Pursued By Stacks 5: Topological Stacks

19 May 2022

The purpose of this post is to single out the topological (or geometric or Artin) stacks.

Atlases

We will say that a stack over \(\mathbf{Top}\) “is a space” if it is isomorphic as a stack to \(\underline{X}\) for some topological space \(X\). Following Lerman, we will in fact just drop the underline, thinking of \(X\) as the stack \(\underline{X}\).

An atlas for a stack \(\mathsf{D}\) is a space \(X\) and a map \(p\colon X \to \mathsf{D}\) such that for any map \(f\colon Y \to \mathsf{D}\) of a space into \(\mathsf{D}\), the fiber product \(Y \times_{\mathsf{D}} X\) is a space and the map \(\pi_1 \colon Y \times_{\mathsf{D}} X \to Y\) is an open surjection which admits local sections.

Apparently it is common to say in the literature that the map \(X \to \mathsf{D}\) is representable. We say that \(\mathsf{D}\) is a topological stack if it has an atlas.

Example: \(\mathsf{B}\mathcal{G}\)

Let \(\mathcal{G}\) be a topological groupoid, and let \(\mathsf{B}\mathcal{G}\) be the category of principal \(\mathcal{G}\)-bundles and \(\mathcal{G}\)-equivariant maps. Lerman shows that \(\mathsf{B}\mathcal{G}\) is fibered in groupoids over \(\mathbf{Top}\); well, strictly speaking he is talking about Lie groupoids, but the argument works just fine for spaces and topological groupoids. In fact it is a stack: just as, you might think, given a topological group \(G\) and a principal \(G\)-bundle on an open cover of a space \(X\) with gluing data, you can glue up to form a principal \(G\)-bundle on \(X\), you can do the same for groupoids, and moreover isomorphisms form a sheaf.

Given a topological groupoid \(\mathcal{G}\), we claim that the map \(p\colon \mathcal{G}_0 \to \mathsf{B}\mathcal{G}\) defined (by the 2-Yoneda lemma) by the principal \(\mathcal{G}\)-bundle \(\omega\colon \mathcal{G}_1 \to \mathcal{G}_0\) is an atlas.

So given any space \(Y\) and a map \(E\colon Y \to \mathsf{B}\mathcal{G}\), we need to show that \(Y\times_{\mathsf{B}\mathcal{G}} \mathcal{G}_0\) is a space. First, note that by the 2-Yoneda lemma, the functor \(E\) is determined by a principal \(\mathcal{G}\)-bundle \(E \to Y\). Given a map \(g\colon Z \to Y\), the bundle \(E(g)\) is canonically isomorphic to the pullback \(g^*E\), so to ease notation, we will assume that it is actually equal to \(g^*E\). Similarly given \(f\colon Z \to \mathcal{G}_0\), we will assume the bundle \(p(f)\) is \(f^*\mathcal{G}_1\). Thus an object of the fiber product over a space \(Z\) is a triple \((g,f,\beta)\) where \(g\colon Z \to Y\) and \(f\colon Z \to \mathcal{G}_0\) are maps of spaces and \(\beta\colon g^*E \to f^*\mathcal{G}_1\) is a \(\mathcal{G}\)-equivariant map of bundles, hence an isomorphism.

Recall that

\[f^*\mathcal{G}_1 = \{(z,h) \in Z \times \mathcal{G}_1 : f(z) = \omega(h) \}.\]

The map \(z \mapsto (z,1_{f(z)})\) defines a global section \(\sigma\) of this bundle. We claim that the isomorphism \(\beta^{-1}\colon f^*\mathcal{G}_1 \to g^*E\) is determined uniquely by the image of \(\sigma\). Indeed, given \(\beta^{-1}(\sigma(z)) = (z,e)\), we must have \(\beta^{-1}(z,h) = \beta^{-1}(\sigma(z).h) = (z,e).h\), for all appropriate \(h \in \mathcal{G}_1\), so fiber by fiber the bundle map \(\beta^{-1}\) is determined by the point \(\beta^{-1}(\sigma(z))\). This gives us a map \(\tau\colon Z \to E\) defined as \(z \mapsto e\) such that \(\pi \tau = g\), where \(\pi\colon E \to Y\) is the defining map of the principal bundle \(E\). Notice that the anchor map of, for instance, \(g^*E\) is \((z,e) \mapsto a(e)\), where \(a \colon E \to \mathcal{G}_0\) is the anchor map. We have

\[f = f\pi\sigma = \omega f_{\mathcal{G}_1} \sigma = \alpha (\cdot)^{-1} f_{\mathcal{G}_1} \sigma = \alpha f_{\mathcal{G}_1}\sigma,\]

since \(\sigma(z) = (z,1_{f(z)})\). But then

\[\alpha f_{\mathcal{G}_1\sigma} = a g_E \beta^{-1} \sigma,\]

since the isomorphism \(\beta\) intertwines the actions and thus preserves anchor maps. Finally,

\[a g_E \beta^{-1} \sigma = a \tau.\]

Conversely, given any map \(\tau \colon Z \to E\), we reconstruct \((g,f,\beta)\) as follows: the map \(g \colon Z \to Y\) is \(\pi\tau\), the map \(f\) is \(a\tau\), Finally we need a map \(\beta \colon g^*E \to f^*\mathcal{G}_1\). By definition we have

\[g^*E = \{(z,e) \in Z\times E : \pi\tau(z) = \pi(e) \} \quad\text{and}\quad f^*\mathcal{G}_1 = \{(z,h) \in Z \times \mathcal{G}_1 : a\tau(z) = \omega(h) \}.\]

Define \(\beta(z,\tau(z)) = (z,1_{a\tau(z)})\) and extend equivariantly. It is clear that these constructions are mutually inverse, and we conclude that objects of the 2-fiber product are (isomorphic to) maps of spaces to \(E\).

Now for arrows! Suppose we have an arrow \((u,v)\colon (g_1,f_1,\beta_1) \to (g_2,f_2,\beta_2)\) over a map of spaces \(h\colon Z_1 \to Z_2\). This is a pair of maps \(u\colon Z_1 \to Z_2\) and \(v\colon Z_1 \to Z_2\) with the property that the following diagrams commute

\[\require{AMScd}\begin{CD} Z_1 @>g_1>> Y \\ @VVuV @| \\ Z_2 @>g_2>> Y \end{CD}\qquad\begin{CD} Z_1 @>f_1>> \mathcal{G}_0 \\ @VVvV @| \\ Z_2 @>f_2>> \mathcal{G}_0 \end{CD}\qquad \begin{CD} g_1^*E @>\beta_1>> f_1^*\mathcal{G}_1 \\ @VV\tilde uV @VV\tilde vV \\ g_2^*E @>\beta_2>> f_2^*\mathcal{G}_1, \end{CD}\]

where \(\tilde u\) and \(\tilde v\) are the unique \(\mathcal{G}\)-equivariant maps satisfying \((g_1)_E = (g_2)_E \tilde u\) and \((f_1)_{\mathcal{G}_1} = (f_2)_{\mathcal{G}_1}\tilde v\). But because each \(\beta_i\) is over the identity map of \(Z_i\), we conclude that in fact \(u = v\). We have

\[\tau_1\pi_1 = (g_1)_E \beta_1^{-1} = (g_2)_E \tilde u \beta_1^{-1} = (g_2)_E \beta_2^{-1}\tilde v = \tau_2\pi_2 \tilde v = \tau_2 u \pi_1,\]

so because \(\pi_1\) is an epimorphism, we conclude \(\tau_1 = \tau_2 u\). Finally, because, for instance, \((g_1)_E(z,\tau_1(z).h) = \tau_1(z).h\) and \((g_2)_E(z,\tau_2(z).h) = \tau_2(z).h\), we must have \(\tilde u(z,\tau_1(z).h) = (u(z),\tau_2u(z)) = (u(z),\tau_1(z))\) and similarly \(\tilde v(z,1_{a\tau_1(z)}) = (v(z),1_{a\tau_2(u(z))}) = (v(z),1_{a\tau_1(z)})\).

Conversely, given \(u \colon Z_1 \to Z_2\) such that \(\tau_1 = \tau_2u\), observe that \(g_1 = \pi\tau_1 = \pi\tau_2 u = g_2u\) and \(f_2 = a\tau_1 = a\tau_2 u = f_2u\). We have, for any \(z \in Z_1\) and appropriate \(h \in \mathcal{G}_1\),

\[\begin{gather*} \tilde v \beta_1(z,\tau_1(z).h) = \tilde v(z,h) = (v(z),h) = (u(z),h) \\ = \beta_2(u(z),\tau_2u(z).h) = \beta_2(u(z),\tau_1(z).h) = \beta_2\tilde u(z,\tau_1(z).h). \end{gather*}\]

Therefore arrows in the 2-fiber product are (isomorphic to) maps in \(\underline{E}\), and we conclude that the 2-fiber product is the space \(E\). The map \(\pi\colon E \to Y\) is an open surjection which admits local sections because \(E\) is a principal \(\mathcal{G}\)-bundle. Therefore the map \(p\colon \mathcal{G}_0 \to \mathsf{B}\mathcal{G}\) is an atlas.