It’s a very basic, cool, and important fact that spaces and categories have a common generalization in ∞-categories. This is the first step to realizing the deep relationship between space and algebra, especially 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.
So, I’m studying higher category theory with Lurie’s Higher Topos Theory as the main reference. That’s a huge book! To make matters worse, it is not really suitable as a standalone read. So I’ll be using lots of additional references.
As a personal project, I’m texing notes about my day-to-day studies for this. The motto is “slow and steady.” I’m not pruning these notes, they’re mainly for my own personal reference. They may serve well for something more intentional and comprehensive in the future.
Update: this spring, some grad student mutuals are arranging a similar project. I may join them for the time. They are following closely Charles Rezk’s notes (under his advision).