Higher Category Theory

Higher category theory is the study of categories in which morphisms have morphisms between them, and those morphisms have morphisms between them, and so on, up to any finite or even infinite level. Where ordinary category theory studies objects and the arrows between them, higher category theory studies the arrows between arrows, the arrows between those arrows, and the higher-dimensional cells that encode equivalences between equivalences. The central insight is that equality is too coarse a relation for describing structural sameness at higher levels, and must be replaced by equivalence — a relation that remembers the data of how two things are the same, not merely that they are.

This is not an abstraction for its own sake. It is the recognition that when you ask whether two constructions are the