Contractibliity as Uniqueness

Speaker: Emily Riehl
Date of Talk: May 17, 2022
Upstream link: UCLA Distinguished Lecture Series

A lot of the details of this talk went over my head, as this was the first talk I had ever attended. Moreover, this was before I had learned any category theory whatsoever, so some of the language was hard to get through. It took me a while to figure out what the heck the talk was really about (at no fault of the speaker!), so perhaps take these notes with a grain of salt.

1. Important Definition: Pushforwards and Direct Images

Definition 1.

Let \(f:X\to Y\) be a function between sets. The pushforward of a subset \(A\) of \(X\) is the set of elements \(y\in Y\) such that \(f(x)=y\) implies \(x\in A\). In other words, \(f ^{-1}(y)\subseteq A\).

In contrast, the usual direct image is just \[f(A)=\lbrace f(x) : x\in A\rbrace.\] The reason the pushforward (sometimes also called the pushout) is used is that it preserves “important properties”, and I think the most natural example is in topology. Continuous maps between topological spaces preserve open sets under the inverse image, but not necessarily under the direct image! The pushforward of an open set, however, is still open.

2. A Question

One of the motivations of this talk was a connection or correspondence between the topological notion of contractibility and the set-theoretic notion of uniqueness, as well as something about connectedness (the details went over my head). It seems that things like holes will “get in the way of this”; do topological notions like genus have an interpretation as well?

3. A Comment about the Presentation

Something that stood out to me about the presentation was that a lot of care was put into colour coding the slides. Beyond just math, I’ve heard many times that using slides in a presentation can easily lead to an overwhelming amount of content sitting in front of the audience, and I’ve been on the receiving end of that in quite a few classes and talks.

Even though this talk had a lot of information on the slides, it felt far more manageable because of the colour coding and general formatting: detailed remarks or insights were grayed out, and recurring constructs or themes were identified with the same colour. This was especially helpful for wading through the dense notation, and though I couldn’t understand it very well, it was at least tractible.

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