It’s a basic, cool, and important fact that spaces and categories have a common generalization in ∞-categories. This is a first step toward realizing an important relationship between space and algebra, and this buys you e.g. the application of higher category theory to algebraic topology, geometry, … And these applications have had serious successes, e.g. the proof of the geometric cobordism hypothesis. Nowadays, ∞-categories are the “right” way to talk about and understand concepts in many parts of modern mathematics. That includes the mathematics I am interested in.

Being such a big and formal and loaded area, maybe a good way to learn higher category theory is to chip away at it slowly and over a long time. As a personal project, I’m going to try this. Probably a serious goal is to read Jacob Lurie’s Higher Topos Theory, but that’s a hard (huge) read, and not all-encompassing. So I’ll be drawing on many references.

I’m texing notes for this. The motto is “slow and steady.” I’m writing these notes as I learn, and I’m not pruning them, they’re for my own reference. Maybe they will be useful for something more intentional and comprehensive in the future.

UPDATE: this spring, some grad students are arranging a reading course on the subject. I am joining them for some time. They are following closely Charles Rezk’s notes (under his advision).