Pursued by Stacks 1: CFGs

May 16, 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) ├ętale 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.

$$\begin{CD} \xi'' @>f>> \xi \\ @. @| \\ \xi' @>h>> \xi \end{CD}$$

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 $\mathbf{Top}$ .