)$ for $f_1, f_2 \in \mathscr {C}(G,\mathbb {Z})$ . Algebras of this form have for instance been studied in [Reference Steinberg87, Reference Clark, Exel, Pardo, Sims and Starling12].

Now, consider

$$ \begin{align*}\mathscr{C}({G^{(0)}},\mathsf{C}) \mathrel{:=} \mathrm{span} \left\{ c_U \text{: } U \subseteq {G^{(0)}} \text{ compact open}, \, c \in \mathsf{C} \right\}. \end{align*} $$

$\mathscr {C}({G^{(0)}},\mathsf {C})$ is a left- and right- $\mathscr {C}(G,\mathbb {Z})$ -module via

$$ \begin{align*}(f m) (x) = \sum_{g^{-1} g = x} f(g^{-1}) m(\mathrm{r}(g)), \qquad (m f) (x) = \sum_{g g^{-1} = x} m(\mathrm{s}(g)) f(g^{-1}), \end{align*} $$

for $f \in \mathscr {C}(G,\mathbb {Z})$ and $m \in \mathscr {C}({G^{(0)}},\mathsf {C})$ . Let us now define groupoid homology in terms of the bar resolution and then explain an alternative approach using $\mathrm {Tor\,}$ .

Let ${G^{(\nu )}} \mathrel {:=} \left \{ (g_1, \dotsc , g_{\nu }) \in G^{\nu } \text {: } \mathrm {s}(g_{\mu + 1}) = \mathrm {r}(g_{\mu }) \right \}$ , equipped with the subspace topology coming from the product topology on $G^{\nu }$ . Let $\mathscr {O}^{(\nu )}$ be the collection of subsets of ${G^{(\nu )}}$ of the form

where $O_{\mu }$ are open bisections with $\mathrm {s}(O_{\mu + 1}) = \mathrm {r}(O_{\mu })$ . Let $\mathscr {C}({G^{(\nu )}},\mathsf {C})$ be defined as above (with $Z = {G^{(\nu )}}$ , $\mathscr {O} = \mathscr {O}^{(\nu )}$ ). Consider the maps $\tilde {d}_{\nu }^{\mu }: \: {G^{(\nu )}} \to {G^{(\nu - 1)}}$ given by $\tilde {d}_1^0 = \mathrm {s}$ , $\tilde {d}_1^1 = \mathrm {r}$ and

$$ \begin{align*}\tilde{d}_{\nu}^{\mu}(g_1, \dotsc, g_{\nu}) = \begin{cases} (g_2, \dotsc, g_{\nu}) & \text{ if } \mu = 0,\\ (g_1, \dotsc, g_{\mu} g_{\mu + 1}, \dotsc, g_{\nu}) & \text{ if } 0 < \mu < \nu,\\ (g_1, \dotsc, g_{\nu - 1}) & \text{ if } \mu = \nu. \end{cases} \end{align*} $$

Since $\tilde {d}_{\nu }^{\mu }$ are local homeomorphisms, they induce homomorphisms $(\tilde {d}_{\nu }^{\mu })_*: \: \mathscr {C}({G^{(\nu )}},\mathsf {C}) \to \mathscr {C}({G^{(\nu - 1)}},\mathsf {C})$ given by $(\tilde {d}_{\nu }^{\mu })_*(f)(z) = \sum _{y \ \in \ (d_{\nu }^{\mu })^{-1}(z)} f(y)$ . Now, define

(1) $$ \begin{align} \tilde{\partial}_{\nu} \mathrel{:=} \sum_{\mu = 0}^{\nu} (-1)^{\mu} (\tilde{d}_{\nu}^{\mu})_*. \end{align} $$

It is straightforward to check that $B_*(G,\mathsf {C}) \mathrel {:=} (\mathscr {C}({G^{(\nu )}},\mathsf {C}), \tilde {\partial }_{\nu })_{\nu }$ is a chain complex. Groupoid homology is defined as the homology of this chain complex, that is,

$$ \begin{align*}H_*(G,\mathsf{C}) \mathrel{:=} H_*(B_*(G,\mathsf{C})). \end{align*} $$

2.3.2 Groupoid homology for examples

Given an AF groupoid G as in § 2.2.1, the $0$ -th homology $H_0(G,\mathsf {C})$ is given by the dimension group, with coefficients in $\mathsf {C}$ , of a Bratteli diagram describing G (the dimension group is independent of the choice of the diagram), and all higher homology groups vanish, that is, $H_*(G,\mathsf {C}) \cong \left \{ 0 \right \}$ for all $*> 0$ (see for instance [Reference Renault76, Reference Krieger47, Reference Farsi, Kumjian, Pask and Sims19, Reference Matui56]).

For a transformation groupoid $G = \Gamma \ltimes X$ as in § 2.2.2, it follows from the definitions that groupoid homology is canonically isomorphic to group homology with coefficients in the $\Gamma $ -module $C_c(X,\mathsf {C})$ , that is, $H_*(G,\mathsf {C}) \cong H_*(\Gamma ,C_c(X,\mathsf {C}))$ (see [Reference Brown6] for the definition of group homology). Here, $C_c(X,\mathsf {C})$ denotes the set of compactly supported continuous functions on X with values in $\mathsf {C}$ , where $\mathsf {C}$ is equipped with the discrete topology. Note that since X is Hausdorff, $C_c(X,\mathsf {C})$ coincides with $\mathscr {C}(X,\mathsf {C})$ from § 2.3.1. Let us describe groupoid homology more explicitly in the case of Cantor minimal systems, following [Reference Giordano, Putnam and Skau31]. In that case, $\Gamma = \mathbb {Z}$ , X is homeomorphic to the Cantor space, and $C_c(X,\mathsf {C}) = C(X,\mathsf {C})$ because X is compact. Let $\varphi \in \mathrm {Homeo}(X)$ be the homeomorphism corresponding to the canonical generator $1$ of $\mathbb {Z}$ . Then

$$ \begin{align*}H_0(\mathbb{Z} \ltimes X,\mathsf{C}) \cong C(X,\mathsf{C}) / \left\{ f - f \circ \varphi^{-1} \text{: } f \in C(X,\mathsf{C}) \right\}, \end{align*} $$

$H_1(\mathbb {Z} \ltimes X,\mathsf {C}) \cong \mathsf {C}$ and all higher homology groups vanish, that is, $H_*(\mathbb {Z} \ltimes X,\mathsf {C}) \cong \left \{ 0 \right \}$ for all $*>1$ .

Let us now consider tiling groupoids as in § 2.2.3. Given a tiling of $\mathbb {R}^d$ , let G be its tiling groupoid and $\Omega $ the hull space of our tiling. As observed in [Reference Proietti and Yamashita72, § 5.2], groupoid cohomology of G can be identified with sheaf cohomology of $\Omega $ . Using the description of groupoid homology of G in terms of group homology (see, for instance, [Reference Proietti and Yamashita72, § 5.2]) and Poincaré duality, we obtain an identification of groupoid homology $H_*(G)$ with the $(d-*)$ -th Čech cohomology $\check {H}^{d-*}(\Omega )$ of $\Omega $ . Explicit homology computations can be found in [Reference Gähler and Kellendonk26, Reference Forrest, Hunton and Kellendonk22, Reference Forrest, Hunton and Kellendonk23, Reference Gähler, Hunton and Kellendonk25].

For an SFT groupoid $G_A$ as in § 2.2.4, it was shown in [Reference Matui56, Theorem 4.14] that

$$ \begin{align*}H_*(G_A,\mathsf{C}) \cong \begin{cases} \mathrm{coker}\,(\mathrm{id} - A^t: \: \bigoplus_{E^0} \mathsf{C} \to \bigoplus_{E^0} \mathsf{C}) & \text{ if } *=0,\\ \mathrm{ker}\,(\mathrm{id} - A^t: \: \bigoplus_{E^0} \mathsf{C} \to \bigoplus_{E^0} \mathsf{C}) & \text{ if } *=1,\\ \left\{ 0 \right\} & \text{ else}. \end{cases} \end{align*} $$

For homology computations for more general graph groupoids, we refer to [Reference Nyland and Ortega67] and the references therein.

For groupoids of higher rank graphs as in § 2.2.5, groupoid homology has been computed for some cases in [Reference Farsi, Kumjian, Pask and Sims19]. Let us briefly summarize the result from [Reference Farsi, Kumjian, Pask and Sims19] in the one vertex case. Let $\Lambda $ be a one vertex k-graph and $G_{\Lambda }$ the corresponding groupoid. Let $\Lambda ^{\varepsilon _i}$ be the elements of $\Lambda $ with degree $\varepsilon _i$ , where $\varepsilon _i$ are the standard generators of $\mathbb {N}^k$ . Write $N_i \mathrel {:=} \lvert \Lambda ^{\epsilon _i}\rvert - 1$ . If $\Lambda $ is row-finite and $N_i \geq 1$ for all i, then

$$ \begin{align*}H_*(G_{\Lambda}) \cong \begin{cases} (\mathbb{Z}/\gcd(N_1, \dotsc, N_k) )^{\binom{k-1}{*}} & \text{ if } 0 \leq * \leq k-1,\\ \left\{ 0 \right\} & \text{ else}. \end{cases} \end{align*} $$

For products of SFT groupoids, which is another particular case of groupoids of higher rank graphs, a complete computation of groupoid homology has been established in [Reference Matui58, Proposition 5.4].

Consider Katsura–Exel–Pardo groupoids $G_{A,B}$ , which are special cases of groupoids attached to self-similar actions on graphs (see § 2.2.6). Let us present the groupoid homology computation in [Reference Ortega69] (see also [Reference Nyland and Ortega68]). We use the same notation as in § 2.2.6. Assume that A and B are row-finite matrices with integer entries and all entries of A are nonnegative. Suppose that for all $1 \leq i, j \leq N$ , $B_{i,j} = 0$ if and only if $A_{i,j} = 0$ . Then

$$ \begin{align*}H_*(G_{A,B}) \cong \begin{cases} \mathrm{coker}\,(\mathrm{id} - A) & \text{ if } * = 0,\\ \mathrm{ker}\,(\mathrm{id} - A) \oplus \mathrm{coker}\,(\mathrm{id} - B) & \text{ if } * = 1,\\ \mathrm{ker}\,(\mathrm{id} - B) & \text{ if } * = 2,\\ \left\{ 0 \right\} & \text{ else}. \end{cases} \end{align*} $$

For groupoids arising from piecewise affine transformations as in § 2.2.7, groupoid homology computations for classes of examples can be found in [Reference Li50].

2.3.3 Description of groupoid homology using derived functors

Our goal now is to show that

$$ \begin{align*}H_*(G,\mathsf{C}) \cong \mathrm{Tor\,}_*^{\mathscr{C}(G,\mathbb{Z})}(\mathscr{C}({G^{(0)}},\mathbb{Z}),\mathscr{C}({G^{(0)}},\mathsf{C})). \end{align*} $$

For Hausdorff groupoids, this is shown in [Reference Bönicke, Dell’Aiera, Gabe and Willett4, Reference Miller61]. We treat the case of non-Hausdorff groupoids. The results of § 2.3.3 are merely included for completeness; they are not needed in the sequel. However, some of the ideas will appear again in § 4.

First of all, note that, following for instance the approach in [Reference Miller61, § 4.1], we will be able to use standard results in homological algebra, even though they are usually formulated for unital rings whereas our ring $\mathscr {C}(G,\mathbb {Z})$ is in general not unital, only locally unital.

First of all, the inversion map induces an involution on $\mathscr {C}(G,\mathbb {Z})$ which flips the order of multiplication, which in turn allows us to interchange left- $\mathscr {C}(G,\mathbb {Z})$ -modules and right- $\mathscr {C}(G,\mathbb {Z})$ -modules and thus leads to the identification $ \mathrm {Tor\,}_*^{\mathscr {C}(G,\mathbb {Z})}(\mathscr {C}({G^{(0)}},\mathbb {Z}),\mathscr {C}({G^{(0)}},\mathsf {C})) \cong \mathrm { Tor\,}_*^{\mathscr {C}(G,\mathbb {Z})}(\mathscr {C}({G^{(0)}},\mathsf {C}),\mathscr {C}({G^{(0)}},\mathbb {Z}))$ . So it suffices to show that $H_*(G,\mathsf {C}) \cong \mathrm { Tor\,}_*^{\mathscr {C}(G,\mathbb {Z})}(\mathscr {C}({G^{(0)}},\mathsf {C}),\mathscr {C}({G^{(0)}},\mathbb {Z}))$ .

Next, we define another chain complex $E_*(G,\mathbb {Z}) \mathrel {:=} (\mathscr {C}({G^{(\nu + 1)}},\mathbb {Z}),\partial _{\nu + 1})_{\nu }$ , where $\partial _{\nu + 1}$ is given as follows: The maps $d_{\nu + 1}^{\mu }: \: {G^{(\nu + 1)}} \to {G^{(\nu )}}$ given by

$$ \begin{align*}d_{\nu + 1}^{\mu}(g_0, \dotsc, g_{\nu}) = \begin{cases} (g_0, \dotsc, g_{\mu} g_{\mu + 1}, \dotsc, g_{\nu}) & \text{ if } 0 \leq \mu < \nu,\\ (g_0, \dotsc, g_{\nu - 1}) & \text{ if } \mu = \nu, \end{cases} \end{align*} $$

are local homeomorphisms, hence induce homomorphisms $(d_{\nu + 1}^{\mu })_*: \: \mathscr {C}({G^{(\nu + 1)}},\mathbb {Z}) \to \mathscr {C}({G^{(\nu )}},\mathbb {Z})$ given by $(d_{\nu + 1}^{\mu })_*(f)(z) = \sum _{y \ \in \ (d_{\nu + 1}^{\mu })^{-1}(z)} f(y)$ . Define $\partial _{\nu + 1} \mathrel {:=} \sum _{\mu = 0}^{\nu } (d_{\nu + 1}^{\mu })_*$ . Now, consider the left G-action on ${G^{(\nu + 1)}}$ with respect to the anchor map $\rho : \: {G^{(\nu + 1)}} \to {G^{(0)}}, \, (g_0, \dotsc , g_{\nu }) \mapsto \mathrm {r}(g_0)$ and the action $g.(g_0, \dotsc , g_{\nu }) \mathrel {:=} (gg_0, g_1, \dotsc , g_{\nu })$ for all $g \in G$ and $(g_0, \dotsc , g_{\nu }) \in {G^{(\nu + 1)}}$ with $\rho (g_0, \dotsc , g_{\nu }) = \mathrm {s}(g)$ . This G-action induces a left- $\mathscr {C}(G,\mathbb {Z})$ -module structure on $\mathscr {C}({G^{(\nu + 1)}},\mathbb {Z})$ via $(fm)(z) \mathrel {:=} \sum _{g \in G, \, y \in {G^{(\nu + 1)}}, \, g.y \, = \, z} f(g) m(y)$ for $f \in \mathscr {C}(G,\mathbb {Z})$ , $m \in \mathscr {C}({G^{(\nu + 1)}},\mathbb {Z})$ . It is straightforward to check that $E_*(G,\mathbb {Z})$ is a chain complex in the category of $\mathscr {C}(G,\mathbb {Z})$ -modules.

Observe that $B_*(G,\mathsf {C}) \cong \mathscr {C}({G^{(0)}},\mathsf {C}) \otimes _{\mathscr {C}(G,\mathbb {Z})} E_*(G,\mathsf {C})$ . This is because the identification ${G^{(0)}} \times _G {G^{(\nu + 1)}} \cong {G^{(\nu )}}, \, (\mathrm {r}(g_0), (g_0, \dotsc , g_{\nu }))) \mapsto (g_1, \dotsc , g_{\nu })$ induces an isomorphism

$$ \begin{align*}\mathscr{C}({G^{(0)}},\mathsf{C}) \otimes_{\mathscr{C}(G,\mathbb{Z})} \mathscr{C}({G^{(\nu + 1)}},\mathbb{Z}) \cong \mathscr{C}({G^{(\nu)}},\mathsf{C}) \end{align*} $$

sending to .

Moreover, $E_*(G,\mathsf {C})$ is exact. The corresponding chain homotopy is induced by the maps $h_{\nu }: \: {G^{(\nu )}} \to {G^{(\nu + 1)}}, \, (g_0, \dotsc , g_{\nu - 1}) \mapsto (\mathrm {r}(g_0), g_0, \dotsc , g_{\nu - 1})$ for $\nu \geq 1$ and the inclusion $h_0: \: {G^{(0)}} \to G$ for $\nu = 0$ (see, for instance, [Reference Bönicke, Dell’Aiera, Gabe and Willett4, Reference Miller61], but note that $EG_{\bullet }$ in [Reference Bönicke, Dell’Aiera, Gabe and Willett4, § 2.3] does not coincide with our $G^{(\bullet )}$ ; instead, use the identification $EG_{\bullet } \ni (g_0, g_1, g_2, \dotsc ) \mapsto (g_0, g_0^{-1} g_1, g_1^{-1} g_2, \dotsc ) \in G^{(\bullet )}$ ).

Therefore, once we show that $\mathscr {C}({G^{(\nu + 1)}},\mathbb {Z})$ are flat left- $\mathscr {C}(G,\mathbb {Z})$ -modules, then we conclude that $H_*(G,\mathsf {C}) \cong \mathrm { Tor\,}_*^{\mathscr {C}(G,\mathbb {Z})}(\mathscr {C}({G^{(0)}},\mathbb {Z}),\mathscr {C}({G^{(0)}},\mathsf {C}))$ .

As a first step, observe that we have an isomorphism

(2) $$ \begin{align} \mathscr{C}(G,\mathbb{Z}) \otimes_{\mathscr{C}({G^{(0)}},\mathbb{Z})} \mathscr{C}({G^{(\nu)}},\mathbb{Z}) \cong \mathscr{C}({G^{(\nu + 1)}},\mathbb{Z}) \end{align} $$

sending $a \otimes f$ to the function $(g_0, g_1, \dotsc , g_{\nu }) \mapsto a(g_0) f(g_1, \dotsc , g_{\nu })$ . The proof is similar as the one for Lemma 2.2.

Now, suppose that M is a right- $\mathscr {C}(G,\mathbb {Z})$ -module. By [Reference Steinberg88], there is a sheaf $\mathcal M$ of $\mathbb {Z}$ -modules over ${G^{(0)}}$ together with a G-action such that $M \cong \Gamma _c({G^{(0)}},\mathcal M)$ as right- $\mathscr {C}(G,\mathbb {Z})$ -modules. Here, $\Gamma _c$ stands for continuous sections with compact support. Using the anchor map $\rho : \: {G^{(\nu )}} \to {G^{(0)}}$ , define the pullback $\rho ^* \mathcal M$ as a sheaf over ${G^{(\nu )}}$ with fibre $(\rho ^* \mathcal M)_z = \mathcal M_{\rho (z)}$ for $z \in {G^{(\nu )}}$ . Let $\mathscr {O}^{(\nu )}$ be the collection of subspaces of the form as above. Note that $\rho $ restricts to a homeomorphism on these subspaces . For a compact open subspace $U \subseteq O \in \mathscr {O}^{(\nu )}$ and $m \in M$ , define $(\rho ^*m)_U$ as the composite

Set

$$ \begin{align*}\Gamma_{\mathscr{C}}({G^{(\nu)}},\rho^*\mathcal M) \mathrel{:=} \mathrm{span} \left\{ (\rho^*m)_U \text{: } U \text{ compact open subspace of some } O \in \mathscr{O}^{(\nu)}, \, m \in M \right\}. \end{align*} $$

By construction, $\Gamma _{\mathscr {C}}$ consists of sections ${G^{(\nu )}} \to \rho ^*\mathcal M$ . Note that since ${G^{(\nu )}}$ is not Hausdorff in general, $\Gamma _{\mathscr {C}}$ does not coincide with $\Gamma _c$ . Now, a similar argument as for Lemma 2.2 implies that the following map,

(3) $$ \begin{align} M \otimes_{\mathscr{C}({G^{(0)}},\mathbb{Z})} \mathscr{C}({G^{(\nu)}},\mathbb{Z}) \cong \Gamma_{\mathscr{C}}({G^{(\nu)}},\rho^*\mathcal M), \end{align} $$

sending $m \otimes f$ to the function $z \mapsto m( \rho (z)) f(z)$ , is an isomorphism.

Now, we arrive at the desired conclusion.

Proposition 2.4. For all $\nu \geq 0$ , $\mathscr {C}({G^{(\nu + 1)}},R)$ is a flat left- $\mathscr {C}(G,\mathbb {Z})$ -module.

Proof. Suppose that is an exact sequence of right- $\mathscr {C}(G,\mathbb {Z})$ -modules. By [Reference Steinberg88], we obtain corresponding sheaves $\mathcal M$ and $\mathcal N$ . Moreover, $\iota $ induces injective homomorphisms $\iota _x: \: \mathcal M_x \hookrightarrow \mathcal N_x$ on the fibres, for all $x \in {G^{(0)}}$ . We want to show that

$$ \begin{align*}\iota \otimes \mathrm{id}: \: M \otimes_{\mathscr{C}(G,\mathbb{Z})} \mathscr{C}({G^{(\nu + 1)}},\mathbb{Z}) \to N \otimes_{\mathscr{C}(G,\mathbb{Z})} \mathscr{C}({G^{(\nu + 1)}},\mathbb{Z}) \end{align*} $$

is injective. Using equations (2) and (3), we obtain the following identification:

$$ \begin{align*} M \otimes_{\mathscr{C}(G,\mathbb{Z})} \mathscr{C}({G^{(\nu + 1)}},\mathbb{Z}) & \cong M \otimes_{\mathscr{C}(G,\mathbb{Z})} \mathscr{C}(G,\mathbb{Z}) \otimes_{\mathscr{C}({G^{(0)}},\mathbb{Z})} \mathscr{C}({G^{(\nu)}},\mathbb{Z})\\ & \cong M \otimes_{\mathscr{C}({G^{(0)}},\mathbb{Z})} \mathscr{C}({G^{(\nu)}},\mathbb{Z})\\ & \cong \Gamma_{\mathscr{C}}({G^{(\nu)}},\rho^*\mathcal M). \end{align*} $$

Similarly, $N \otimes _{\mathscr {C}(G,\mathbb {Z})} \mathscr {C}({G^{(\nu + 1)}},\mathbb {Z}) \cong \Gamma _{\mathscr {C}}({G^{(\nu )}},\rho ^*\mathcal N)$ . Identifying elements of $\Gamma _{\mathscr {C}}({G^{(\nu )}},\rho ^*\mathcal M)$ and $\Gamma _{\mathscr {C}}({G^{(\nu )}},\rho ^*\mathcal N)$ as functions on ${G^{(\nu )}}$ with values in $\mathcal M$ and $\mathcal N$ , respectively, we see that, under the identifications above, $\iota \otimes \mathrm {id}$ sends $f \in \Gamma _{\mathscr {C}}({G^{(\nu )}},\rho ^*\mathcal M)$ to the map $z \mapsto \iota _{\rho (z)} (f(z))$ . And since $\iota _x$ is injective for all $x \in {G^{(0)}}$ , we deduce that $\iota \otimes \mathrm {id}$ must be injective as well, as desired.

As explained above, using [Reference Miller61, Proposition 4.21], this leads to the desired description of groupoid homology in terms of $\mathrm {Tor\,}$ .

Theorem 2.5. $H_*(G,\mathsf {C}) \cong \mathrm {Tor\,}_*^{\mathscr {C}(G,\mathbb {Z})}(\mathscr {C}({G^{(0)}},\mathbb {Z}),\mathscr {C}({G^{(0)}},\mathsf {C}))$ .

2.4 Topological full groups

In the following, let G be an ample groupoid with locally compact Hausdorff unit space ${G^{(0)}}$ .

If G is effective, that is, when the interior of the isotropy subgroupoid $\left \{ g \in G \text {: } \mathrm {r}(g) = \mathrm {s}(g) \right \}$ coincides with ${G^{(0)}}$ , then the map sending $\sigma \in \boldsymbol {F}(G_U^U) \subseteq \boldsymbol {F}(G)$ to the homeomorphism ${G^{(0)}} \to {G^{(0)}}$ given by $x \mapsto \sigma .x$ on U and identity on ${G^{(0)}} \setminus U$ is injective so that we may view $\boldsymbol {F}(G)$ as a subgroup of $\mathrm {Homeo}({G^{(0)}})$ . Here, we use the notation that $\sigma .x$ denotes $\mathrm {r}(g)$ for the unique element $g \in \sigma $ with $\mathrm {s}(g) = x$ .

Topological full groups first appeared in [Reference Krieger47, Reference Giordano, Putnam and Skau32] (for special classes of groupoids). Several subgroups of $\boldsymbol {F}(G)$ have been constructed, for instance, the alternating full group $\boldsymbol {A}(G)$ (see [Reference Nekrashevych66]). It is known that for almost finite or purely infinite groupoids G which are minimal, effective and Hausdorff, with unit space ${G^{(0)}}$ homeomorphic to the Cantor space, the alternating full group coincides with the commutator subgroup $\boldsymbol {D}(G)$ of $\boldsymbol {F}(G)$ (see [Reference Matui57, Reference Nekrashevych66]).

Nekrashevych showed in [Reference Nekrashevych66] that for every minimal, effective groupoid G whose unit space ${G^{(0)}}$ is homeomorphic to the Cantor space, the alternating full group $\boldsymbol {A}(G)$ is simple. Moreover, again for minimal, effective groupoids G with unit space ${G^{(0)}}$ homeomorphic to the Cantor space, it is possible to reconstruct the groupoid G from the topological full group $\boldsymbol {F}(G)$ (see [Reference Matui57, Reference Nekrashevych66]). A far-reaching generalization of these results has been obtained in [Reference Matte Bon55].

Matui formulated the AH-conjecture, which describes the first homology group $H_1(\boldsymbol {F}(G))$ in terms of groupoid homology of G. He constructed an index map $I: \: H_1(\boldsymbol {F}(G)) \to H_1(G)$ and conjectured for every minimal, effective groupoid G whose unit space ${G^{(0)}}$ is homeomorphic to the Cantor space, there is an exact sequence

Note that Matui restricts his discussion to second countable Hausdorff groupoids. Moreover, he formulated the AH-conjecture in terms of the abelianization $\boldsymbol {F}(G)^{\mathrm {ab}}$ of $\boldsymbol {F}(G)$ , which is isomorphic to $H_1(\boldsymbol {F}(G))$ . The AH-conjecture has been verified for all principal, almost finite, second countable Hausdorff groupoids as well as groupoids arising from shifts of finite type, products of groupoids from shifts of finite type, graph groupoids and Katsura–Exel–Pardo groupoids as well as transformation groupoids of of odometers and Cantor minimal dihedral systems (see [Reference Matui56, Reference Matui57, Reference Matui58, Reference Nyland and Ortega67, Reference Nyland and Ortega68, Reference Scarparo79, Reference Scarparo80]).

2.5 Examples of topological full groups

For an AF groupoid G, the topological full group $\boldsymbol {F}(G)$ is the increasing union of finite direct sums of finite symmetric groups.

For a transformation groupoid G of a Cantor minimal $\mathbb {Z}$ -system, Juschenko and Monod showed that $\boldsymbol {F}(G)$ is amenable [Reference Juschenko and Monod39]. By taking alternating full groups $\boldsymbol {A}(G)$ in the case of minimal subshifts (in this case $\boldsymbol {A}(G)$ coincides with the commutator subgroup $\boldsymbol {D}(G)$ of $\boldsymbol {F}(G)$ ), this produces the first examples of finitely generated amenable infinite simple groups, answering an open problem in group theory.

Now, let $D_{\infty }$ be the infinite dihedral group $\mathbb {Z} \rtimes (\mathbb {Z} / 2) \cong (\mathbb {Z} / 2) * (\mathbb {Z} / 2)$ . Starting with a Cantor minimal $D_{\infty }$ -system, Nekrashevych constructed another Cantor minimal $\Gamma $ -system (for some new group $\Gamma $ ) and shows that, under certain conditions, the alternating full group of the groupoid of germs for the $\Gamma $ -action is a finitely generated simple periodic group of intermediate growth (see [Reference Nekrashevych65]). This again answers an open problem in group theory.

Other examples include topological full groups of transformation groupoids of Cantor minimal $\mathbb {Z}^d$ -systems, tiling groupoids or topological full groups of interval exchange transformations (the latter have been studied in [Reference Chornyi, Juschenko and Nekrashevych11]).

The groupoids discussed so far are almost finite (with the exception of Nekrashevych’s examples, where the groupoids could be non-Hausdorff and almost finiteness is not known). Let us now turn to topological full groups of purely infinite groupoids.

Let $G_2$ be the groupoid attached to the one-sided full shift on two symbols. In the language of § 2.2.4, the graph we consider consists of one vertex and two edges (which must then be loops), the corresponding adjacency matrix A is given by the $1 \times 1$ -matrix with entry $2$ , and we set $G_2 \mathrel {:=} G_A$ . Then $\boldsymbol {F}(G_2) \cong V$ , where V is Thompson’s group V (see [Reference Cannon, Floyd and Parry9]). This was first observed in [Reference Nekrashevych63] (see also [Reference Matui57]). More generally, if we consider the one-sided full shift on k symbols, its graph given by one vertex and k edges, the adjacency matrix given by the $1 \times 1$ -matrix with entry k and let the corresponding groupoid be $G_k$ , then $\boldsymbol {F}(\mathcal R_r \times G_k^n) \cong n V_{k,r}$ . Here, $\mathcal R_r$ is the groupoid given by the full equivalence relation on the finite set $\left \{ 1, \dotsc , r \right \}$ , and $n V_{k,r}$ are the Brin–Higman–Thompson groups (see, for instance, [Reference Higman37, Reference Brin5] and also [Reference Matui58]).

Szymik and Wahl show that the group homology $H_*(V_{k,r})$ does not depend on r and that $V_{k,r}$ is acyclic for $k=2$ . They also produce further computations of parts of $H_*(V_{k,r})$ (see [Reference Szymik and Wahl90]). Their work is based on Cantor algebras, which lead to the construction of a small permutative category and hence an algebraic K-theory spectrum $\mathbb {K}$ such that $H_*(V_{n,r}) \cong H_*(\Omega ^{\infty }_0 \mathbb {K})$ . Here, $\Omega ^{\infty } \mathbb {K}$ is the infinite loop space corresponding to $\mathbb {K}$ , and $\Omega ^{\infty }_0 \mathbb {K}$ denotes the path component of the base point of $\Omega ^{\infty } \mathbb {K}$ (these notions are introduced in § 2.6). The results in [Reference Szymik and Wahl90] are then obtained by analysing the homotopy groups of $\mathbb {K}$ . It is not immediate how to carry over the constructions in [Reference Szymik and Wahl90] to more general groupoids because the notion of Cantor algebras is tailored to the situation of Higman–Thompson groups.

Topological full groups for SFT groupoids and products of SFT groupoids have been studied in detail in [Reference Matui57, Reference Matui58].

For groupoids G arising from self-similar actions on trees, the topological full groups $\boldsymbol {F}(G)$ are isomorphic to Röver-Nekrashevych groups (see [Reference Röver77, Reference Nekrashevych63, Reference Nekrashevych64] as well as [Reference Skipper, Witzel and Zaremsky83]). These groups have interesting finiteness properties. We say that a group is of type $\mathrm {F}_n$ if it admits a classifying space with a compact n-skeleton. These finiteness properties play an important role in group homology and specialize to familiar notions in low dimensions (a group is of type $\mathrm {F}_1$ if and only if it is finitely generated and of type $\mathrm {F}_2$ if and only if it is finitely presented). [Reference Skipper, Witzel and Zaremsky83] shows that topological full groups arising from classes of self-similar actions give rise to first examples of infinite simple groups which are of type $\mathrm {F}_{n-1}$ but not of type $\mathrm {F}_n$ , for each n.

For groupoids G arising from piecewise affine transformations [Reference Li50], the topological full groups $\boldsymbol {F}(G)$ are isomorphic to groups considered in [Reference Stein86] (for the parameters $l=1$ , $A = \mathbb {Z}[\lambda ,\lambda ^{-1}]$ , $P=\left \langle \lambda \right \rangle $ in the terminology of [Reference Stein86]). Moreover, a similar construction as in [Reference Li50] leads to ample groupoids whose topological full groups coincide with the groups denoted by $G(l,A,P)$ in [Reference Stein86], for arbitrary parameters $l, A, P$ (see [Reference Tanner91]).

As these examples show, topological full groups and the closely related notion of alternating full groups lead to new examples in group theory with interesting properties. They provide a rich supply of infinite simple groups. However, even though we have seen much progress regarding particular example classes and spectacular advances have been made in our understanding of these groups, general results about topological full groups are rare and seem to be difficult to obtain. For instance, not much is known regarding analytic properties of topological full groups in general. All in all, it is a fascinating yet challenging problem to develop a better understanding of the interplay between group-theoretic properties of topological full groups and dynamical properties of the underlying topological groupoids.

2.6 Algebraic K-theory spectra of small permutative categories

Let us now describe the construction of algebraic K-theory spectra from small permutative categories as in [Reference Segal82, Reference Thomason92]. We follow the exposition in [Reference Elmendorf and Mandell17]. We will use the language of simplicial sets (see, for instance, [Reference Gabriel and Zisman24, Reference Goerss and Jardine34]) and of spectra in the sense of algebraic topology (see, for instance, [Reference Switzer89, Reference Adams2]).

Definition 2.7. A small permutative category is a small category $\mathfrak B$ with object set $\mathrm {obj}\, \mathfrak B$ , morphism set $\mathrm {mor}\, \mathfrak B$ , together with a functor $\oplus : \: \mathfrak B \times \mathfrak B \to \mathfrak B$ , an object $0 \in \mathrm {obj}\, \mathfrak B$ and natural isomorphisms

$$ \begin{align*}\left\{ \pi_{u,u'} \in \mathrm{mor}\, \mathfrak B: \: \pi_{u,u'}: \: u \oplus u' \cong u' \oplus u \right\}_{u,u' \, \in \, \mathrm{obj}\, \mathfrak B} \end{align*} $$

such that $\oplus $ is associative with unit $0$ , and we have $\pi _{0,u} = \pi _{u,0} = \mathrm {id}_u$ , $\pi _{u',u} \pi _{u,u'} = \mathrm {id}_{u \oplus u'}$ , that is, the diagram

commutes, and $(\pi _{u,u"} \oplus \mathrm {id}_{u'}) (\mathrm {id}_u \oplus \pi _{u',u"}) = \pi _{u \oplus u',u"}$ , that is, the diagram

commutes, for all objects $u, u', u"$ of $\mathfrak B$ .

Given $\sigma \in \mathrm {mor}\, \mathfrak B$ , we write $\mathfrak t(\sigma )$ for its target and $\mathfrak d(\sigma )$ for its domain, and we denote by $\mathfrak B(v,u)$ the set of morphisms of $\mathfrak B$ from u to v, that is, $\mathfrak B(v,u) = \left \{ \sigma \in \mathrm {mor}\, \mathfrak B \text {: } \mathfrak t(\sigma ) = v, \, \mathfrak d(\sigma ) = u \right \}$ .

Now, let A be a finite based set, that is, a finite set with a choice of an element called the base point.

Definition 2.8. Given a small permutative category $\mathfrak B$ and a finite based set A, let $\mathfrak B(A)$ be the category with objects of the form $\left \{ u_S, \varphi _{T,T'} \right \}_{S,T,T'}$ , where $S, T, T'$ run through all subsets of A not containing the base point with $T \cap T' = \emptyset $ , $u_S \in \mathrm {obj}\, \mathfrak B$ for all S and $\varphi _{T,T'} \in \mathfrak B(u_{T \cup T'}, u_T \oplus u_{T'})$ are isomorphisms for all $T, T'$ . We require that for $S = \emptyset $ , $u_{\emptyset } = 0$ , and for $T = \emptyset $ , $\varphi _{\emptyset ,T'} = \mathrm {id}_{u_{T'}}$ . Moreover, for all pairwise disjoint $T, T', T"$ , the following diagrams should commute:

A morphism $f: \: \left \{ u_S, \varphi _{T,T'} \right \} \to \left \{ \tilde {u}_S, \tilde {\varphi }_{T,T'} \right \}$ consists of $f_S \in \mathfrak B(\tilde {u}_S,u_S)$ for all S such that $f_{\emptyset } = \mathrm {id}_0$ and the following diagram commutes for all disjoint $T, T'$ :

The following result is for instance explained in [Reference Elmendorf and Mandell17, Theorem 4.2].

Next, we construct a $\Gamma $ -space in the sense of [Reference Segal82] (see also [Reference Bousfield and Friedlander3]), that is, a functor from the category of finite based sets to the category of simplicial sets, sending the trivial based set to the simplicial set which is constantly given by one point. Given a finite based set A, let $\mathfrak N \mathfrak B (A)$ be the simplicial set with p-simplices $\mathfrak N_p \mathfrak B(A)$ consisting of elements of the form $(f_1, \dotsc , f_p)$ , where are morphisms in $\mathfrak B(A)$ such that . The face maps are given by $\delta ^1_0(f) = \mathfrak d(f)$ , $\delta ^1_1(f) = \mathfrak t(f)$ and

Degeneracy maps are also part of the structure of a simplicial set, but since these are not needed for homology, we do not recall their definition here (see for instance [Reference Gabriel and Zisman24, Reference Goerss and Jardine34]).

Given a map $\alpha : \: A \to \bar {A}$ of finite based sets, define a map of simplicial sets $\mathfrak N \mathfrak B(\alpha ): \: \mathfrak N \mathfrak B(A) \to \mathfrak N \mathfrak B(\bar {A})$ by setting $\mathfrak N_p \mathfrak B(\alpha )(f_1, \dotsc , f_p) \mathrel {:=} (f^{\alpha }_1, \dotsc , f^{\alpha }_p)$ .

The topological space $S^1$ is modelled by the simplicial set, also denoted by $S^1$ , given by $S^1_q = \left \{ 0, \dotsc , q \right \}$ with base point $0$ and face maps

$$ \begin{align*}d^q_{\bullet}: \: S^1_q \to S^1_{q-1}, \, a \mapsto \begin{cases} a & \text{ if } a \leq \bullet,\\ a-1 & \text{ if } a> \bullet. \end{cases} \end{align*} $$

For $\bullet = q$ , $d^q_q$ sends q to $0$ . Again, we do not need the precise form of the degeneracy maps. Note that we have $S^1 \cong \Delta ^1 / \partial \Delta ^1$ (see for instance [Reference Gabriel and Zisman24, Reference Goerss and Jardine34]).

To obtain simplicial sets describing $S^n$ , set $S^n = S^1 \wedge \dotso \wedge S^1$ (n factors), as simplicial sets. Here, the smash product $X \wedge S^1$ (where X is some simplicial set) is given by

$$ \begin{align*}(X \wedge S^1)_q = (X_q \times S^1_q) / ((\left\{ 0_{X_q} \right\} \times S^1_q) \cup (X_q \times \left\{ 0_{S^1_q} \right\})) \cong X_q^\times \times (S^1_q)^\times \cup \left\{ 0 \right\}, \end{align*} $$

where $0$ stands for base point and, for a based set A with base point $0$ , $A^\times $ denotes $A \setminus \left \{ 0 \right \}$ . The face maps are induced from the face maps of X and $S^1$ . Concretely,

$$ \begin{align*}S^n_q = \left\{ (a_1, \dotsc, a_n) \text{: } a_m \in \left\{ 1, \dotsc, q \right\} \right\} \cup \left\{ 0 \right\}, \end{align*} $$

and $d^q_{\bullet }: \: S^n_q \to S^n_{q-1}, \, (a_1, \dotsc , a_n) \mapsto (b_1, \dotsc , b_n)$ is given by

$$ \begin{align*}b_m = \begin{cases} a_m & \text{ if } a_m \leq \bullet,\\ a_m - 1 & \text{ if } a_m> \bullet. \end{cases} \end{align*} $$

For $\bullet = q$ , $b_m$ is defined to be $0$ if $a_m = q$ . For $n = 0$ , we set $S^0_q \mathrel {:=} \left \{ 0,1 \right \}$ for all q and $d^q_{\bullet } = \mathrm {id}_{\left \{ 0,1 \right \}}$ .

Now, consider the bisimplicial set $(p,q) \mapsto \mathfrak N_p \mathfrak B(S^n_q)$ , with face maps and $\mathfrak N_q \mathfrak B(d_{\bullet }): \: \mathfrak N_p \mathfrak B(S^n_q) \to \mathfrak N_p \mathfrak B(S^n_{q-1})$ . Furthermore, form the diagonal, that is, the simplicial set $\mathfrak N \mathfrak B(S^n)$ given by $\mathfrak N \mathfrak B(S^n)_q \mathrel {:=} \mathfrak N_q \mathfrak B(S^n_q)$ and the face maps, for $0 \leq \bullet \leq q$ , given by the composites

Now, we are ready for the definition of the algebraic K-theory spectrum $\mathbb {K}(\mathfrak B)$ . The n-th simplicial set is given by $X_n = \mathfrak N \mathfrak B(S^n)$ . We need to define the structure maps $\varsigma _n: \: \Sigma X_n \to X_{n+1}$ . $\Sigma X_n$ is given by the smash product $\mathfrak N \mathfrak B(S^n) \wedge S^1$ . As explained above, we have $(\mathfrak N \mathfrak B(S^n) \wedge S^1)_q \cong (\mathfrak N \mathfrak B(S^n)_q)^\times \times (S^1_q)^\times \cup \left \{ 0 \right \}$ , and the face maps are induced by the ones of $\mathfrak N \mathfrak B(S^n)$ and $S^1$ . Every $b \in (S^1_q)^\times $ induces the map

$$ \begin{align*}\iota_b: \: S^n_q \to S^{n+1}_q, \, \boldsymbol{a} \mapsto (\boldsymbol{a},b), \, 0 \mapsto 0. \end{align*} $$

This in turn induces $\mathfrak N_q \mathfrak B(\iota _b): \: \mathfrak N_q \mathfrak B(S^n_q) \to \mathfrak N_q \mathfrak B(S^{n+1}_q)$ . Therefore, we obtain the based maps

$$ \begin{align*}(\mathfrak N \mathfrak B(S^n) \wedge S^1)_q \to \mathfrak N \mathfrak B(S^{n+1})_q, \, (\boldsymbol{f},b) \mapsto \mathfrak N_q \mathfrak B(\iota_b)(\boldsymbol{f}), \, 0 \mapsto 0. \end{align*} $$

In this way, we obtain the simplicial map $\varsigma _n: \: \mathfrak N \mathfrak B(S^n) \wedge S^1 \to \mathfrak N \mathfrak B(S^{n+1})$ , as desired.

We have constructed a spectrum $\mathbb {K}(\mathfrak B)$ of simplicial sets. By taking geometric realizations, we also obtain a spectrum consisting of topological spaces.

It turns out that $\mathbb {K}(\mathfrak B)$ is a symmetric spectrum which is also a connective positive $\Omega $ -spectrum, that is, the adjoint maps $X_n \to \Omega X_{n+1}$ of $\varsigma _n$ are homotopy equivalences for all $n \geq 1$ . The infinite loop space attached to $\mathbb {K}(\mathfrak B)$ is given by $\Omega ^{\infty } \mathbb {K}(\mathfrak B) \mathrel {:=} \Omega X_1$ (see [Reference Segal82]). Note that $\Omega ^{\infty } \mathbb {K}(\mathfrak B)$ coincides up to homotopy with $\Omega B \vert \mathfrak B \vert $ . Here, $\vert \mathfrak B \vert $ is the nerve or classifying space of $\mathfrak B$ , and $B \vert \mathfrak B \vert $ is the bar construction of the monoid $\vert \mathfrak B \vert $ , where the monoid structure is induced by the operation $\oplus $ . We refer the reader to [Reference Adams2] for more information about infinite loop space theory.

Next, we briefly recall the definition of homology groups for simplicial sets and spectra. Let $\mathsf {C}$ be an abelian group as above. Let X be a simplicial set with face maps $d_{\bullet }$ . Define a chain complex $(C_* X, C_* d)$ by $C_q X \mathrel {:=} \bigoplus _{X_q} \mathsf {C}$ , setting $C_q d \mathrel {:=} \sum _{\bullet = 0}^q (-1)^{\bullet } C_q d_{\bullet }$ , where $C_q d_{\bullet }$ is the homomorphism $C_q X \to C_{q-1} X$ induced by $d_{\bullet }$ . The homology $H_*(X,\mathsf {C})$ is by definition the homology of the chain complex $(C_* X, C_* d)$ .

Note that by the Eilenberg–Zilber theorem (see for instance [Reference Goerss and Jardine34, Chapter IV, § 2.2]), given a bisimplicial set like $(p,q) \mapsto \mathfrak N_p \mathfrak B(S^n_q)$ , the homology of the diagonal (in our case $\mathfrak N \mathfrak B(S^n)$ ) is naturally isomorphic to the homology of the total complex (denoted by $C_{p,q} \mathfrak N_p \mathfrak B(S^n_q)$ in our case) associated to the bisimplicial set. Moreover, $H_*(X,\mathsf {C})$ is canonically isomorphic to the singular homology with coefficients in $\mathsf {C}$ of the geometric realization of X (see, for instance, [Reference Gabriel and Zisman24, Appendix Two, § 1]).

Let us now define the homology of $\mathbb {K}(\mathfrak B)$ . Applying the above definition of homology groups to $X_n$ , we obtain the homology groups $H_{*+n}(X_n,\mathsf {C})$ .

Definition 2.10. $H_*(\mathbb {K}(\mathfrak B),\mathsf {C}) \mathrel {:=} \varinjlim _n H_{*+n}(X_n,\mathsf {C})$ , where the inductive limit is taken with respect to the connecting maps

Here, the first map is the suspension isomorphism.

Let us also introduce the (stable) homotopy groups of $\mathbb {K}(\mathfrak B)$ .

Definition 2.11. $\pi _*(\mathbb {K}(\mathfrak B)) \mathrel {:=} \varinjlim _n \pi _{*+n}(X_n)$ , where the inductive limit is taken with respect to the connecting maps

Here, the first map is the suspension homomorphism.

Note that $\pi _*(\mathbb {K}(\mathfrak B)) \cong \pi _*(\Omega ^{\infty } \mathbb {K}(\mathfrak B))$ (see, for instance, [Reference Schwede81, Chapter I, § 1]).

The construction of algebraic K-theory spectra for small permutative spectra is functorial with respect to permutative functors, that is, functors $\Phi : \: \mathfrak B \to \mathfrak C$ between small permutative categories which are compatible with all the structures, that is, $\Phi (0) = 0$ , $\Phi (u \oplus v) = \Phi (u) \oplus \Phi (v)$ , similarly for morphisms, and $\Phi (\pi _{u,v}) = \pi _{\Phi (u),\Phi (v)}$ . Indeed, given such a functor $\Phi : \: \mathfrak B \to \mathfrak C$ and a finite based set A, then we obtain a functor $\Phi (A): \: \mathfrak B(A) \to \mathfrak C(A)$ sending $\left \{ u_S, \varphi _{T,T'} \right \}$ to $\left \{ \Phi (u_S), \Phi (\varphi _{T,T'}) \right \}$ and a morphism $\left \{ f_S \right \}$ to $\left \{ \Phi (f_S) \right \}$ . Moreover, a based map $\alpha : \: A \to \bar {A}$ between finite based sets induces functors $\mathfrak B(\alpha ): \: \mathfrak B(A) \to \mathfrak B(\bar {A})$ and $\mathfrak C(\alpha ): \: \mathfrak C(A) \to \mathfrak C(\bar {A})$ such that the following diagram commutes:

So $\Phi $ induces a map of bisimplicial sets between $(p,q) \mapsto \mathfrak N_p \mathfrak B(S^n_q)$ and $(p,q) \mapsto \mathfrak N_p \mathfrak C(S^n_q)$ . It is straightforward to check that these maps are compatible with the structure maps defining the spectra $\mathbb {K}(\mathfrak B)$ and $\mathbb {K}(\mathfrak C)$ .

3.1 Functoriality of our construction

Our construction of small permutative categories of bisections is functorial for two types of maps, open embeddings and fibrewise bijective proper surjections. These types of maps also appear in [Reference Li51, § 5]. In the following, let G and $\tilde {G}$ be ample groupoids with locally compact Hausdorff unit spaces. Denote by $\mathrm {s},\mathrm {r}$ the source and range maps of G and by $\tilde {\mathrm {s}},\tilde {\mathrm {r}}$ the source and range maps of $\tilde {G}$ . Let Z be a G-space with anchor map $\rho : \: Z \to {G^{(0)}}$ and $\tilde {Z}$ be a $\tilde {G}$ -space with anchor map $\tilde {\rho }: \: \tilde {Z} \to \tilde {G}^{(0)}$ . Assume that $\rho $ and $\tilde {\rho }$ are local homeomorphisms. Let $\mathscr {O}$ be a family of open Hausdorff subspaces of Z covering Z and $\tilde {\mathscr {O}}$ a family of open Hausdorff subspaces of $\tilde {Z}$ covering $\tilde {Z}$ . As above, construct the small permutative categories $\mathfrak B_{G \curvearrowright Z}$ and $\mathfrak B_{\tilde {G} \curvearrowright \tilde {Z}}$ .

3.1.1 The case of open embeddings

Suppose that $\phi : \: G \to \tilde {G}$ is a groupoid homomorphism which is an embedding with open image and that $\psi : \: Z \to \tilde {Z}$ is a continuous map such that for all $O \in \mathscr {O}$ there exists $\tilde {O} \in \tilde {\mathscr {O}}$ such that $\psi (O) \subseteq \tilde {O}$ and that $\psi $ restricts to a homeomorphism $\psi \vert _O: \: O \cong \psi (O)$ . Furthermore, we require that the diagram

commutes and that $\psi (g.z) = \phi (g).\psi (z)$ for all $g \in G$ and $z \in Z$ with $\mathrm {s}(g) = \rho (z)$ . In this situation, we define a functor $F_{\phi ,\psi }: \: \mathfrak B_{G \curvearrowright Z} \to \mathfrak B_{\tilde {G} \curvearrowright \tilde {Z}}$ as follows: On objects, set $F_{\phi ,\psi }(\coprod _i (i,U_i)) \mathrel {:=} \coprod _i (i,\psi (U_i))$ . Given a morphism $\sigma = \coprod _{j,i} (\mathbb {s}_{j,i}, \sigma _{j,i}, U_{j,i})$ in $\mathfrak B_{G \curvearrowright Z}$ , set $F_{\phi ,\psi }(\sigma ) \mathrel {:=} \coprod _{j,i} (\mathbb {s}_{j,i}, \phi (\sigma _{j,i}), \psi (U_{j,i}))$ . It is straightforward to check that this is well defined, that is, this defines a morphism in $\mathfrak B_{\tilde {G} \curvearrowright \tilde {Z}}$ . For instance, we have that

$$ \begin{align*}\tilde{\mathrm{s}}(\phi(\sigma_{j,i})) = \phi(\mathrm{s}(\sigma_{j,i})) = \phi(\rho(U_{j,i})) = \tilde{\rho}(\psi(U_{j,i})), \end{align*} $$

and moreover, the restriction

$$ \begin{align*}\tilde{\rho} \vert_{\psi(U_{j,i})}: \: \psi(U_{j,i}) \to \tilde{\rho}(\psi(U_{j,i})) = \phi(\rho(U_{j,i})) \end{align*} $$

is a homeomorphism because the composite

coincides with

and $\psi $ as well as $\phi \circ \rho $ are homeomorphisms.

Given two morphisms $\tau = \coprod _{k,j} (\mathbb {s}_{k,j}, \tau _{k,j}, V_{k,j})$ and $\sigma = \coprod _{j,i} (\mathbb {s}_{j,i}, \sigma _{j,i}, U_{j,i})$ with $\mathfrak d(\tau ) = \mathfrak t(\sigma )$ , we have

$$ \begin{align*} F_{\phi,\psi}(\tau) F_{\phi,\psi}(\sigma) &= \coprod_{k,i} \big( \mathbb{s}_{k,i}, \coprod_j \phi(\tau_{k,j}) \phi(\sigma_{j,i}), \coprod_j \phi(\sigma_{j,i})^{-1}.(\psi(V_{k,j}) \cap \phi(\sigma_{j,i}).\psi(U_{j,i})) \big)\\ &= \coprod_{k,i} \big( \mathbb{s}_{k,i}, \coprod_j \phi(\tau_{k,j} \sigma_{j,i}), \coprod_j \psi(\sigma_{j,i}^{-1}.(V_{k,j} \cap \sigma_{j,i}.U_{j,i})) \big)\\ &= F_{\phi,\psi}(\tau \sigma). \end{align*} $$

Hence, $F_{\phi ,\psi }$ respects composition. Furthermore, $F_{\phi ,\psi }$ also respects $\oplus $ by construction. This shows that $F_{\phi ,\psi }$ is a permutative functor. Moreover, our construction is (covariantly) functorial in $(\phi ,\psi )$ , in the sense that $F_{\phi ',\psi '} F_{\phi ,\psi } = F_{\phi ' \phi ,\psi ' \psi }$ .

3.1.2 The case of fibrewise bijective proper surjections

Now, suppose that $\phi : \: \tilde {G} \to G$ is a groupoid homomorphism which is a fibrewise bijective proper surjection. ‘Fibrewise bijective’ means that for all $\tilde {x} \in \tilde {G}^{(0)}$ , $\phi $ restricts to a bijection $\tilde {\mathrm {r}}^{-1}(\tilde {x}) \to \mathrm {r}^{-1}(\phi (\tilde {x}))$ . (Equivalently, we could consider fibres of the source maps.) Moreover, suppose that $\psi : \: \tilde {Z} \to Z$ is a continuous map such that for all $O \in \mathscr {O}$ there exists $\tilde {O} \in \tilde {\mathscr {O}}$ such that $\psi ^{-1}(O) \subseteq \tilde {O}$ and that $\psi $ restricts to a proper map $\psi \vert _{\psi ^{-1}(O)}: \: \psi ^{-1}(O) \to O$ . Furthermore, we require that the diagram

commutes and that $\psi (g.z) = \phi (g).\psi (z)$ for all $g \in G$ and $z \in Z$ with $\mathrm {s}(g) = \rho (z)$ . In addition, assume that for all $\tilde {x} \in \tilde {G}^{(0)}$ , $\psi $ restricts to a bijection

$$ \begin{align*}\tilde{\rho}^{-1}(\tilde{x}) \cap \psi^{-1}(O) \to \rho^{-1}(\phi(\tilde{x})) \cap O. \end{align*} $$

In this situation, we define a functor $F^{\phi ,\psi }: \: \mathfrak B_{G \curvearrowright Z} \to \mathfrak B_{\tilde {G} \curvearrowright \tilde {Z}}$ as follows: On objects, set

$$ \begin{align*}F^{\phi,\psi}(\coprod_i (i,U_i)) \mathrel{:=} \coprod_i (i,\psi^{-1}(U_i)). \end{align*} $$

Given a morphism $\sigma = \coprod _{j,i} (\mathbb {s}_{j,i}, \sigma _{j,i}, U_{j,i})$ in $\mathfrak B_{G \curvearrowright Z}$ , set

$$ \begin{align*}F^{\phi,\psi}(\sigma) \mathrel{:=} \coprod_{j,i} (\mathbb{s}_{j,i}, \phi^{-1}(\sigma_{j,i}), \psi^{-1}(U_{j,i})). \end{align*} $$

Let us now check that this is well defined, that is, this defines a morphism in $\mathfrak B_{\tilde {G} \curvearrowright \tilde {Z}}$ .

First, observe that for all compact open subspaces U contained in some $O \in \mathscr {O}$ , we have $\phi ^{-1}(\rho (U)) = \tilde {\rho }(\psi ^{-1}(U))$ . Indeed, ‘ $\supseteq $ ’ is clear. Given $x \in \rho (U)$ , let $\tilde {x} \in \tilde {G}^{(0)}$ with $\phi (\tilde {x}) = x$ be arbitrary and choose $z \in \rho ^{-1}(x) \cap O$ . As $\psi $ restricts to a bijection $\tilde {\rho }^{-1}(\tilde {x}) \cap \psi ^{-1}(O) \to \rho ^{-1}(\phi (\tilde {x})) \cap O$ , there exists $\tilde {z} \in \tilde {\rho }^{-1}(\tilde {x}) \cap \psi ^{-1}(O)$ with $\psi (\tilde {z}) = z$ . It follows that $\tilde {z} \in \psi ^{-1}(U)$ . Hence, $\tilde {x} = \tilde {\rho }(\tilde {z}) \in \tilde {\rho }(\psi ^{-1}(U))$ .

Secondly, observe that for every compact open bisection $\sigma \subseteq G$ , $\tilde {\mathrm {s}}(\phi ^{-1}(\sigma )) = \phi ^{-1}(\mathrm {s}(\sigma ))$ . Indeed, ‘ $\subseteq $ ’ is clear. Given $x \in \mathrm {s}(\sigma )$ , let $\tilde {x} \in \tilde {G}^{(0)}$ be arbitrary with $\phi (\tilde {x}) = x$ . Choose $g \in \mathrm {s}^{-1}(x) \cap \sigma $ . As $\phi $ restricts to a bijection $\tilde {\mathrm {s}}^{-1}(\tilde {x}) \to \mathrm {s}^{-1}(x)$ , there exists $\tilde {g} \in \tilde {\mathrm {s}}^{-1}(\tilde {x})$ with $\phi (\tilde {g}) = g$ . Hence, $\tilde {g} \in \phi ^{-1}(\sigma )$ . Thus, $\tilde {x} = \tilde {\mathrm {s}}(\tilde {g}) \in \tilde {\mathrm {s}}(\phi ^{-1}(\sigma ))$ . This shows ‘ $\supseteq $ ’.

It follows that $\tilde {\mathrm {s}}(\phi ^{-1}(\sigma _{j,i})) = \phi ^{-1}(\mathrm {s}(\sigma _{j,i})) = \phi ^{-1}(\rho (U_{j,i})) = \tilde {\rho }(\psi ^{-1}(U_{j,i}))$ .

Moreover, the restriction $\tilde {\rho } \vert _{\psi ^{-1}(U_{j,i})}: \: \psi ^{-1}(U_{j,i}) \to \tilde {\rho }(\psi ^{-1}(U_{j,i})) = \phi ^{-1}(\rho (U_{j,i}))$ is a homeomorphism. It suffices to show that this restriction is bijective. Given $\tilde {z}_1, \tilde {z}_2 \in \psi ^{-1}(U_{j,i})$ with $\tilde {\rho }(\tilde {z}_1) = \tilde {\rho }(\tilde {z}_2) = \tilde {x}$ , we have $\rho (\psi (\tilde {z}_1)) = \phi (\tilde {\rho }(\tilde {z}_1)) = \phi (\tilde {x}) = \phi (\tilde {\rho }(\tilde {z}_2)) = \rho (\psi (\tilde {z}_2))$ . As $\rho $ is injective on $U_{j,i}$ , we deduce that $\psi (\tilde {z}_1) = \psi (\tilde {z}_2)$ . But $\psi $ restricts to a bijection $\tilde {\rho }^{-1}(\tilde {x}) \cap \psi ^{-1}(O) \to \rho ^{-1}(\phi (\tilde {x})) \cap O$ , where $O \in \mathscr {O}$ is such that $U_{j,i} \subseteq O$ . Hence, we conclude that $\tilde {z}_1 = \tilde {z}_2$ , as desired.

Given two morphisms $\tau = \coprod _{k,j} (\mathbb {s}_{k,j}, \tau _{k,j}, V_{k,j})$ and $\sigma = \coprod _{j,i} (\mathbb {s}_{j,i}, \sigma _{j,i}, U_{j,i})$ with $\mathfrak d(\tau ) = \mathfrak t(\sigma )$ , we have

$$ \begin{align*} & F^{\phi,\psi}(\tau) F^{\phi,\psi}(\sigma) \\ = \ & \coprod_{k,i} \big( \mathbb{s}_{k,i}, \coprod_j \phi^{-1}(\tau_{k,j}) \phi^{-1}(\sigma_{j,i}), \coprod_j \phi^{-1}(\sigma_{j,i})^{-1}.(\psi^{-1}(V_{k,j}) \cap \phi^{-1}(\sigma_{j,i}).\psi(U_{j,i})) \big)\\ = \ & \coprod_{k,i} \big( \mathbb{s}_{k,i}, \coprod_j \phi^{-1}(\tau_{k,j} \sigma_{j,i}), \coprod_j \psi^{-1}(\sigma_{j,i}^{-1}.(V_{k,j} \cap \sigma_{j,i}.U_{j,i})) \big)\\ = \ & F^{\phi,\psi}(\tau \sigma). \end{align*} $$

Hence, $F^{\phi ,\psi }$ respects composition. Furthermore, $F^{\phi ,\psi }$ also respects $\oplus $ by construction. This shows that $F^{\phi ,\psi }$ is a permutative functor. Moreover, our construction is (contravariantly) functorial in $(\phi ,\psi )$ , in the sense that $F^{\phi ',\psi '} F^{\phi ,\psi } = F^{\phi \phi ',\psi \psi '}$ .

4.1 Functors inducing homotopy equivalences of classifying spaces

First, we establish a criterion for certain functors to induce homotopy equivalences of classifying spaces. Let us start by describing the setting. Let $\Phi : \: \mathfrak C \to \mathfrak G$ be a functor between small categories. Assume that $\mathfrak G$ is a groupoid and that $\Phi $ is faithful, that is, the induced maps $\mathfrak C(*,\bullet ) \to \mathfrak G(\Phi (*),\Phi (\bullet ))$ are injective for all $\bullet , * \in \mathrm {obj}\, \mathfrak C$ . Given an object u of $\mathfrak G$ , we first recall the definition of the category $u \backslash \Phi $ from [Reference Quillen73, § 1]. Objects of $u \backslash \Phi $ consist of pairs $(v,\sigma )$ , where $v \in \mathrm {obj}\, \mathfrak C$ and $\sigma \in \mathfrak G(\Phi (v),u)$ . A morphism in $u \backslash \Phi $ from $(v,\sigma )$ to $(w,\tau )$ is given by $f \in \mathfrak C(w,v)$ such that $\Phi (f) \sigma = \tau $ , that is, the diagram

commutes in $\mathfrak G$ .

We now define a new category $\mathfrak F_{u,\Phi }$ as a quotient of $u \backslash \Phi $ . Objects of $\mathfrak F_{u,\Phi }$ are equivalence classes $[v,\sigma ]$ of objects $(v,\sigma )$ of $u \backslash \Phi $ , where $(v,\sigma )$ and $(w,\tau )$ are equivalent if there exists an invertible morphism of $u \backslash \Phi $ from $(v,\sigma )$ to $(w,\tau )$ , that is, there exists an invertible element $a \in \mathfrak C(w,v)$ such that $\Phi (a) \sigma = \tau $ , that is, the diagram

commutes in $\mathfrak G$ . Morphisms of $\mathfrak F_{u,\Phi }$ are equivalence classes $[f]$ of morphisms f of $u \backslash \Phi $ , where $f: \: (v,\sigma ) \to (w,\tau )$ and $f': \: (v',\sigma ') \to (w',\tau ')$ are equivalent if there exist invertible elements $b \in \mathfrak C(w',w)$ , $b' \in \mathfrak C(v',v)$ , which are invertible morphisms in $u \backslash \Phi $ , such that $b f = f' b'$ in $\mathfrak C$ (here we view f and $f'$ as morphisms in $\mathfrak C$ ), that is, the diagram

commutes in $\mathfrak C$ .

Note that a morphism f in $u \backslash \Phi $ from $(v,\sigma )$ to $(w,\tau )$ , if it exists, is unique. This is because $\mathfrak G$ is a groupoid and $\Phi $ is faithful.

Proof. For every $c \in \mathrm {obj}\, \mathfrak F_{u,\Phi }$ , choose $x_c \in \mathrm {obj}\, (u \backslash \Phi )$ such that $[x_c] = c$ . Given $[f] \in \mathrm {mor}\, \mathfrak F_{u,\Phi }$ , let $\mathring {f} \in \mathrm {mor}\, (u \backslash \Phi )$ be the unique morphism from $x_{\mathfrak d(f)}$ to $x_{\mathfrak t(f)}$ . This defines a functor $\mathfrak F_{u,\Phi } \to u \backslash \Phi $ . We also have the canonical functor $u \backslash \Phi \to \mathfrak F_{u,\Phi }$ given by forming equivalence classes. By construction, the composite $\mathfrak F_{u,\Phi } \to u \backslash \Phi \to \mathfrak F_{u,\Phi }$ is the identity on $\mathfrak F_{u,\Phi }$ . Now, let $\Theta $ be the composite $u \backslash \Phi \to \mathfrak F_{u,\Phi } \to u \backslash \Phi $ . By construction, $\Theta (o) = x_{[o]}$ on objects and $\Theta (f) = \mathring {f}$ on morphisms. We claim that there is a natural transformation $T: \: \mathrm {id}_{u \backslash \Phi } \Rightarrow \Theta $ . Indeed, given an object $o \in \mathrm {obj}\, (u \backslash \Phi )$ , let $T_o \in (u \backslash \Phi )(x_{[o]},o)$ be the unique morphism in $u \backslash \Phi $ from o to $x_{[o]}$ . Given a morphism $f \in (u \backslash \Phi )(\tilde {o},o)$ , the diagram

commutes because of uniqueness of morphisms in $u \backslash \Phi $ . Now, our proof is complete because of [Reference Quillen73, Proposition 2].

The following is now an immediate consequence of Proposition 4.1 and [Reference Quillen73, Theorem A].

4.2 Homology for certain free and proper actions

Now, let $\nu \geq 1$ and consider $\mathfrak B_{{G^{(0)}} \curvearrowright {G^{(\nu - 1)}}}$ and $\mathfrak B_{G \curvearrowright {G^{(\nu )}}}$ as defined in § 3. We set out to define a functor $I: \: \mathfrak B_{{G^{(0)}} \curvearrowright {G^{(\nu - 1)}}} \to \mathfrak B_{G \curvearrowright {G^{(\nu )}}}$ . Given $U \in \mathcal C \mathcal O^{(\nu - 1)}$ , let

$$ \begin{align*}I(U) \mathrel{:=} \left\{ (\rho(z),z) \in {G^{(\nu)}} \text{: } z \in U \right\}. \end{align*} $$

Now, define

$$ \begin{align*}I\left(\coprod_i (i,U_i)\right) \mathrel{:=} \coprod_i (i,I(U_i)). \end{align*} $$

On morphisms, define

$$ \begin{align*}I\left(\coprod_{j,i} (\mathbb{s}_{j,i},\sigma_{j,i},U_{j,i})\right) \mathrel{:=} \coprod_{j,i} (\mathbb{s}_{j,i},\sigma_{j,i},I(U_{j,i})). \end{align*} $$

Note that $\sigma _{j,i} = \rho (U_{j,i})$ because $\coprod _{j,i} (\mathbb {s}_{j,i},\sigma _{j,i},U_{j,i})$ is a morphism in $\mathfrak B_{{G^{(0)}} \curvearrowright {G^{(\nu - 1)}}}$ .

It is straightforward to check that I is a permutative functor and that I is faithful. Therefore, we are in the setting of Proposition 4.1 and Corollary 4.2.

The following is an immediate consequence of Corollary 4.2 and Proposition 4.3.

Now, let C be a compact Hausdorff subspace of ${G^{(\nu - 1)}}$ . Let $\mathfrak B_C$ be the full subcategory of $\mathfrak B_{{G^{(0)}} \curvearrowright {G^{(\nu - 1)}}}$ whose objects are of the form $\coprod _i (i,U_i)$ , where $U_i \subseteq C$ . It follows that morphisms of $\mathfrak B_C$ are of the form $\coprod _{j,i} (\mathbb {s}_{j,i},\sigma _{j,i},U_{j,i})$ , with $U_{j,i} \subseteq C$ (and hence $\sigma _{j,i}.U_{j,i} = U_{j,i} \subseteq C$ ). As C is totally disconnected, we can describe C as $C \cong \varprojlim _{l \in \mathfrak L} \mathcal U_l$ , where $\mathcal U_l$ consist of finitely many compact open subspaces which partition C, and $\mathfrak L$ is the index set given by all these partitions, partially ordered by refinement. For $l \in \mathfrak L$ , let $\mathfrak B_l$ be the small permutative category with objects of the form $\coprod _i (i,U_i)$ with $U_i \in \mathcal U_l$ , and morphisms of the form $\coprod _{j,i} (\mathbb {s}_{j(i),i}, \rho (U_i), U_i)$ from $\coprod _i (i,U_i)$ to $\coprod _j (j,U_j)$ , where $i \mapsto j(i)$ is a bijection. In other words, the only morphisms in $\mathfrak B_l$ are given by permutations. Note that $\mathfrak B_l$ is a subcategory of $\mathfrak B_C$ .

Lemma 4.5. For all $l \in \mathfrak L$ , we have a canonical isomorphism

$$ \begin{align*}H_*(\mathbb{K}(\mathfrak B_l),\mathsf{C}) \cong \begin{cases} (\bigoplus_{\mathcal U_l} \mathsf{C}) \oplus \mathsf{C} & \text{ if } * = 0,\\ \left\{ 0 \right\} & \text{ else}. \end{cases} \end{align*} $$

Proof. It is easy to see that $\mathfrak B_l$ coincides with the free permutative category $P \mathcal U_l$ in the sense of [Reference Thomason92, § 1], where $\mathcal U_l$ is the category with object set $\mathcal U_l$ and only identity morphisms. Moreover, it is observed in [Reference Thomason92, § 1] that $P \mathcal U_l$ is equivalent to the free symmetric monoidal category $S \mathcal U_l$ on $\mathcal U_l$ so that [Reference Thomason92, Lemma 2.3 and Lemma 2.5] imply that $\mathbb {K}(P \mathcal U_l)$ is weakly homotopy equivalent to $\Sigma ^{\infty }(B \mathcal U_l)^+$ . Hence, we obtain

$$ \begin{align*}H_*(\mathbb{K}(\mathfrak B_l),\mathsf{C}) \cong H_*(\mathbb{K}(P \mathcal U_l),\mathsf{C}) \cong H_*(\Sigma^{\infty}(B \mathcal U_l)^+,\mathsf{C}), \end{align*} $$

and now our claim follows from $H_*(B \mathcal U_l, \mathsf {C}) \cong H_*(\mathcal U_l, \mathsf {C})$ . Here, we view $\mathcal U_l$ as a discrete space.

For $k \leq l$ , define a permutative functor $I_{l,k}: \: \mathfrak B_k \to \mathfrak B_l$ by $U \mapsto \bigoplus V(U)$ for $U \in \mathcal U_k$ , where the sum is taken over all $V(U) \in \mathcal U_l$ contained in U, so that $U = \coprod V(U)$ as subspaces in C, and by extending this to all objects via $\coprod _i (i,U_i) \mapsto \bigoplus _i (\bigoplus V(U_i))$ . Here, we are working with a fixed ordering of $\left \{ V(U) \right \}$ for every U. On morphisms, let $I_{l,k}(\coprod _{j,i} (\mathbb {s}_{j(i),i}, \rho (U_i), U_i))$ be the morphism

$$ \begin{align*}\bigoplus_i (\bigoplus V(U_i)) \to \bigoplus_j (\bigoplus V(U_j)) \end{align*} $$

induced by the permutation $i \mapsto j(i)$ . Let us now form the homotopy colimit, in the sense of [Reference Thomason92, Construction 3.22], of the following functor F from $\mathfrak L$ to permutative categories: We define $F(l) \mathrel {:=} \mathfrak B_l$ on objects $l \in \mathfrak L$ , and a morphism $k \to l$ (i.e., $k, l \in \mathfrak L$ satisfying $k \leq l$ ) is mapped to $I_{l,k}$ under F. We recall the construction of $\mathfrak H \mathrel {:=} \mathrm {hocolim} \, F$ from [Reference Thomason92, Construction 3.22]. Objects of $\mathfrak H$ are of the form $(l_1,u_1) \oplus \dotso \oplus (l_n,u_n)$ , where $l_i \in \mathfrak L$ and $u_i \in F(l_i)$ . Morphisms from $(l_1,u_1) \oplus \dotso \oplus (l_n,u_n)$ to $(l^{\prime }_1,u^{\prime }_1) \oplus \dotso \oplus (l^{\prime }_m,u^{\prime }_m)$ are given by $(\lambda _i, \psi , \chi _j)$ , where $\psi : \: \left \{ 1, \dotsc , n \right \} \to \left \{ 1, \dotsc , m \right \}$ is a surjective map, $\lambda _i: \: l_i \to l^{\prime }_{\psi (i)}$ are morphisms in $\mathfrak L$ (in our case this simply means $l_i \leq l^{\prime }_{\psi (i)}$ ), and $\chi _j: \: \bigoplus _{\psi (i) = j} F(\lambda _i)(u_i) \to u^{\prime }_j$ are morphisms in $F(l^{\prime }_j)$ .

Now, assume that C is contained in some $O \in \mathscr {O}^{(\nu - 1)}$ , and define a permutative functor $H: \: \mathfrak H \to \mathfrak B_C$ as follows: Given $u = \coprod _{\bullet } (\bullet ,U_{\bullet })$ with $U_{\bullet } \in \mathcal U_l$ , set $H(l,u) \mathrel {:=} \coprod _{\bullet } (\bullet ,U_{\bullet })$ viewed as an object in $\mathfrak B_C$ . Define H on a morphism given by data $(\lambda _i, \psi , \chi _j)$ as above by sending it to the morphism given by the composition

$$ \begin{align*}\bigoplus_i H(l_i,u_i) \to \bigoplus_i H(l^{\prime}_{\psi(i)},F(\lambda_i)(u_i)) \to \bigoplus_j H(l^{\prime}_j,u^{\prime}_j). \end{align*} $$

Here, the first map is given as follows: If $u_i = \coprod _{\bullet } (\bullet ,U_{\bullet })$ , then $F(\lambda _i)(u_i) = \bigoplus _{\bullet } (\bigoplus V(U_{\bullet }))$ , and on each component, the first map is given by the morphism $\coprod _{V(U)} (\mathbb {s}_{U, V(U)}, \rho (V(U)), V(U))$ . The second map is given by

where we view $\bigoplus _j \chi _j$ as a morphism in $\mathfrak B_C$ .

Our goal is to show that H induces a homotopy equivalence of classifying spaces. To do so, let us first describe $\mathfrak F_{u,H}$ for $u \in \mathrm {obj}\, \mathfrak B_C$ . Two objects $(v,\sigma )$ and $(w,\tau )$ in $u \backslash H$ are equivalent (with respect to the relation defining $\mathfrak F_{u,H}$ ) if there exists a (necessarily unique) morphism a from v to w which is invertible in $\mathfrak H$ , that is, a is given by $(\lambda _i,\psi ,\chi _j)$ such that $\lambda _i = \mathrm {id}$ and $l^{\prime }_{\psi (i)} = l_i$ for all i. Hence, $\mathfrak F_{u,H}$ is a poset, where we define $[v,\sigma ] \geq [w,\tau ]$ if there exists a morphism from $(v,\sigma )$ to $(w,\tau )$ in $u \backslash H$ given by $f \in \mathfrak H(w,v)$ such that $H(f) \sigma = \tau $ . We want to show that the classifying space of $\mathfrak F_{u,H}$ is contractible. This will follow from the next observation.

Proof. We want to show that for all $[v,\sigma ]$ and $[w,\tau ]$ , there exists $[x,\alpha ]$ such that $[v,\sigma ] \geq [x,\alpha ]$ and $[w,\tau ] \geq [x,\alpha ]$ . Indeed, suppose that $u = \coprod _i (i,U_i)$ , $v = \bigoplus _{\bullet } (l_{\bullet },v_{\bullet })$ , $H(v) = \coprod _j (j,V_j)$ and that $\sigma = \coprod _{j,i} (\mathbb {s}_{j,i},\rho (U_{j,i}),U_{j,i})$ . Then $U_i = \coprod _j U_{j,i}$ and $V_j = \coprod _i U_{j,i}$ . Find an index $l'$ with $l_{\bullet } \leq l'$ for all $\bullet $ such that $U_{j,i}$ can be written as a disjoint union $U_{j,i} = \coprod V_{j,i,\zeta }$ for some $V_{j,i,\zeta } \in \mathcal U_{l'}$ . Define $v' \mathrel {:=} (l', \bigoplus _{j,i,\zeta } V_{j,i,\zeta })$ . Construct a morphism e in $\mathfrak H$ from v to $v'$ given by data $(\lambda _{\bullet }, \psi , \chi )$ as above (note that $m=1$ for $v'$ ), with $\psi (\bullet ) = 1$ for all $\bullet $ , $\lambda _{\bullet }$ is the morphism $l_{\bullet } \to l'$ in $\mathfrak L$ , which exists because $l_{\bullet } \leq l'$ , and

$$ \begin{align*}\chi: \: \bigoplus_j \bigoplus V(V_j) \to \bigoplus_{j,i,\zeta} V_{j,i,\zeta} \end{align*} $$

given by the obvious permutation, induced by the decomposition $V_j = \coprod _i U_{j,i} = \coprod _i \coprod V_{j,i,\zeta }$ . By construction, we have

$$ \begin{align*}H(e) = \coprod_{j,i,\zeta} (\mathbb{s}_{(j,i,\zeta),j},\rho(V_{j,i,\zeta}),V_{j,i,\zeta}). \end{align*} $$

Similarly, suppose that $w = \bigoplus _{\circ } (l_{\circ },w_{\circ })$ , $H(w) = \coprod _k (k,W_k)$ and $\tau = \coprod _{k,i} (\mathbb {s}_{k,i},\rho (U_{k,i}),U_{k,i})$ . Then $W_k = \coprod _i U_{k,i}$ . Find an index $\tilde {l}$ with $l_{\circ } \leq \tilde {l}$ for all $\circ $ such that $U_{k,i}$ can be written as a disjoint union $U_{k,i} = \coprod W_{k,i,\eta }$ for some $W_{k,i,\eta } \in \mathcal U_{\tilde {l}}$ . Define $\tilde {w} \mathrel {:=} (\tilde {l}, \bigoplus _{k,i,\eta } W_{k,i,\eta })$ , and construct a morphism f in $\mathfrak H$ from w to $\tilde {w}$ as above such that

$$ \begin{align*}H(f) = \coprod_{k,i,\eta} (\mathbb{s}_{(k,i,\eta),k},\rho(W_{k,i,\eta}),W_{k,i,\eta}). \end{align*} $$

Now, find $l \in \mathfrak L$ such that $l', \tilde {l} \leq l$ . It follows that all $V_{j,i,\zeta }$ and $W_{k,i,\eta }$ can be written as disjoint unions of elements in $\mathcal U_l$ . This leads to morphisms $e'$ and $f'$ in $\mathfrak H$ such that

$$ \begin{align*} H(e') &= \coprod_{k,j,i,\zeta,\eta} (\mathbb{s}_{(k,j,i,\zeta,\eta),(j,i,\zeta)},\rho(U_{k,j,i,\zeta,\eta}),U_{k,j,i,\zeta,\eta})\\ H(f') &= \coprod_{k,j,i,\zeta,\eta} (\mathbb{s}_{(k,j,i,\zeta,\eta),(k,i,\eta)},\rho(\tilde{U}_{k,j,i,\zeta,\eta}),\tilde{U}_{k,j,i,\zeta,\eta}) \end{align*} $$

satisfying $\mathfrak t(e') = \mathfrak t(f')$ and $H(e') H(e) \sigma = H(f') H(f) \tau $ . Setting $x \mathrel {:=} \mathfrak t(e') = \mathfrak t(f')$ and $\alpha \mathrel {:=} H(e') H(e) \sigma = H(f') H(f) \tau $ , we indeed have $[v,\sigma ] \geq [x,\alpha ]$ and $[w,\tau ] \geq [x,\alpha ]$ , as desired.

We obtain the following consequence of Corollary 4.2 and Lemma 4.6 because posets with the property that any two elements have a common lower bound have contractible classifying spaces.

Proposition 4.8. We have

$$ \begin{align*}H_*(\mathbb{K}(\mathfrak B_C),\mathsf{C}) \cong \begin{cases} \mathscr{C}(C,\mathsf{C}) \oplus \mathsf{C} & \text{ if } * = 0,\\ \left\{ 0 \right\} & \text{ else}. \end{cases} \end{align*} $$

Proof. Corollary 4.7 implies that $H_*(\mathbb {K}(\mathfrak B_C)) \cong H_*(\mathbb {K}(\mathfrak H))$ because of [Reference Thomason92, Lemma 2.3]. Thus, we obtain

$$ \begin{align*}H_*(\mathbb{K}(\mathfrak B_C), \mathsf{C}) \cong H_*(\mathbb{K}(\mathfrak H), \mathsf{C}) \cong H_*(\mathrm{hocolim}_l \, \mathbb{K}(\mathfrak B_l), \mathsf{C}) \cong \varinjlim_l H_*(\mathbb{K}(\mathfrak B_l), \mathsf{C}). \end{align*} $$

Here, we used [Reference Thomason92, Theorem 4.1] for the second isomorphism, and we obtain the third isomorphism because we are taking a filtered colimit. Thus, applying Lemma 4.5, we derive

$$ \begin{align*}H_*(\mathbb{K}(\mathfrak B_C), \mathsf{C}) \cong \begin{cases} \varinjlim_l (\bigoplus_{\mathcal U_l} \mathsf{C}) \oplus \mathsf{C} \cong \mathscr{C}(C,\mathsf{C}) \oplus \mathsf{C}& \text{ if } * = 0,\\ \left\{ 0 \right\} & \text{ else}, \end{cases} \end{align*} $$

as desired.

Now, suppose that $O \in \mathscr {O}^{(\nu - 1)}$ . Let $\mathfrak B_O$ be the full subcategory of $\mathfrak B_{{G^{(0)}} \curvearrowright {G^{(\nu - 1)}}}$ whose objects are of the form $\coprod _i (i,U_i)$ , where $U_i \subseteq O$ .

Proposition 4.9. We have

$$ \begin{align*}H_*(\mathbb{K}(\mathfrak B_O),\mathsf{C}) \cong \begin{cases} \mathscr{C}(O,\mathsf{C}) \oplus \mathsf{C} & \text{ if } * = 0,\\ \left\{ 0 \right\} & \text{ else}. \end{cases} \end{align*} $$

Proof. Let us use the same notation as in § 2.6. Since $O = \bigcup _U U$ , where U runs through all compact open subspaces of O, we conclude that for all $p, q$ , we have $\mathfrak N_p \mathfrak B_O(S^n_q) = \bigcup _U \mathfrak N_p \mathfrak B_U(S^n_q)$ . Hence, after taking diagonals, we obtain $\mathfrak N \mathfrak B_O(S^n) = \bigcup _U \mathfrak N \mathfrak B_U(S^n)$ . Since homology is compatible with inductive limits, we obtain $H_*(X_n) \cong \varinjlim _U H_*(X_{U,n})$ , where $X_n$ is the n-th simplicial set of $\mathbb {K}(\mathfrak B_O)$ and $X_{U,n}$ is the n-th simplicial set of $\mathbb {K}(\mathfrak B_U)$ . Therefore, using Proposition 4.8, we conclude that

$$ \begin{align*}H_*(\mathbb{K}(\mathfrak B_O),\mathsf{C}) \cong \varinjlim_U H_*(\mathbb{K}(\mathfrak B_U),\mathsf{C}) \cong \varinjlim_U \mathscr{C}(U,\mathsf{C}) \oplus \mathsf{C} \cong \mathscr{C}(O,\mathsf{C}) \oplus \mathsf{C}, \end{align*} $$

for $* = 0$ , and $H_*(\mathbb {K}(\mathfrak B_O),\mathsf {C}) \cong \left \{ 0 \right \}$ for $*> 0$ .

Now, suppose that we are given $O_1, \dotsc , O_N \in \mathscr {O}^{(\nu - 1)}$ . Let $\mathfrak B_{\cap }$ be the full subcategory of $\mathfrak B_{{G^{(0)}} \curvearrowright {G^{(\nu - 1)}}}$ whose objects are of the form $\coprod _i (i,U_i)$ , where $U_i \subseteq O_1 \cap O_n$ for some $2 \leq n \leq N$ . Let $\mathfrak B_1$ be the full subcategory of $\mathfrak B_{{G^{(0)}} \curvearrowright {G^{(\nu - 1)}}}$ whose objects are of the form $\coprod _i (i,U_i)$ , where $U_i \subseteq O_1$ . Let $\mathfrak B_2$ be the full subcategory of $\mathfrak B_{{G^{(0)}} \curvearrowright {G^{(\nu - 1)}}}$ whose objects are of the form $\coprod _i (i,U_i)$ , where $U_i \subseteq O_n$ for some $2 \leq n \leq N$ . Finally, let $\mathfrak B$ be the full subcategory of $\mathfrak B_{{G^{(0)}} \curvearrowright {G^{(\nu - 1)}}}$ whose objects are of the form $\coprod _i (i,U_i)$ , where $U_i \subseteq O_n$ for some $1 \leq n \leq N$ . Let $\mathfrak L$ be the category with three objects $\mathfrak l_{\cap }$ , $\mathfrak l_1$ and $\mathfrak l_2$ , the corresponding identity morphisms, one morphism $\mathfrak l_{\cap } \to \mathfrak l_1$ and another morphism $\mathfrak l_{\cap } \to \mathfrak l_2$ . Let F be the functor from $\mathfrak L$ to small permutative categories sending $\mathfrak l_{\cap }$ to $\mathfrak B_{\cap }$ , $\mathfrak l_1$ to $\mathfrak B_1$ , $\mathfrak l_2$ to $\mathfrak B_2$ , the morphism $\mathfrak l_{\cap } \to l_1$ to the inclusion $\mathfrak B_{\cap } \hookrightarrow \mathfrak B_1$ and the morphism $\mathfrak l_{\cap } \to l_2$ to the inclusion $\mathfrak B_{\cap } \hookrightarrow \mathfrak B_2$ .