Pursued by Stacks 6: Stacks and Groupoids

May 19, 2022

In writing this post, I started to see what’s kind of cool about stacks: you can treat them almost as if they were spaces in the sense that, as you’ll see, objects of a stack $\mathsf{D}$ over a space $Y$ are the same thing as maps of stacks $Y \to \mathsf{D}$ , just like the situation of $\underline{X}$ for a space $X$ . The difference is that $\mathsf{D}$ may have many isomorphisms stacked over $Y \to \mathsf{D}$ while $\underline{X}$ has only the identity of $Y$ .

Pursued by Stacks 5: Topological Stacks

May 19, 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.

Pursued by Stacks 4: Fiber Products

May 18, 2022

Okay, we’re closing in on the definition of a geometric stack over $\mathbf{Top}$ . Actually, the nLab tells me such stacks are called topological, so I’ll try and say that. But! Before we get there, we need the construction of the 2-fiber product of categories fibered in groupoids over $\mathbf{Top}$ . The purpose of this post is to discuss this construction, which I think is adorable. I cannot shake the feeling that I’ve talked through the 2-fiber product of categories before, but I have no idea where or why I might have done this except possibly the last time I tried to learn about stacks.

Pursued by Stacks 3: We Need a Lemma

May 17, 2022

In the previous two posts we met stacks for the first time. We’d like to know what a geometric or Artin stack over $\mathbf{Top}$ is, since these are the stacks that correspond to, e.g. graphs of groups. To get there, I’m told, Yoneda lemma (sorry sorry). If you’re trying to follow who I’m following, we’re back to Lerman, for no particular reason other than I find it helpful to bounce.

Pursued by Stacks 2: Descent

May 16, 2022

A stack, so I’m told, is a category fibered in groupoids satisfying some extra conditions. The stacks we’re interested in are geometric stacks, so this won’t be the last post on stacks, but with any luck we’ll see the definition in this post. Let $\mathsf{C}$ be a category fibered in groupoids over $\mathbf{Top}$ . We say an object $\xi \in \mathsf{C}$ is over a space $X$ if $\pi(\xi) = X$ .

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) 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.

More Maps: Lerman's Critique

May 15, 2022

Lerman critiques the category of étale groupoids with arrows generalized maps by arguing that there is a problem with gluing maps. The purpose of this blog post is to try to digest this critique. We follow the notation set up in the previous two blog posts. Lerman’s critique takes place in the realm of bibundles, but as we saw in the previous post, it’s possible to translate from the bibundles point of view to the generalized maps point of view.

Maps of Étale Groupoids

May 15, 2022

The purpose of this post is to explore two or three notions of a map between étale groupoids. We follow the notation for étale groupoids established in the last post. Functors If $\mathcal{G}$ and $\mathcal{H}$ are étale groupoids, which are in particular categories, a natural class of maps of étale groupoids to consider would be the continuous functors. A functor $f\colon \mathcal{G} \to \mathcal{H}$ is a pair of maps $f_0\colon \mathcal{G}_0 \to \mathcal{H}_0$ and $f_1\colon \mathcal{G}_1 \to \mathcal{H}_1$ that commute with the various structure maps.

Graphs of Groups as Étale Groupoids

May 14, 2022

It’s been a while since I talked about étale groupoids! Time to rectify this situation. The purpose of this post is to explain how to see graphs of groups as étale groupoids. In a later post I’d like to explore the notion of maps between étale groupoids. Groupoids A groupoid is a small category $\mathcal{G}$ in which all arrows are invertible. Thus there are two sets, $\mathcal{G}_0$ and $\mathcal{G}_1$ of objects and arrows respectively, and several structure maps:

A Cut Vertex Lemma

May 10, 2022

Building on ideas from the previous post regarding the star graph, I wrote a short paper generalizing a classical lemma of Whitehead. The purpose of this post is to describe Whitehead’s lemma and my generalization. In the previous post, we met the star graph associated to a collection of conjugacy classes of elements of a virtually free group. It should be reasonably clear that one can extend the notion of a star graph to conjugacy classes of elements of a group that acts cocompactly on a tree.