2. The main construction

This section is devoted to proving Theorem 1.1 for orbispaces which are presented by a simplicial complex of groups. This special case (stated as Theorem 2.13) carries the essential topological content of Theorem 1.1. For general background on orbispaces and topological stacks, we refer the reader to § 3.

We begin by describing the basic idea of the proof, which we then implement in detail. Our orbispace $X$ comes with a filtration by skeleta

\[ \varnothing=X_{-1}\subseteq X_0\subseteq X_1\subseteq\cdots, \]

where $X_k$ is obtained from $X_{k-1}$ by attaching cells of the form $(D^k,\partial D^k)\times \mathbb {B} G$ for finite groups $G$ (here $\mathbb {B} G=*/G$ is the orbispace quotient). We construct the desired vector bundle by induction on skeleta.

A direct implementation of this strategy runs immediately into the following obstruction. A (complex) vector bundle $V$ on $\partial D^k\times \mathbb {B} G$ decomposes canonically as $V=\bigoplus _{\rho \in \hat G}V_\rho \otimes \rho$ for vector bundles $V_\rho$ on $\partial D^k$ indexed by the complex irreducible representations $\rho$ of $G$. Thus, $V$ is classified by an element of $\prod _{\rho \in \hat G}\pi _{k-1}(\coprod _nBU(n))$, which vanishes if and only if $V$ extends to $D^k\times \mathbb {B} G$. We seek to detect these obstructions using Chern characters. According to Bott periodicity, the homotopy groups of $BU$ are given by

\[ \pi_i(BU)=\begin{cases}\mathbb{Z} & i\text{ even,}\\0 & i\text{ odd,}\end{cases} \]

and the Chern character of a generator of $\pi _{2i}(BU)$ is nonzero. Now the Chern character of $V=\bigoplus _{\rho \in \hat G}V_\rho \otimes \rho$ is given by $\operatorname {ch}(V)=\sum _{\rho \in \hat G}\operatorname {ch}(V_\rho )\dim \rho$, so triviality of $\operatorname {ch}(V)$ does not imply triviality of each $\operatorname {ch}(V_\rho )$ or of the aforementioned obstructions to extending $V$.

To capture the information we need about $V$, we consider the more refined characteristic class which we call the inertial Chern character Footnote ¹ $\operatorname {ch}^I$ (studied previously by Adem and Ruan [Reference Adem and RuanAR03]), which is a cohomology class on the inertia stack

\[ IX:=\operatorname{Eq}\bigl(X\rightrightarrows X\bigr)=X\times_{X\times X}X. \]

In local coordinates $X=Y/G$, we have $IX=\bigl (\bigsqcup _{g\in G}Y^g\bigr )/G$. In such coordinates, the inertial Chern character is defined by recalling the Chern–Weil description of the usual Chern character $\operatorname {ch}(V)=\operatorname {tr}\exp (\Omega /(-2\pi i))$, for $\Omega$ the curvature of a hermitian connection on $V$, and writing

\[ \operatorname{ch}^I(V)=\operatorname{tr}\biggl[g\exp\biggl(\frac\Omega{-2\pi i}\biggr)\biggr]. \]

Now the inertia stack of $\partial D^k\times \mathbb {B} G$ is $\partial D^k\times G/G$ (quotient by the conjugation action), and the inertial Chern character of $V=\bigoplus _{\rho \in \hat G}V_\rho \otimes \rho$ is given by $\operatorname {ch}^I(V)=\sum _{\rho \in \hat G}\operatorname {ch}(V_\rho )\chi _\rho$. Thus, if $\operatorname {ch}^I(V)$ is trivial, then so is each $\operatorname {ch}(V_\rho )$ by linear independence of characters, and hence $V$ extends to $D^k\times \mathbb {B} G$ as desired.

We are thus led to consider the modified problem of constructing a vector bundle on $X$ with the desired fibers and with trivial inertial Chern character. It suffices to show that if $X_{k-1}$ admits such a vector bundle, then so does $X_k$. The vanishing of $\operatorname {ch}^I(V)\in H^*(IX_{k-1})$ guarantees that the obstructions to extending $V$ to $X_k$ vanish, by the above discussion. It thus remains to show that this extension can be taken to have trivial inertial Chern character. The inertial Chern character of any extension is an element of $\ker (H^*(IX_k)\to H^*(IX_{k-1}))$. By the long exact sequence

\[ \cdots\to H^*(IX_k,IX_{k-1})\to H^*(IX_k)\to H^*(IX_{k-1})\to\cdots \]

it is thus in the image of $H^*(IX_k,IX_{k-1})$. The inertial Chern character is an even degree class, so we may assume that $k$ is even. By modifying how we extend $V$ from $X_{k-1}$ to $X_k$ by an element of $\prod _{\rho \in \hat G}\pi _k(BU)$ over a given $k$-simplex, we can shift its inertial Chern character by any integral linear combination of characters in $\operatorname {Hom}(G/G,\mathbb {C})\subseteq H^*(IX_k,IX_{k-1})$ (for $G$ the isotropy group of that given $k$-simplex). By replacing $V$ with $V^{\oplus a}$, we can multiply its inertial Chern character by any positive integer $a$. A combination of these two operations suffice to kill the inertial Chern character, provided that it is $\chi$-rational (rational with respect to a certain $\mathbb {Q}$-structure on $H^*(IX)$ differing from the usual one). This rationality is not at all obvious given the transcendental definition we give of the inertial Chern character, but it is true, and thus the proof is complete.

The remainder of this section is devoted to making the above outline precise. We begin by recalling the definition of a simplicial complex of groups (see also Haefliger [Reference HaefligerHae91], Corson [Reference CorsonCor92], or Bridson and Haefliger [Reference Bridson and HaefligerBH99]).

A simplicial complex of groups $(Z,G)$ presents an orbispace $\|(Z,G)\|$ called its geometric realization. A precise definition of this geometric realization is given in § 4. For now, it will suffice to know that the coarse space of $\|(Z,G)\|$ is (the geometric realization of) $Z$ itself, and that over the open star $\operatorname {st}(\sigma )\subseteq Z$ of a simplex $\sigma \subseteq Z$, the geometric realization $\|(Z,G)\|$ is given by the orbispace quotient

\[ \biggl(\bigsqcup_{\tau\supseteq\sigma}\tau\times(G_\tau\setminus G_\sigma)\biggr)\biggm/G_\sigma, \]

where the pieces for $\tau \supseteq \tau '\supseteq \sigma$ are glued together via the map $G_\tau \setminus G_\sigma \to G_{\tau '}\setminus G_\sigma$ induced by the map $G_\tau \to G_{\tau '}$ and the element of $G_\sigma$ conjugating the composition $G_\tau \to G_{\tau '}\to G_\sigma$ to the map $G_\tau \to G_\sigma$.

We often abuse terminology and say ‘simplicial complex of groups’ when we really mean its geometric realization. Thus, we refer to $X=\|(Z,G)\|$ as a simplicial complex of groups. Its coarse space $|X|$ is the geometric realization of $Z$.

To apply the methods of differential topology to a given simplicial complex of groups, we fix a family of (germs of) smooth retractions $\tau \to \sigma$ for every pair of simplices $\sigma \subseteq \tau$ such that the maps $\tau \to \sigma \to \rho$ and $\tau \to \rho$ agree for $\rho \subseteq \sigma \subseteq \tau$ (such a family of smooth retractions may be constructed by induction). Given this data, objects of differential topology on $\tau$ (functions, differential forms, bundles, connections, etc.) are required to be pulled back under $\tau \to \sigma$ in a(n unspecified) neighborhood of every $\sigma \subseteq \tau$. For bundles, this requirement consists of the data of a compatible family of isomorphisms with the pullback bundles (one could equivalently consider only vector bundles built out of transition functions which satisfy the given pullback conditions).

In particular, the de Rham complex $\Omega ^*(X)$ of a simplicial complex of groups $X$ is defined, and it coincides with the de Rham complex of its coarse space $\Omega ^*(|X|)$. There is a natural map $\Omega ^*(|X|)\to C^*(|X|;\mathbb {R})=C^*(X;\mathbb {R})$ from the de Rham complex to the simplicial cochain complex over $\mathbb {R}$, given by integrating differential forms over oriented simplices. It is a standard fact that this map is a quasi-isomorphism (proof: filter $X$ by skeleta and invoke the five lemma to reduce to showing that $\Omega ^*(\Delta ^k,\partial \Delta ^k)\to C^*(\Delta ^k,\partial \Delta ^k;\mathbb {R})$ is a quasi-isomorphism, which follows from the Poincaré lemma).

Definition 2.2 (Chern character)

Let $X$ be a simplicial complex of groups, and let $V\to X$ be a complex vector bundle. Given a hermitian metric $\mu$ and hermitian connection $\theta$ on $V$, the curvature of $\theta$ is a $2$-form $\Omega (V,\theta )$ valued in $\mathfrak u(V,\mu )\subseteq \mathfrak {gl}(V)=\operatorname {End}(V)$, and the Chern character form

\begin{align*} \operatorname{ch}(V,\theta):=\operatorname{tr}\exp \biggl(\frac{\Omega(V,\theta)}{-2\pi i}\biggr)\in\Omega^{\mathrm{even}}(X) \end{align*}

is closed. Its class in cohomology $\operatorname {ch}(V)\in H^\mathrm {even}(X;\mathbb {R})$ is called the Chern character of $V$. This class is independent of $\mu$ and $\theta$ by an interpolation argument (interpolate on $X\times [0,1]$ between any two metric/connection pairs on $X\times 0$ and $X\times 1$, and use the fact that $H^*(X\times [0,1])=H^*(X)$).

For any stack $X$, its inertia stack is the stack

\[ IX:=\operatorname{Eq}\bigl(X\rightrightarrows X\bigr)=X\times_{X\times X}X. \]

When $X=Y/G$, we have $IX=\bigl (\bigsqcup _{g\in G}Y^g\bigr )/G$. In particular, the local description of the geometric realization of a simplicial complex of groups provides a local description of its inertia stack as well. In fact, if $X$ is a simplicial complex of groups then so is $IX$: a simplex of $IX$ is a pair $(\sigma,[g])$ where $\sigma \subseteq X$ is a simplex and $[g]\subseteq G_\sigma$ is a conjugacy class, and $G_{(\sigma,[g])}$ is the centralizer of any element of the conjugacy class $[g]\subseteq G_\sigma$, etc.

Definition 2.3 (Inertial Chern character)

Let $X$ be a simplicial complex of groups, and let $V\to X$ be a complex vector bundle. The inertial Chern character $\operatorname {ch}^I(V)\in H^\mathrm {even}(IX;\mathbb {C})$ is represented by the closed form

\[ \operatorname{ch}^I(V,\theta):=\operatorname{tr}\biggl[g\exp\biggl(\frac\Omega{-2\pi i}\biggr)\biggr]\in\Omega^\mathrm{even}(IX;\mathbb{C}) \]

for any choice of hermitian metric and connection $\theta$ on $V$. Closedness can be seen by splitting the pullback of $V$ to $IX$ into isotypic pieces for the action of the cyclic group generated by $g$ and observing that the contribution of each such piece is a (usual) Chern character form. Independence of the metric and connection follows from the same interpolation argument as before.

Despite admitting the aforementioned transcendental definition in terms of differential forms, the (usual) Chern character $\operatorname {ch}(V)\in H^*(X;\mathbb {R})$ is well known be rational, i.e. it lies in the subspace $H^*(X;\mathbb {Q})\subseteq H^*(X;\mathbb {R})$. The inertial Chern character does not always lie in the subspace $H^*(IX;\mathbb {Q})\subseteq H^*(IX;\mathbb {C})$, rather it is rational with respect to a different $\mathbb {Q}$-structure, which we now introduce.

As the coefficient systems on $X$ appearing in Definition 2.7 are finite free modules over $\mathbb {Z}$, we may dualize to define homology groups $H_*(IX;\mathbb {Z}_\chi )$ as well.

Lemma 2.9 There are canonical isomorphisms

\begin{align*} H^*(IX;\mathbb{C})&=\operatorname{Hom}(H_*(IX;\mathbb{Z}_\chi),\mathbb{C}),\\ H^*(IX;\mathbb{Q}_\chi)&=\operatorname{Hom}(H_*(IX;\mathbb{Z}_\chi),\mathbb{Q}), \end{align*}

and a canonical short exact sequence

\[ 0\to\operatorname{Ext}^1(H_{*-1}(IX;\mathbb{Z}_\chi),\mathbb{Z})\to H^*(IX;\mathbb{Z}_\chi)\to\operatorname{Hom}(H_*(IX;\mathbb{Z}_\chi),\mathbb{Z})\to 0. \]

It is worth remarking that the proof we give does not provide an explicit expression for $N(m,d,n)$ other than for $n=1$.

Proof. We first show that the inertial Chern character of any line bundle $L/X$ is $\chi$-rational. We have $\operatorname {tr}(ab)=\operatorname {tr}(a)\operatorname {tr}(b)$ for endomorphisms $a,b$ of $\mathbb {C}$, so the inertial Chern character of a line bundle $L$ splits as the product

\[ \operatorname{ch}^I(L)=\operatorname{ch}_0^I(L)\exp(\operatorname{ch}_1(L)), \]

where $\operatorname {ch}_0^I(L)\in H^0(IX;\mathbb {Z}_\chi )$ is the fiberwise character of $L$ and $\operatorname {ch}_1(L)\in H^2(X;\mathbb {C})$ is the Chern character in degree one. It thus suffices to show that $\operatorname {ch}_1(L)$ is rational. We have $k\operatorname {ch}_1(L)=\operatorname {ch}_1(L^{\otimes k})$, so it suffices to show that $\operatorname {ch}_1(L^{\otimes k})$ is integral for some positive integer $k$. As all isotropy groups of $X$ have order $\leqslant m$, the tensor power $L^{\otimes m!}$ descends to a line bundle $M$ on the coarse space. As $M$ is an ordinary line bundle over a space (rather than an orbispace), its first Chern class $c_1(M)=\operatorname {ch}_1(M)$ is integral. We have thus shown that $\operatorname {ch}^I(L)$ is $\chi$-rational (in fact, we have shown that $d!(m!)^d\operatorname {ch}^I_d(L)$ is $\chi$-integral).

To treat the case of vector bundles of dimension greater than one, we use the splitting principle. Given a vector bundle $V/X$ with hermitian metric, let $\mathbb {F}(V)\to X$ denote the fibration whose fiber over $x\in X$ is the space of decompositions of $V_x$ into ordered orthogonal one-dimensional subspaces. In other words, $\mathbb {F}(V)=P/U(1)^n$ where $P\to X$ is the principal $U(n)$-bundle associated to $V/X$ with its chosen metric. We claim that the pullback map

\[ H^*(IX;\mathbb{C})\to H^*(I\mathbb{F}(V);\mathbb{C}) \]

is injective. The fiber of $I\mathbb {F}(V)\to IX$ over a given point $(x,g)\in IX$ is the space of $g$-invariant ordered decompositions of $V_x$ into one-dimensional subspaces. As the group generated by $g$ is abelian, its irreducible representations are one-dimensional, and hence there are plenty of such decompositions which are $g$-invariant, namely when each one-dimensional subspace is contained in some $g$-isotypic piece of $V_x$. Thus, $I\mathbb {F}(V)\to IX$ is a disjoint union of iterated projective space bundles, from which the desired injectivity statement follows (for any projective space bundle $\mathbb {P}(W)\to Z$ over an orbispace $Z$, the pullback map $H^*(Z)\to H^*(\mathbb {P}(W))$ is split by the map $\alpha \mapsto \int (c_1(L)^{\dim W-1}\cup \alpha )$ where $L/\mathbb {P}(W)$ denotes the tautological line bundle and $\int :H^{*+2(\dim W-1)}(\mathbb {P}(W))\to H^*(Z)$ denotes fiberwise integration).

As $H^*(IX;\mathbb {C})\to H^*(I\mathbb {F}(V);\mathbb {C})$ is injective and $\operatorname {Hom}_\mathbb {Z}(-,\mathbb {C})$ is exact, we conclude from Lemma 2.9 that $\operatorname {Hom}(H_*(IX;\mathbb {Z}_\chi )/H_*(I\mathbb {F}(V);\mathbb {Z}_\chi ),\mathbb {C})=0$, and hence that the domain is torsion. From this and Lemma 2.9 again, it follows that a class in $H^*(IX;\mathbb {C})$ is $\chi$-rational if and only if its pullback to $H^*(I\mathbb {F}(V);\mathbb {C})$ is $\chi$-rational. Now consider the pullback of $V$ to $\mathbb {F}(V)$. This pullback splits as a direct sum of line bundles, so because the inertial Chern character is additive under direct sum, we conclude that the pullback of $\operatorname {ch}^I(V)$ to $I\mathbb {F}(V)$ is $\chi$-rational, and hence that $\operatorname {ch}^I(V)$ is itself $\chi$-rational.

It remains to produce an integer $N=N(m,d,n)$ such that $N\cdot \operatorname {ch}^I_d(V)$ is $\chi$-integral for $\dim V=n$. We claim that there exists a finite orbi-complex $\mathbb {B}_{m,d}U(n)$ carrying a principal $U(n)$-bundle

\[ \mathbb{E}_{m,d}U(n)\to\mathbb{B}_{m,d}U(n) \]

with the property that every principal $U(n)$-bundle over a simplicial complex of groups $X$ of dimension $\leqslant d$ and isotropy $\leqslant m$ is a pullback of $\mathbb {E}_{m,d}U(n)\to \mathbb {B}_{m,d}U(n)$. As the inertial Chern character of $\mathbb {E}_{m,d}U(n)\to \mathbb {B}_{m,d}U(n)$ is $\chi$-rational and $\mathbb {B}_{m,d}U(n)$ is finite, there exists an integer $N=N(m,d,n)$ such that $N$ times this inertial Chern character is $\chi$-integral in cohomological degree $d$. By pullback, the same integer $N$ works for any vector bundle of rank $n$ over a simplicial complex of groups of order $\leqslant m$. Note that this argument gives no explicit bound on the integer $N$.

It remains to construct $\mathbb {E}_{m,d}U(n)\to \mathbb {B}_{m,d}U(n)$. We seek $U(n)$-spaces $\mathbb {E}_{m,d}U(n)$ such that

\begin{align*} \mathbb{E}_{m,-1}U(n)&=\varnothing,\\ \mathbb{E}_{m,d}U(n)&=\mathbb{E}_{m,d-1}U(n)\cup\biggl(\,\begin{matrix}\text{finitely many cells of the}\hfill\\\text{form }(D^d,\partial D^d)\times U(n)/G\hfill\end{matrix}\,\biggr),\\ \pi_r((\mathbb{E}_{m,d}U(n))^G)&=0\quad\text{for }r< d, \end{align*}

where $G\subseteq U(n)$ ranges over all finite subgroups of $U(n)$ of order $\leqslant m$. Note that there are finitely many conjugacy classes of such subgroups, and it suffices to consider just one representative of each conjugacy class. To show that $\mathbb {E}_{m,d}U(n)$ exists given $\mathbb {E}_{m,d-1}U(n)$, argue as follows. We have $\pi _{d-1}((\mathbb {E}_{m,d-1}U(n))^G)=H_{d-1}((\mathbb {E}_{m,d-1}U(n))^G)$ by Hurewicz, and the latter group is finitely generated because $\mathbb {E}_{m,d-1}U(n)$ is made up of finitely many cells. We define $\mathbb {E}_{m,d}U(n)$ by attaching cells $(D^d,\partial D^d)\times U(n)/G$ along a choice of finitely many generators of $\pi _{d-1}((\mathbb {E}_{m,d-1}U(n))^G)$, for each of the finitely many subgroups $G\subseteq U(n)$ on our fixed set of representatives. Note that $(D^d\times U(n)/H)^G=D^d\times \{a\in U(n)\,|\,a^{-1}Ga\subseteq H\}$, so any map $S^r\to (\mathbb {E}_{m,d}U(n))^G$ for $r< d$ can be homotoped to land inside $(\mathbb {E}_{m,d-1}U(n))^G$. It follows that the homotopy groups of $(\mathbb {E}_{m,d}U(n))^G$ satisfy the desired vanishing property.

Let us show that any principal $U(n)$-bundle $P\to X$ is pulled back from $\mathbb {E}_{m,d}U(n)\to \mathbb {B}_{m,d}U(n)$ when $\dim X\leqslant d$ with isotropy groups of order $\leqslant m$. That is, we should construct an equivariant map $P\to \mathbb {E}_{m,d}U(n)$ (note that since the target is a space, this is the same as an equivariant map from the coarse space $|P|$). By induction on the cells of $X$, it suffices to solve the extension problem for equivariant maps from $(D^r,\partial D^r)\times U(n)/G$ to $\mathbb {E}_{m,d}U(n)$ for $r\leqslant d$ and $|G|\leqslant m$. This is equivalent to the extension problem for maps $(D^r,\partial D^r)\to (\mathbb {E}_{m,d}U(n))^G$, whose positive solution for $r\leqslant d$ is one of the defining properties of $\mathbb {E}_{m,d}U(n)$.

Lemma 2.12 (Bott)

The image of the Chern character map

\[ \operatorname{ch}_n:\pi_{2n}(BU)\to H^{2n}(S^{2n};\mathbb{Q}) \]

is precisely $H^{2n}(S^{2n};\mathbb {Z})$.

We now have all the ingredients we need to prove the main result of this section, namely Theorem 1.1 for simplicial complexes of groups.

Proof. The condition on the fibers of $V$ amounts to the assertion that $\operatorname {ch}^I_0(V)=n\chi _1$ where $\chi _1\in H^0(IX;\mathbb {Q}_\chi )$ denotes ‘the characteristic function of the identity’ as a function $G_x/G_x\to \mathbb {C}$. We construct $V$ satisfying

\[ \operatorname{ch}^I(V)=n\chi_1 \]

(meaning all higher inertial Chern characters vanish). Certainly such a vector bundle exists over the $0$-skeleton $X_0$ of $X$, with rank $m!$, because all isotropy groups have order $\leqslant m$. It therefore suffices to show that any such vector bundle on $X_{k-1}$ extends to $X_k$.

To extend $V$ as a vector bundle from $X_{k-1}$ to $X_k$ amounts to doing an extension from $\partial D^k\times \mathbb {B} G$ to $D^k\times \mathbb {B} G$ for each $k$-simplex. The obstruction to doing this lies in $\prod _{\rho \in \hat G}\pi _{k-1}(BU(n_\rho ))$, where $n_\rho =(\dim V)(\dim \rho )/|G|$. The map $BU(n_\rho )\to BU$ is an isomorphism on $\pi _{k-1}$ provided $k-1<2n_\rho +2$, which is guaranteed by taking say $\dim V>dm$. The Chern character detects $\pi _*(BU)$ by Lemma 2.12, so vanishing of the inertial Chern character and linear independence of characters means that these obstructions all vanish. Thus, $V$ extends to $X_k$.

It remains to show that the extension of $V$ to $X_k$ can be taken to have trivial inertial Chern character. The space of extensions over a given $k$-simplex is a torsor for $\prod _{\rho \in \hat G}\pi _k(BU(n_\rho ))$, which is again $\prod _{\rho \in \hat G}\pi _k(BU)$ once we take $\dim V>dm$. We use this freedom to ensure that the inertial Chern character of $V$ on $X_k$ vanishes. Consider the long exact sequence

\[ \cdots\to H^*(IX_k,IX_{k-1};\mathbb{Q}_\chi)\to H^*(IX_k;\mathbb{Q}_\chi)\to H^*(IX_{k-1};\mathbb{Q}_\chi)\to\cdots. \]

As the inertial Chern character in $H^*(IX_k;\mathbb {Q}_\chi )$ maps to zero in $H^*(IX_{k-1};\mathbb {Q}_\chi )$, it is in the image of $H^*(IX_k,IX_{k-1};\mathbb {Q}_\chi )$. In particular, it may be nonzero only in degree $k$. Let us now replace $V$ with $V^{\oplus a}$ (which multiplies its inertial Chern character by $a$) for an appropriate positive integer $a$ so that by Proposition 2.10 its inertial Chern character is $\chi$-integral. Modifying our bundle by an element of (the product over the $k$-simplices of) $\prod _{\rho \in \hat G}\pi _k(BU)$ shifts its inertial Chern character by anything in $H^k(IX_k,IX_{k-1};\mathbb {Z}_\chi )$ by Lemma 2.12. We are done because $H^k(IX_k,IX_{k-1};\mathbb {Z}_\chi )\to H^k(IX_k;\mathbb {Z}_\chi )$ is surjective by the long exact sequence.

3. Topological stacks

We review some basic facts about stacks (on the category of topological spaces), we give a precise definition of what we mean by an ‘orbispace’, and we establish some of their basic properties. For further background, the reader may wish to consult Noohi [Reference NoohiNoo05], Gepner and Henriques [Reference Henriques and GepnerHG07], Behrend [Reference BehrendBeh14], Metzler [Reference MetzlerMet03], Behrend and Noohi [Reference Behrend and NoohiBN06], Heinloth [Reference HeinlothHei05], or Laumon and Moret-Bailly [Reference Laumon and Moret-BaillyLMB00].

Let $\mathsf {Top}$ denote the category of topological spaces and continuous maps, and let $\mathsf {Grpd}$ denote the 2-category of (essentially) small groupoids. A stack is a functor $F:\mathsf {Top}^\mathsf {op}\to \mathsf {Grpd}$ which satisfies descent, i.e. such that for every topological space $U$ and every open cover $\{U_i\to U\}_i$, the natural functor

is an equivalence. Stacks form a 2-category, with morphisms given by natural transformations of functors. The 2-category of stacks is complete, meaning all (small) limits exist; furthermore, these limits may be calculated pointwise in the sense that $(\lim _\alpha F_\alpha )(U)=\lim _\alpha (F_\alpha (U))$. Note that, as we are working in a 2-categorical context, all functors are 2-functors, all diagrams are 2-diagrams, all limits are 2-limits, etc. (though we usually omit the prefix ‘2-’).

The Yoneda lemma implies that the Yoneda functor $X\mapsto \operatorname {Hom}(-,X)$ embeds the category of topological spaces fully faithfully into the 2-category of stacks, and moreover that the natural map from $F(X)$ to the groupoid of maps of stacks $\operatorname {Hom}(-,X)\to F(-)$ is an equivalence. The category of topological spaces is complete, and the Yoneda embedding is continuous (commutes with limits). Hence, we make no distinction between a topological space $X$ and the associated stack $\operatorname {Hom}(-,X)$ of maps to $X$, nor between objects of $F(X)$ and maps $\operatorname {Hom}(-,X)\to F(-)$ (which we simply write as $X\to F$).

Every stack $X$ has a coarse space $|X|$ (a topological space) which is initial in the category of maps from $X$ to topological spaces. Concretely, the points of $|X|$ are the isomorphism classes of maps $*\to X$, and a subset $U\subseteq |X|$ is open if and only if for every map $Y\to X$ from a topological space $Y$, the inverse image of $U$ is an open subset of $Y$.

A stack is called representable if and only if it is in the essential image of the Yoneda embedding (i.e. it is isomorphic to a topological space). A morphism of stacks $F\to G$ is called representable if and only if for every map $X\to G$ from a topological space $X$, the fiber product $F\times _GX$ is representable.

For any property $\mathcal {P}$ of morphisms of topological spaces which is preserved under pullback, a representable morphism of stacks $F\to G$ is said to have property $\mathcal {P}$ if and only if the pullback $F\times _GX\to X$ has $\mathcal {P}$ for every map $X\to G$ from a topological space $X$. The following are examples of properties $\mathcal {P}$ of morphisms $f:X\to Y$ which are preserved under pullback:

• • $f$ is injective;

• • $f$ is surjective;

• • $f$ is open, meaning that the image of any open set is open. In contrast, the property of being closed, meaning that the image of any closed set is closed, is not preserved under pullback;

• • $f$ is an embedding, meaning that it is a homeomorphism onto its image;

• • $f$ is a closed embedding;

• • $f$ is étale, meaning that for every $x\in X$ there exists an open neighborhood $x\in U\subseteq X$ such that $f|_U:U\to Y$ is an open embedding;

• • $f$ is separated, meaning that for every distinct pair $x_1,x_2\in X$ with $f(x_1)=f(x_2)$, there exist open neighborhoods $x_i\in U_i\subseteq X$ which are disjoint $U_1\cap U_2=\varnothing$ (this is equivalent to the relative diagonal $X\to X\times _YX$ being a closed embedding);

• • $f$ is universally closed, meaning that $X\times _YZ\to Z$ is closed for every $Z\to Y$; this is equivalent to the assertion that for every $y\in Y$ and every collection of open sets $\{U_i\subseteq X\}_i$ covering $f^{-1}(y)$, there exists a finite subcollection which covers $f^{-1}(V)$ for some open neighborhood $y\in V\subseteq Y$ (one proof of this equivalence goes via yet a third equivalent condition, namely that every net $\{x_\alpha \in X\}_\alpha$ with $f(x_\alpha )\to y$ has a subnet converging to some $x\in f^{-1}(y)$);

• • $f$ is proper, meaning that it is separated and universally closed;

• • $f$admits local sections, meaning that there is an open cover $\{U_i\subseteq Y\}_i$ such that every restriction $f|_{f^{-1}(U_i)}:f^{-1}(U_i)\to U_i$ admits a section;

• • $f$ is a finite covering space, meaning that there is an open cover $\{U_i\subseteq Y\}_i$ such that every restriction $f|_{f^{-1}(U_i)}:f^{-1}(U_i)\to U_i$ is isomorphic to $U_i^{\sqcup n_i}\to U_i$ for some integer $n_i\geqslant 0$.

Each of these properties $\mathcal {P}$ is also closed under composition, and thus also under fiber products, meaning that for maps $X\to X'$ and $Y\to Y'$ over $Z$, if both $X\to X'$ and $Y\to Y'$ have $\mathcal {P}$, then $X\times _ZY\to X'\times _ZY'$ also has $\mathcal {P}$ (indeed, $X\times _ZY\to X\times _ZY'$ is a pullback of $Y\to Y'$).

For any stack $X$, open (respectively, closed) embeddings $Y\hookrightarrow X$ are in natural bijection with open (respectively, closed) subsets of $|X|$.

To check that a given map of spaces satisfies one of the properties $\mathcal {P}$ above, it is often helpful to make use of the fact that these $\mathcal {P}$ are all local on the target, meaning that for every open cover $\{U_i\subseteq Y\}_i$, if $X\times _YU_i\to U_i$ has $\mathcal {P}$ for every $i$, then so does $X\to Y$. This leads to the following generalization for maps of stacks: if $F\to G$ is a representable morphism of stacks and $G'\to G$ is a representable morphism of stacks admitting local sections, then $F\to G$ has $\mathcal {P}$ if and only if $F\times _GG'\to G'$ has $\mathcal {P}$. In fact, in this statement we need not assume that $G'\to G$ is representable, just that it admit local sections in the generalized sense that for every map $X\to G$ from a topological space $X$, there exists an open cover $\{U_i\subseteq X\}_i$ such that each $G'\times _GU_i\to U_i$ admits a section. We thus say that ‘$\mathcal {P}$ descends along maps admitting local sections’. The same descent property holds for representability itself.

A complex vector bundle over a stack $X$ is a representable map $V\to X$ together with maps $V\times _XV\to V$ and $\mathbb {C}\times V\to V$ (both over $X$) such that for every map $U\to X$ from a topological space $U$, there exists an open cover $\{U_i\subseteq U\}_i$ and integers $n_i\geqslant 0$ such that $V\times _XU_i\to U_i$ is isomorphic to $\mathbb {C}^{n_i}\times U_i\to U_i$ equipped with its fiberwise addition and scaling maps. A pullback of a complex vector bundle is naturally a complex vector bundle.

The class of so called topological stacks (those which admit a presentation via a topological groupoid) are somewhat better behaved than general stacks. A topological groupoid $M\rightrightarrows O$ consists of a pair of topological spaces $O$ (‘objects’) and $M$ (‘morphisms’), two maps $M\rightrightarrows O$ (‘source’ and ‘target’), a map $O\to M$ (‘identity’), an involution $M\to M$ (‘inverse’), and a map $M\times _OM\to M$ (‘composition’) satisfying the axioms of a groupoid. A topological groupoid $M\rightrightarrows O$ presents a stack $[M\rightrightarrows O]$ defined as follows. An object of $[M\rightrightarrows O](X)$ is an open cover $\{U_i\subseteq X\}_i$ together with maps $U_i\to O$ and $U_i\cap U_j\to M$ satisfying a compatibility condition, and an isomorphism in $[M\rightrightarrows O](X)$ consists of maps $U_i\cap U_{i'}'\to M$ satisfying a compatibility condition. There is a natural map $O\to [M\rightrightarrows O]$ (take the trivial open cover $\{X\subseteq X\}$) and the fiber product $O\times _{[M\rightrightarrows O]}O$ is naturally identified with $M$. The morphism $O\to [M\rightrightarrows O]$ admits local sections (by definition), so because $O\times _{[M\rightrightarrows O]}O\to O$ is representable, it follows by descent that $O\to [M\rightrightarrows O]$ is representable. Conversely, a representable map $U\to X$ admitting local sections from a topological space $U$ to a stack $X$ determines a topological groupoid $U\times _XU\rightrightarrows U$ presenting $X$. Indeed, the fiber product $U\times _XU$ is representable (because $U$ and $U\to X$ are), it admits two maps to $U$ (the two projections), an involution (exchanging the two factors), and a composition map $(U\times _XU)\times _U(U\times _XU)=U\times _XU\times _XU\to U\times _XU$ (forgetting the middle factor), and one can check using the stack property that the natural map $[U\times _XU\rightrightarrows U]\to X$ is an equivalence. A stack $X$ for which there exists a representable map $U\to X$ admitting local sections from a space $U$ is called topological, and such $U\to X$ is called an atlas for $X$.

For a topological stack $X$ with atlas $U\to X$, for any property $\mathcal {P}$ which descends along maps admitting local sections, the map $U\to X$ has $\mathcal {P}$ if and only if both maps $U\times _XU\to U$ have $\mathcal {P}$, and the diagonal $X\to X\times X$ has $\mathcal {P}$ if and only if the map $U\times _XU\to U\times U$ has $\mathcal {P}$. In particular, for any topological stack $X$, the diagonal $X\to X\times X$ is representable, and thus every map $Z\to X$ from a topological space $Z$ is representable. More generally, for any map of topological stacks $X\to Y$, the relative diagonal $X\to X\times _YX$ is representable (by descent from its pullback $V\times _XV\to V\times _YV$ for an atlas $V\to X$). If $X\to Y$ is representable and $Y$ is topological, then so is $X$ (if $U\to Y$ is an atlas for $Y$, then its pullback $V=U\times _YX\to X$ is an atlas for $X$), and the relative diagonal $X\to X\times _YX$ has $\mathcal {P}$ if and only if the relative diagonal $V\to V\times _UV$ of atlases has $\mathcal {P}$ (the latter is the pullback of the former under the map $V\times _UV\to X\times _YX$, which is representable and admits local sections because it is a pullback of $U\to Y$).

For a topological space $V$, a topological group $G$, and a continuous group action $G\curvearrowright V$, we may consider the action groupoid $G\times V\rightrightarrows V$ with source and target maps $(g,x)\mapsto x$ and $(g,x)\mapsto gx$. The stack associated to this groupoid is denoted $V/G$ and is called the stack quotient of the action $G\curvearrowright V$. If $G$ is discrete, the two maps $G\times V\to V$ are étale, so $V\to V/G$ is étale. If $G$ is compact and $V$ is Hausdorff, the map $G\times V\to V\times V$ is universally closed (factor as $G\times V\to G\times V\times V$ which is a closed embedding because $V$ is Hausdorff and $G\times V\times V\to V\times V$ which is universally closed because $G$ is compact), so $V/G$ has universally closed diagonal. If $G$ is Hausdorff, then the map $G\times V\to V\times V$ is separated, so $V/G$ has separated diagonal. Thus, if $G$ is compact Hausdorff and $V$ is Hausdorff, then $V/G$ has proper diagonal.

A groupoid presentation of a topological stack $X$ also gives a description of its coarse space $|X|$ as follows. For an atlas $U\to X$, consider the equivalence relation $\sim _X$ on $U$ given by the image of $U\times _XU\to U\times U$. There is a map $X\to U/{\sim _X}$ (which is tautological once we regard $X$ as $[U\times _XU\rightrightarrows U]$), inducing a map $|X|\to U/{\sim _X}$, which is a bijection, essentially by definition. As an open substack of $X$ pulls back to an open subset of $U$ invariant under $\sim _X$, it follows that $|X|\xrightarrow \sim U/{\sim _X}$ is open and is, thus, a homeomorphism. In particular, it follows that the coarse space $\big |V/G\big |$ of the stack quotient $V/G$ is the usual topological quotient of $V$ by $G$.

An action $G\curvearrowright V$ is called locally trivial if and only if $V$ admits a cover by $G$-invariant open sets $\{G\times Z_i\subseteq V\}_i$ where $G\curvearrowright G\times Z_i$ acts by left multiplication on $G$ and trivially on $Z_i$. If $G\curvearrowright V$ is locally trivial, then the natural map from the stack quotient to the topological quotient $V/G\to |V/G|$ is an equivalence. Indeed, this assertion is local on $|V/G|$, so it suffices to consider the case of $G\curvearrowright G\times Z$, where it holds by inspection.

(Similar results include [Reference BehrendBeh14, Theorem 1.108] and [Reference NoohiNoo05, Proposition 14.10].)

4. Coverings and nerves

We show how Theorem 2.13 implies Theorem 1.1. It is enough to show that a given orbispace admits a representable map to a simplicial complex of groups; this is Proposition 4.9. This simplicial complex of groups is basically just the nerve of a suitable open cover, however its construction is somewhat more delicate than one might initially expect.

A sieve on a topological space $X$ is a subset $S\subseteq 2^X$ consisting of open sets such that $U'\subseteq U\in S$ implies $U'\in S$. A covering sieve on $X$ is a sieve $S$ such that $\bigcup _{U\in S}U=X$. An open cover $\{U_i\subseteq X\}_i$ is said to generate the covering sieve on $X$ consisting of those open sets which are contained in some $U_i$.

Note that for sieves satisfying condition (1), condition (2) is equivalent to condition (2$'$) for $U,V\in S$ either $U\cap V=\varnothing$ or $f(U\cap V)=f(U)\cap f(V)$. If $S'\subseteq S$ is an inclusion of covering sieves and $S$ is a connection sieve on $X\to Y$, then so is $S'$. In particular, if $S$ and $S'$ are connection sieves on $X\to Y$, then so is $S\cap S'$. To check that an open cover $\{U_i\subseteq X\}_i$ generates a connection sieve, it is enough to check axioms (1) and (2$'$) for the open sets $U_i$.

Open embeddings are strongly étale, and strongly étale maps are separated and étale (however, the converse is false by Example 4.3). Being strongly étale is preserved under pullback (take the connection sieve generated by the pullback of the original connection sieve), and the class of strongly étale maps is closed under composition (a connection sieve $S_{X/Z}$ for a composition $X\to Y\to Z$ is given by those elements of a fixed connection sieve $S_{X/Y}$ for $X\to Y$ whose image lies in a fixed connection sieve $S_{Y/Z}$ for $Y\to Z$). A disjoint union $\bigsqcup _iX_i\to \bigsqcup _iY_i$ of strongly étale maps $X_i\to Y_i$ is strongly étale (take disjoint union of connection sieves), and the projection $A\times Y\to Y$ is strongly étale for any discrete space $A$. Being strongly étale is not local on the target, as shown by the following example.

Recall that a topological space is called paracompact if and only if every open cover admits a locally finite refinement [Reference DieudonnéDie44]. If $X$ is paracompact and Hausdorff, then there exists a partition of unity subordinate to any given locally finite open cover $\{U_i\subseteq X\}_i$, namely functions $f_i:X\to \mathbb {R}_{\geqslant 0}$ with $\operatorname {supp} f_i\subseteq U_i$ such that $\sum _if_i\equiv 1$ (recall that the support $\operatorname {supp} f$ of a function $f:X\to \mathbb {R}_{\geqslant 0}$ is, by definition, the complement of the largest open set over which $f$ vanishes identically).

An orbispace will be called paracompact (respectively, coarsely finite-dimensional, $d$-dimensional) if and only if its coarse space is.

A simplicial complex is a pair $X=(V,S)$ consisting of a set $V$ (‘vertices’) and a set $S\subseteq 2^V\setminus \{\varnothing \}$ (‘simplices’) of finite subsets of $V$ such that $S$ contains all singletons and $\varnothing \ne A\subseteq B\in S$ implies $A\in S$. The star $\operatorname {st}(X,\sigma )\subseteq X$ of a simplex $\sigma$ in a simplicial complex $X$ is the subcomplex consisting of all simplices $\tau \subseteq X$ with $\sigma \cup \tau \in S(X)$.

A map of simplicial complexes $X\to Y$ is a map of vertex sets $V(X)\to V(Y)$ which maps simplices to simplices (the image of an element of $S(X)$ is an element of $S(Y)$). A map of simplicial complexes is called injective if and only if the map on vertex sets (hence, also the map on simplices) is injective. A map of simplicial complexes $f:X\to Y$ is called étale (respectively, locally injective) if and only if the induced maps on stars $\operatorname {st}(X,\sigma )\to \operatorname {st}(Y,f(\sigma ))$ are isomorphisms (respectively, injective). We call a map of simplicial complexes $X\to Y$ sufficiently étale if and only if every simplex $\sigma \subseteq Y$ (equivalently, every vertex) is the image of a simplex $\tau \subseteq X$ at which $X\to Y$ is étale (this is a useful weakening of the condition of being surjective and étale, which in the context of simplicial complexes is too strong).

The geometric realization $\|X\|$ of a simplicial complex $X$ is the set of tuples $t\in \mathbb {R}_{\geqslant 0}^{V(X)}$ with $\sum _vt_v=1$ such that $\{v:t_v>0\}\in S(X)$, topologized by declaring that the realization of the complete simplex on $k+1$ vertices has the usual topology and that a realization $\|X\|$ is given the strongest topology for which (the realization of) every map from a complete simplex to $X$ is continuous. The geometric realization of an étale map of simplicial complexes is an étale map of spaces.

A locally injective simplicial complex groupoid $M\rightrightarrows O$ consists of simplicial complexes $O$ and $M$ together with structure maps satisfying the axioms of a groupoid, where both maps $M\rightrightarrows O$ are locally injective. Local injectivity of the two maps $M\rightrightarrows O$ implies that the natural map $\big \|M\times _OM\big \|\xrightarrow \sim \|M\|\times _{\|O\|}\|M\|$ is a homeomorphism, and thus the geometric realization $\|M\|\rightrightarrows \|O\|$ is a topological groupoid. If $\partial O\subseteq O$ denotes the subcomplex consisting of those simplices $\sigma \subseteq O$ for which it is not the case that the first projection $M\to O$ is étale at every simplex $\tau \subseteq M$ mapped to $\sigma$ under the second projection, then the natural map $\|O\|\setminus \|\partial O\|\to [\|M\|\rightrightarrows \|O\|]$ is étale. A locally injective simplicial complex groupoid $M\rightrightarrows O$ is called sufficiently étale if and only if this map is surjective (equivalently, every vertex of $O$ is $M$-isomorphic to one not in $\partial O$).

The abstract simplex category $\mathsf {Simp}$ has objects finite totally ordered sets and has morphisms weakly order-preserving maps; every object of $\mathsf {Simp}$ is isomorphic to $[n]:=\{0<\cdots < n\}$ for a unique integer $n\geqslant 0$. A simplicial object in a category $\mathsf {C}$ is a functor $\mathsf {Simp}^\mathsf {op}\to \mathsf {C}$, and the category of simplicial objects in $\mathsf {C}$ is denoted $\mathsf {s}\mathsf {C}$. If $\mathsf {C}$ is complete (respectively, cocomplete), then so is $\mathsf {s}\mathsf {C}$, and limits (respectively, colimits) are calculated pointwise.

We consider only simplicial sets (objects of the category $\mathsf {s}\mathsf {Set}$) and simplicial groupoids (objects of the category $\mathsf {s}\mathsf {Grpd}$). A simplicial set (or groupoid) will be denoted $X_\bullet$, where $X_n$ is its set (or groupoid) of $n$-simplices. We denote by $\Delta ^n_\bullet \in \mathsf {s}\mathsf {Set}$ the standard $n$-simplex, given by $[m]\mapsto \operatorname {Hom}([m],[n])$. The Yoneda lemma implies that $X_n=\operatorname {Hom}(\Delta ^n_\bullet,X_\bullet )$.

A map of simplicial sets $X_\bullet \to Y_\bullet$ is called injective if and only if it is so levelwise (i.e. every $X_n\to Y_n$ is injective). A map $X_\bullet \to Y_\bullet$ is called étale (respectively, locally injective) if and only if for every map $[n]\to [m]$ in $\mathsf {Simp}$, the induced map $X_m\xrightarrow \sim Y_m\times _{Y_n}X_n$ is a bijection (respectively, injective) (it is equivalent to impose this condition only for $n=0$). A map $X_\bullet \to Y_\bullet$ is called étale (respectively, locally injective) at a given $n$-simplex $\sigma$ of $X_\bullet$ if and only if $X_m\xrightarrow \sim Y_m\times _{Y_n}\{\sigma \}$ is a bijection (respectively, injective) (this condition at a given $\sigma$ implies the same at any preimage of $\sigma$ under any structure map of $X_\bullet$). A map $X_\bullet \to Y_\bullet$ is called sufficiently étale if and only if the $n$-simplices of $X_\bullet$ at which the map is étale surject onto the $n$-simplices of $Y_\bullet$ (it is equivalent to impose this condition only for $n=0$). These notions generalize to maps of simplicial groupoids by replacing ‘injectivity’ and ‘surjectivity’ for maps of sets with ‘full faithfulness’ and ‘essential surjectivity’ for functors of groupoids. These properties are all preserved under pullback and closed under composition.

The geometric realization $\|X_\bullet \|$ of a simplicial set $X_\bullet$ is the colimit of $\Delta ^n$ over all maps $\Delta ^n_\bullet \to X_\bullet$. Geometric realization is cocontinuous, and the natural map $\|\lim _\alpha (X_\bullet )_\alpha \|\to \lim _\alpha \|(X_\bullet )_\alpha \|$ is bijective, however it need not be a homeomorphism even for finite limits (for example, the case of binary products is discussed in [Reference MayMay67, Theorem 14.3, Remark 14.4] and [Reference WhiteheadWhi49, Reference MilnorMil56, Reference Brooke-TaylorBT17]). The map $\|X_\bullet \times _{Y_\bullet }Z_\bullet \|\to \|X_\bullet \| \times _{\|Y_\bullet \|}\|Z_\bullet \|$ is a homeomorphism if at least one of the maps $X_\bullet \to Y_\bullet$ and $Z_\bullet \to Y_\bullet$ is locally injective. Thus, a locally injective simplicial set groupoid $M_\bullet \rightrightarrows O_\bullet$ determines a topological groupoid $\|M_\bullet \|\rightrightarrows \|O_\bullet \|$.

Let us now introduce the geometric realization $\|X_\bullet \|$ of any simplicial groupoid $X_\bullet$ which is étale, meaning that it admits a locally injective sufficiently étale map $U_\bullet \to X_\bullet$ from a simplicial set $U_\bullet$. For any simplicial groupoid $X_\bullet$ and any locally injective map $U_\bullet \to X_\bullet$, the pair of simplicial sets $U_\bullet \times _{X_\bullet }U_\bullet \rightrightarrows U_\bullet$ forms a locally injective simplicial set groupoid, whose geometric realization $\|U_\bullet \times _{X_\bullet }U_\bullet \|\rightrightarrows \|U_\bullet \|$ thus defines a topological stack. The geometric realization of $X_\bullet$ is defined as this topological stack $[\|U_\bullet \times _{X_\bullet }U_\bullet \|\rightrightarrows \|U_\bullet \|]$ associated to any locally injective sufficiently étale map $U_\bullet \to X_\bullet$.

A simplicial complex $X$ gives rise to a simplicial set $b_\bullet X$ (its barycentric subdivision) whose $n$-simplices are chains of simplices $\sigma _0\subseteq \cdots \subseteq \sigma _n\subseteq X$ (in other words, $b_\bullet X$ is the nerve of $S(X)$). Barycentric subdivision preserves injectivity, local injectivity, étale, and sufficiently étale. There is a natural identification of geometric realizations $\|X\|=\|b_\bullet X\|$. Moreover, for a locally injective sufficiently étale simplicial complex groupoid $M\rightrightarrows O$, there is a natural identification $[\|M\|\rightrightarrows \|O\|]=\|[b_\bullet M\rightrightarrows b_\bullet O]\|$ (the simplicial groupoid $[b_\bullet M\rightrightarrows b_\bullet O]$ is étale because $M\rightrightarrows O$ is sufficiently étale).

A simplicial complex of groups $(Z,G)$ also admits a barycentric subdivision $b_\bullet (Z,G)$ which is a simplicial groupoid. In fact, we define $b_\bullet (Z,G)$ for any simplicial complex $Z$ equipped with a functor $G:S(Z)^\mathsf {op}\to \mathsf {Grpd}$ from the face poset to groupoids (a simplicial complex of groups $(Z,G)$ determines such a functor which sends $\sigma$ to $\mathbb {B} G_\sigma$). The groupoid of $n$-simplices in the barycentric subdivision $b_\bullet (Z,G)$ is now defined as the groupoid of functors from the category $0\to \cdots \to n$ to the category whose objects are pairs $\sigma \in S(Z)$ and $o\in G_\sigma$ and whose morphisms are inclusions $\sigma _1\subseteq \sigma _2$ covered by maps $o_1\to o_2|_{\sigma _1}$. The barycentric subdivision of a simplicial complex of groups is étale, as can be seen as follows. For any $\sigma \subseteq Z$ and $o\in G_\sigma$, we consider the functor $S(\operatorname {st}(Z,\sigma ))^\mathsf {op}\to \mathsf {Grpd}$ given by $\tau \mapsto G_{\sigma \cup \tau }\times _{G_\sigma }\{o\}$. Applying the nerve construction from just above to this functor, we obtain a simplicial set mapping to $b_\bullet (Z,G)$, which is the required locally injective map which is étale over $(\sigma,o)\in b_0(Z,G)$. An essentially equivalent discussion (albeit without barycentrically subdividing) appears in [Reference Bridson and HaefligerBH99, 12.24–12.25]. As $b_\bullet (Z,G)$ is étale, it has a geometric realization $\|b_\bullet (Z,G)\|$ which we also write as $\|(Z,G)\|$. When $G$ are finite groups (or, more generally, groupoids with finite isotropy), then the geometric realization $\|(Z,G)\|$ is an orbispace by Lemma 4.7, and this is what we have been calling the orbispace presented by the simplicial complex of finite groups $(Z,G)$.

The nerve $N(X,\{U_i\}_i)$ of a collection of open sets $\{U_i\subseteq X\}_i$ is the simplicial complex whose vertices $V$ are the indices $i$ with $U_i\ne \varnothing$, and in which a collection $I$ of indices spans a simplex (i.e. $I\in S$) if and only if $\bigcap _{i\in I}U_i\ne \varnothing$. A partition of unity $\{f_i:X\to \mathbb {R}_{\geqslant 0}\}_i$ subordinate to a locally finite open cover $\{U_i\subseteq X\}_i$ defines a map from $X$ to the geometric realization $\|N(X,\{U_i\}_i)\|$ of the nerve of the open cover.

Proof. We begin with an open cover $\{V_i/G_i\subseteq X\}_i$ with the properties guaranteed by Proposition 4.5, and we set $U:=\bigsqcup _iV_i$. As $|X|$ is paracompact, by $G_i$-equivariantly shrinking the spaces $V_i$, we may assume that the associated open cover $\{|V_i/G_i|\subseteq |X|\}_i$ of coarse spaces is locally finite. We fix a covering sieve $S$ on $U\times _XU$ which is invariant under the ‘exchange’ (i.e. ‘inverse’) involution of $U\times _XU$ and is a connection sieve for both projections $U\times _XU\rightrightarrows U$. We also fix a partition of unity $\{f_i:|X|\to \mathbb {R}_{\geqslant 0}\}_i$ subordinate to the open cover $\{|V_i/G_i|\subseteq |X|\}_i$.

Denote by $o(x)\subseteq U$ and $o_i(x)\subseteq V_i$ the fibers over $x\in |X|$, so $o(x)=\bigsqcup _io_i(x)$; similarly define $m(x)\subseteq U\times _XU$ and $m_{ij}(x)\subseteq V_i\times _XV_j$ with $m(x)=\bigsqcup _{i,j}m_{ij}(x)$. These sets are finite because the $G_i$ are finite and $\{|V_i/G_i|\subseteq |X|\}_i$ is locally finite. Furthermore, they have the discrete topology, because $U$ is Hausdorff and $U\times _XU$ is Hausdorff (because $U$ is Hausdorff and $X\to X\times X$ is separated).

The Hausdorff property implies that the inclusions $o_i(x)\subseteq V_i$ and $m_{ij}(x)\subseteq V_i\times _XV_j$ admit retractions defined in some open neighborhood. Now the inverse images of small open neighborhoods $x\in |Z_x|\subseteq |X|$ (i.e. open substacks $Z_x\subseteq X$) form a basis of neighborhoods of $o_i(x)$ and $m_{ij}(x)$, so for sufficiently small $Z_x$, these inverse images are naturally disjoint unions $U_x=\bigsqcup _{o\in o(x)}U_o$ and $U_x\times _XU_x=\bigsqcup _{m\in m(x)}U_m$. Note that this applies only inside $V_i$ and $V_i\times _XV_j$ for which $o_i(x)$ and $m_{ij}(x)$ are nonempty: the full inverse image of $Z_x$ inside $U$ may intersect other $V_i$ nontrivially. By shrinking $Z_x$ further, we may ensure that the retraction $(U_x\times _XU_x\rightrightarrows U_x)\to (m(x)\rightrightarrows o(x))$ is a map of groupoids. For later purposes, let us also take $Z_x$ small enough so that:

• • if $x\in |V_i/G_i|$, then $|Z_x|\subseteq |V_i/G_i|$;

• • if $x\notin \operatorname {supp} f_i$, then $|Z_x|\cap \operatorname {supp} f_i=\varnothing$;

• • if $f_i(x)>0$, then $f_i>0$ over all of $|Z_x|$.

Each of these conditions can be ensured on its own, and because the open cover $\{|V_i/G_i|\subseteq |X|\}_i$ is locally finite, we can ensure all at once.

We also shrink $Z_x$ so as to ensure that each $U_m\in S$ (our chosen connection sieve), which has the following implication: given $o,o'\in U$ with $U_o\cap U_{o'}\ne \varnothing$, the relation $U_m\cap U_{m'}\ne \varnothing$ is a partial bijection between lifts $m,m'\in U\times _XU$ of $o$ and $o'$. Furthermore, the domain of this bijection is as large as possible: for $U_o\cap U_{o'}\ne \varnothing$ (so $o\in o_i(x)$ and $o'\in o_i(x')$ for some $i$) with $x,x'\in |V_j/G_j|$, we get a full bijection between the inverse images of $o$ and $o'$ inside $m_{ij}(x)$ and $m_{ij}(x')$ (this follows because the projection $V_i\times _XV_j\to V_i$ is a finite covering space of degree $|G_j|$ over $U_o\cup U_{o'}$).

We now consider the nerves $N(U,\{U_o\}_{o\in U})$ and $N(U\times _XU,\{U_m\}_{m\in U\times _XU})$. Note that for simplices in these nerves, namely subsets $O\subseteq U$ or $M\subseteq U\times _XU$ with $\bigcap _{o\in O}U_o\ne \varnothing$ or $\bigcap _{m\in M}U_m\ne \varnothing$, the maps $O\to |X|$ or $M\to |X|$ are injective. The natural maps on index sets $U\times _XU\rightrightarrows U$ determine maps of nerves

\[ N(U\times_XU,\{U_m\}_{m\in U\times_XU})\rightrightarrows N(U,\{U_o\}_{o\in U}). \]

These maps are locally injective; indeed, local injectivity means that for every $o,o'\in U$ with $U_o\cap U_{o'}\ne \varnothing$ and every $m\in U\times _XU$ projecting to $o$, there is at most one lift $m'\in U\times _XU$ of $o'$ with $U_m\cap U_{m'}\ne \varnothing$, and this is a direct consequence of our assumption that every $U_m\in S$. Let us now argue that there is a natural composition map

\begin{align*} &N(U\times_XU,\{U_m\}_{m\in U\times_XU})\times_{N(U,\{U_o\}_{o\in U})}N(U\times_XU,\{U_m\}_{m\in U\times_XU})\\ &\quad \longrightarrow N(U\times_XU,\{U_m\}_{m\in U\times_XU}). \end{align*}

More precisely, we claim that for nonempty finite subsets $M\subseteq U\times _XU$ with $\bigcap _{m\in M}U_m\ne \varnothing$ and $M'\subseteq U\times _XU$ with $\bigcap _{m\in M'}U_m\ne \varnothing$ projecting to $O\subseteq U$ (under the first and second projections, respectively), the subset $M''\subseteq U\times _XU$ defined by applying the composition map $(U\times _XU)\times _U(U\times _XU)\to U\times _XU$ to $M$ and $M'$ also satisfies $\bigcap _{m\in M''}U_m\ne \varnothing$. This claim follows from the property that every $U_m\in S$ (indeed, this property implies that $\bigcap _{m\in M}U_m\xrightarrow \sim \bigcap _{o\in O}U_o\xleftarrow \sim \bigcap _{m\in M'}U_m$). We have thus defined a locally injective simplicial complex groupoid

\[ M:=N(U\times_XU,\{U_m\}_{m\in U\times_XU})\rightrightarrows N(U,\{U_o\}_{o\in U})=:O. \]

We now show that this locally injective simplicial complex groupoid is sufficiently étale. Every isomorphism class of vertex (equivalently, every $x\in |X|$ with $|Z_x|\ne \varnothing$) has a representative $o\in V_i\subseteq U$ with $f_i(o)=f_i(x)>0$. To show that these representatives are étale, let $m\in U\times _XU$ be a morphism with source $o$ and target $o'$, and let $U_{o'}\cap U_{o''}\ne \varnothing$. We must show that there is a bijection between morphisms $o\to o'$ and $o\to o''$. The morphisms in question all have source inside $V_i$, so we really can consider just $V_i\times U$ for the present purpose. Now the bulleted conditions on $|Z_x|$ from above imply that because $U_{o'}\cap U_{o''}\ne \varnothing$ and $f_i(o')>0$, we have $U_{o'}\cup U_{o''}\subseteq |V_i/G_i|\subseteq |X|$. Thus, over $U_{o'}\cap U_{o''}$ the map $V_i\times _XU\to U$ is a finite covering space of degree $|G_i|$. Thus, all points have the same number of lifts, so it follows that the connection sieve property gives us a bijection between lifts.

We have already seen above that our sufficiently étale locally injective simplicial complex groupoid satisfies the first hypothesis of Lemma 4.8, and the second hypothesis follows from the partial bijection property derived above from the connection sieve. Thus, by Lemma 4.8, there is a simplicial complex of groups $(Z,G)$ giving rise to the same simplicial groupoid $b_\bullet (Z,G)=[b_\bullet M\rightrightarrows b_\bullet O]$ and thus (because $M\rightrightarrows O$ is sufficiently étale) to the same geometric realization $\|(Z,G)\|=[\|M\|\rightrightarrows \|O\|]$. The groups $G_z$ associated to vertices $z\in Z$ are, by definition, isotropy groups of the vertex groupoid $V(M)\rightrightarrows V(O)$, which are, by definition, isotropy groups of points of $X$.

To conclude, it remains to define a map $X\to \|(Z,G)\|$ which is injective on isotropy groups. To define this map, we shrink the $Z_x$ so that the open cover $\{|Z_x|\subseteq |X|\}_{x\in |X|}$ is locally finite, and choose a partition of unity $\{g_x:|X|\to \mathbb {R}_{\geqslant 0}\}_{x\in |X|}$ subordinate to the open cover $\{|Z_x|\subseteq |X|\}_{x\in |X|}$. These maps $g_x$ lift to maps $g_o:U\to \mathbb {R}_{\geqslant 0}$ supported inside $U_o$ and $g_m:U\times _XU\to \mathbb {R}_{\geqslant 0}$ supported inside $U_m$. The collection of these lifts defines a map of topological groupoids

\[ (U^0\times_XU^0\rightrightarrows U^0)\to\bigl(\big\|N(U\times_XU,\{U_m\}_{m\in U\times_XU})\big\|\rightrightarrows\big\|N(U,\{U_o\}_{o\in U})\big\|\bigr), \]

where $U^0$ is the disjoint union of the open loci $V_i^0\subseteq V_i$ where $f_i>0$ (the maps $g_o$ and $g_m$ do not define a map over all of $U\times _XU\rightrightarrows U$ due to the fact that $U_x=\bigsqcup _{o\in o(x)}U_o$ and $U_x\times _XU_x=\bigsqcup _{m\in m(x)}U_m$ may not be the full inverse images of $Z_x$ inside $U$ and $U\times _XU$). It remains to check that this map is injective on isotropy groups; in other words, for $o,o'\in U^0$ we must show that the map $\{o\}\times _X\{o'\}\to \|N(U\times _XU,\{U_m\}_{m\in U\times _XU})\|$ is injective. Distinct elements of $\{o\}\times _X\{o'\}$ are, in particular, distinct lifts of $o$, which therefore cannot lie in any common $U_m$ because $U_m\to U$ is injective, so we see that the map is indeed injective on isotropy groups.