Infinity-categories

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....

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....