Higher category theory

This semester I am taking part in a seminar on $\infty$-categories, administered by Rune Haugseng. So far we have covered roughly: the basic definitions, fibrations, limits, colimits, Joyal’s lifting theorem, equivalences, straightening, Yoneda lemma, adjunctions and Kan extensions. This week it is my turn to give a talk on stable $\infty$-categories, and this blog post will hopefully be some sort of lecture notes for this talk. The intersection of things in this post and the contents of the talk should at least be non-empty....

How can we use travel-journaling to study some highly abstract and highly complicated mathematical machinery? Can we get anything out of such an analogy? In this post we do just that, study the homotopy hypothesis and infinity groupoids through the lens of a travel ledger.

A couple weeks ago I held a talk on introductory higher category theory. Most of the talk was based upon thing we already have discussed on this blog, such as the strict $2$-category $Cat$, bicategories, and why strictness fails for the category of topological spaces. The inly thing I talked about which I haven’t yet featured on this blog is the notion of quasi-categories, so I though that I would do that today....

You may be thinking, what the heck is a monoid, and why the heck is it vertical? To explain this we will need some insight into classical categories and $2$-categories, which we luckily have been developing for the last few posts. First off, to let the familiar readers know, the objects of study today is called monads, not vertical monoids. But, I like to visualize them and think about them as somehow vertical, or at least something not strictly horizontal or one-dimensional....

Last fall I held a talk about functors, natural transformations and equivalences of categories. This talk was part two of five in a student seminar on introductory category theory. There was mostly second year students attending but also a couple more experienced students. To make the talk a bit interesting for them as well I said that an equivalence of categories is the correct notion of “sameness” of categories, and not isomorphisms due to the fact that categories naturally lie in a $2$-category....

A litmus test is a question asked in politics to a potential candidate for high office in which the answer determines if the person gets nominated or not. If a person or a committee holds the power of nominating candidates, they can use that power to make sure that a potential candidate holds their view on a certain matter. So, what does this have to do with mathematics, or especially with homotopy theory?...