Category Theory
categories objects morphisms
category of sets sets functions
category of groups groups group homomorphisms
category of topological spaces topological spaces continuous maps
category of manifolds manifolds smooth functions

A category is itself a type of mathematical structure, so we can look for "processes" which preserve this structure in some sense; such a process is called a functor.

Diagram chasing is a visual method of arguing with abstract "arrows" joined in diagrams.

Want to read: wikipedia: glossary of category theory

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