Category Theory
Category theory studies mathematical structure by how objects map to one another, not by what they contain. An object is opaque; everything you can say about it is said with arrows.
A category is objects, arrows between them, and a way to compose arrows that is associative and has identities. That is the whole definition — and almost every structure in mathematics is an instance of it.
Instead of peering inside an object, you characterize it by a universal property: the maps into or out of it that make it the best solution to a problem, unique up to a single isomorphism.
Functors carry one category into another, preserving composition and identities, so a whole theory can be transported and compared against another intact.
Natural transformations compare two functors arrow by arrow. When their components make every square commute, the comparison is canonical — independent of any arbitrary choice.
The core results build on these three levels. The Yoneda lemma says an object is fully known by the arrows into it; limits and colimits unify products, pullbacks, and quotients; and adjunctions pair functors that are optimal inverses of one another.
From adjunctions come monads and cartesian closed categories — the structures that carry the subject into computer science, modelling effects, the typed lambda calculus, and the semantics of programs.
The reward is leverage: prove something once about arrows and it holds in every category at once, from sets and groups to spaces, logics, and types.