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 that sets up modern “homotopy theory” and “higher algebra.” The modern language for big chunks of algebraic topology, geometry, K-theory, … is decidedly ∞-categorical, so I should learn about ∞-categories.
The subject is very big and formal. Maybe a good way to learn it is to chip away slowly and consistently. 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 big 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 trying to not prune them. Maybe they will be useful for something more intentional and comprehensive in the future.