Pursued By Stacks 1: Cfgs

16 May 2022

I’ve spent the past couple years on and off understanding how to think of graphs of groups and orbifolds as (being represented by) étale groupoids. In view of Lerman’s critique, to which his answer is the category of geometric stacks over manifolds, I thought it might be relevant to try and understand what a stack is. This is the first in what will likely be several posts towards understanding the definition of a stack; this one is focused on the concept of a category fibered in groupoids.

Categories fibered in groupoids

Let \(\mathbf{Top}\) denote the category of compactly generated topological spaces and continuous maps. (All spaces I tend to think about are compactly generated, being either manifolds or CW complexes, so this is not a serious restriction for me. That being said, I don’t immediately understand the desire to restrict to compactly generated spaces, because my algebraic topology is not strong in that direction.)

A functor \(\pi \colon \mathsf{C} \to \mathbf{Top}\) is a category fibered in groupoids over \(\mathbf{Top}\) if the following conditions hold.

\[\require{AMScd}\begin{CD} \xi'' @>f>> \xi \\ @. @| \\ \xi' @>h>> \xi \end{CD}\]

in \(\mathsf{C}\) such that there exists a continuous map \(g\colon \pi(\xi'') \to \pi(\xi')\) making the following diagram commute

\[\begin{CD} \pi(\xi'') @>\pi(f)>> \pi(\xi) \\ @VgVV @| \\ \pi(\xi') @>\pi(h)>> \pi(\xi), \end{CD}\]

There exists a unique arrow \(\tilde g\colon \xi'' \to \xi'\) such that \(h\tilde g = f\) and \(\pi(\tilde g) = g\). In other words, there is a unique way to lift the map \(g\) to make the diagram commute in \(\mathsf{C}\).

In understanding stacks, I’m trying to follow Lerman’s paper and also Fantechi’s survey paper, so there are two main examples: Lerman gives the example of the category \(\mathsf{B}\mathcal{G}\) of principal \(\mathcal{G}\)-bundles with arrows \(\mathcal{G}\)-equivariant maps. Fantechi gives the example of vector bundles of rank \(r\) and bundle isomorphisms. I’ll give Fantechi’s example.

Example: Vector bundles of constant rank

Recall that a vector bundle of rank \(r\) over a topological space \(X\) is a space \(E\) equipped with a continuous surjection \(\pi\colon E \to X\) such that for each point \(x \in X\), the space \(\pi^{-1}(x)\) is a real vector space of dimension \(r\) and such that for each \(x \in X\), there exists an open neighborhood \(U\) of \(x\) and a homeomorphism \(\varphi\cong U\times \mathbb{R}^r \to \pi^{-1}(U)\) with the property that \(\pi\varphi(y,\vec v) = y\) and the property that the map \(\vec v \mapsto \varphi(y,\vec v)\) is a linear isomorphism from \(\mathbb{R}^r\) to \(\pi^{-1}(y)\) for all \(y \in U\).

There is a category \(\mathsf{Vect}_r\) of rank-\(r\) vector bundles where the arrows are vector bundle maps of rank \(r\): commutative diagrams of the form

\[\begin{CD} E_1 @>\tilde f>> E_2 \\ @VV\pi_1V @VV\pi_2V \\ X_1 @>f>> X_2 \end{CD}\]

with the property that each map \(\pi_1^{-1}(x) \to \pi_2^{-1}(f(x))\) is a linear isomorphism of vector spaces. We will abuse notation, writing \(\tilde f\colon E_1 \to E_2\) to mean the bundle map above.

The claim is that the functor (also called \(\pi\)) sending a rank-\(r\) vector bundle \(\pi\colon E \to X\) to its base \(X\) defines a category fibered in groupoids over \(\mathbf{Top}\).

To see this, suppose that \(\pi\colon E \to X\) is a vector bundle and \(f\colon Y \to X\) is a continuous map. We need a pullback \(\tilde f \colon \xi \to E\) for some vector bundle \(\xi\) over \(Y\). Let us take the literal pullback, that is,

\[\xi = \{ (y,e) \in Y \times E : f(y) = \pi(e) \}\]

There is an obvious projection \(\pi\colon \xi \to Y\) given by projection to the first factor. Given \(x \in X\), let \(U\) be the neighborhood of \(x\) with the homeomorphism \(\varphi \colon U \times \mathbb{R}^r \to \pi^{-1}(U)\). Write \(V = f^{-1}(U)\) and define \(\psi\colon V \times \mathbb{R}^r \to \pi^{-1}(V)\) by the rule

\[\psi(y,\vec v) = (y,(\varphi(f(y),\vec v)).\]

It is clear that \(\pi\psi(y,\vec v) = y\) for all \(y \in V\). Conversely, given \((y,e) \in \pi^{-1}(V)\), note that \(\varphi^{-1}(e) = (f(y),\vec v)\) for some \(\vec v \in \mathbb{R}^r\); put another way, the map \((y,e) \mapsto (y,\vec v)\), where \(\vec v\) is the \(\mathbb{R}^r\) component of \(\varphi^{-1}(e)\), is an inverse homeomorphism for \(\psi\). The map \(\vec v \mapsto \psi(y,\vec v)\) is a linear isomorphism because \(\vec v \mapsto \varphi(f(y),\vec v)\) is a linear isomorphism. Therefore \(\xi\) is a rank-\(r\) vector bundle over \(Y\).

There is a map \(\tilde f \colon \xi \to E\) given by projection to the second factor. The map \(\pi^{-1}(y) \to \pi^{-1}(f(y))\) given by \((y, e) \mapsto e\) is a linear isomorphism. To see this, note that we have the following commutative diagram

\[\begin{CD} \pi^{-1}(y) @>f>> \pi^{-1}(f(y)) \\ @A\psi AA @A\varphi AA \\ \mathbb{R}^r @= \mathbb{R}^r \end{CD} \qquad \begin{CD} \psi(y,\vec v) @>>> \varphi(f(y),\vec v) \\ @AAA @AAA \\ \vec v @= \vec v \end{CD}\]

and the vertical maps are isomorphisms. Therefore the map \(\tilde f\colon \xi \to E\) is an arrow of \(\mathsf{Vect}_r\).

Given maps of vector bundles \(\tilde f\colon \xi'' \to \xi\) and \(\tilde h\colon \xi' \to \xi\) such that there exists a continuous map \(g \colon \pi(\xi'') \to \pi(x')\) making the following diagram commute

\[\begin{CD} \pi(\xi'') @>f>> \pi(\xi) \\ @VVgV @| \\ \pi(\xi') @>h>> \pi(\xi), \end{CD}\]

we claim that there is a unique bundle map \(\tilde g\colon \xi'' \to \xi'\) making the appropriate diagram commute. Given \(x \in \pi(\xi'')\), note that \(\tilde f\) and \(\tilde g\) induce linear isomorphisms \(\pi^{-1}(x) \to \pi^{-1}(f(x))\) and \(\pi^{-1}(g(x)) \to \pi^{-1}(hg(x)) = \pi^{-1}(f(x))\). Call these isomorphisms \(f_x\) and \(h_{g(x)}\). Therefore define \(\tilde g\) as

\(\tilde g(\tilde x) = h_{g(\pi(\tilde x))}^{-1}f_{\pi(\tilde x)}(\tilde x)\).

It is clear that \(\tilde g\) is a bundle map making the relevant diagram commute and that conversely \(\tilde g\) is the only definition we could have made. Therefore \(\mathsf{Vect}_r\) defines a category fibered in groupoids over \(\mathbf{Top}\).