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 opens for "homotopy theory" and "higher algebra." These are complicated subjects, but are also the decided language for a sizeable chunk of algebraic topology, geometry, K-theory, ...

In undergraduate, I started a journal to record my ∞-category learning throes. I have learned a lot about the subject since then, and a lot about learning. My notes are evolving accordingly. For self-accountability and reference, here are my notes (which I caution away from taking very seriously):