## Nonexamples of Coarsely Bounded Generation

### August 29, 2022

This is the sixth 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 discuss when a Polish group is generated by a coarsely bounded set, and give examples of mapping class groups which are locally coarsely bounded but fail this criterion. The general case We have the following theorem of Rosendal. Theorem 1.2 (Rosendal). Let $G$ be a Polish group.

## Tameness and More on Ends

### August 27, 2022

This is the fifth post in a series on Katie Mann and Kasra Rafi’s paper Large-scale geometry of big mapping class groups. At the beginning, I mentioned that Mann and Rafi provide a classification of which $\Sigma$ have $\operatorname{Map}(\Sigma)$ generated by a coarsely bounded set under a hypothesis. The purpose of this post is to explore that hypothesis, which is called tameness. Additionally, I want to report on recent progress by Mann and Rafi in their note Two results on end spaces of infinite-type surfaces.

## Locally Coarsely Bounded Mapping Class Groups

### August 17, 2022

This is the fourth post in a series on Katie Mann and Kasra Rafi’s paper Large-scale geometry of big mapping class groups. Truth be told, I’ve made many abortive attempts at writing this post. I want to describe the classification of locally coarsely bounded mapping class groups in a way that is useful and not too technical. I’m going to try and talk through two examples that Mann–Rafi give at the beginning of the paper, and then state the classification theorem at the end of the section.

## Self-similar End Spaces

### August 14, 2022

This is the third post in a series on Katie Mann and Kasra Rafi’s paper Large-scale geometry of big mapping class groups. In the previous post we explored an obstruction to coarse boundedness, while in this one we will introduce a sufficient condition for $\operatorname{Map}(\Sigma)$ to be coarsely bounded. This post includes proofs that closely follow those in the paper, which while elementary, get a little involved. I imagine this will be one of very few moments in the series where I give full proofs.

## Nondisplaceable Subsurfaces

### August 12, 2022

This is the second 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 main obstruction to coarse boundedness of subsets of a big mapping class group: nondisplaceable finite-type subsurfaces. I also define the mapping class group in this post. The mapping class group Mapping class groups of finite-type surfaces are central objects in modern mathematics, bridging fields as disparate as algebraic geometry and low-dimensional topology.

## 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 $\pi\colon \mathsf{D} \to \mathsf{C}$ .

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
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.