Ends of Surfaces

August 11, 2022

This is the first post in a series on Katie Mann and Kasra Rafi’s paper “Large-scale geometry of big mapping class groups.” The purpose of this post is to introduce the classification of (possibly) infinite-type surfaces by their genus and end spaces, and to introduce Mann and Rafi’s preorder on the end space. By a surface, I mean a 2-dimensional manifold (which I will usually assume to be orientable and without boundary) which is Hausdorff and second countable.

Groups as Metric Spaces

August 10, 2022

A week ago I attended Big Mapping Class Groups at BC, a workshop where participants, myself included, presented on papers about mapping class groups of infinite-type surfaces that we did not write. I’d like to spend a couple of posts talking about the paper I was assigned, Large scale geometry of big mapping class groups by Katie Mann and Kasra Rafi. This is post “zero” in that series; I want to describe the project of large scale geometry of groups.

The Grothendieck Construction

July 6, 2022

I think that right after you learn the Yoneda lemma, the Grothendieck construction might be one of the most surprisingly useful techniques in basic category theory. The purpose of this post is to introduce the Grothendieck construction and illustrate a situation where it came up for me. Let $\mathsf{C}$ be a category and $F\colon \mathsf{C} \to \mathsf{Cat}$ a functor to the category of small categories. The Grothendieck construction takes this situation and produces a new category $\mathsf{D}$ equipped with a functor $\pi\colon \mathsf{D} \to \mathsf{C}$ .

Adjoints

June 1, 2022

I realized I haven’t, to my knowledge, talked about adjoint functors yet! Let’s do that now, and introduce compactly generated topological spaces in the process. Adjoint functors A pair of functors $F\colon \mathsf{C} \to \mathsf{D}$ and $G\colon \mathsf{D} \to \mathsf{C}$ are an adjoint pair, or more accurately, $F$ is left adjoint to $G$ and $G$ is right adjoint to $F$ , written $F \dashv G$ , if there is an isomorphism

Pursued by Stacks 7: Appendix

May 19, 2022

This post is, strictly speaking, a follow up to part three of my series on stacks, on the 2-Yoneda lemma. The classical Yoneda lemma has built into it a naturality statement, while our 2-Yoneda lemma proved essentially only the fact of the isomorphism. Let’s fix that. Theorem. (The 2-Yoneda Lemma) The functor $\operatorname{よ}\colon \operatorname{Hom}(\underline{X},\mathsf{D}) \to \mathsf{D}(X)$ defined on objects as $(F\colon \underline{X} \to \mathsf{D}) \mapsto F(1_X)$ and on arrows as $(\alpha\colon F \Rightarrow G) \mapsto \alpha(1_X)$ is an equivalence of categories which is pseudonatural in both arguments.

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 $\pi(\xi) = X$ .