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What is category theory, Man?

At a certain level, mathematician study structures. In a very small nutshell, category theory allows you to talk about structures recurring throughout other disciplines of mathematics; it's a kind of unifying theory. Much like Algebra studies various arithmetic structures and distills their properties without having to worry about whether you're talking about integers, matrices, families of functions, etc. which in the specifics are quite different, so category wishes to distill properties across various disciplines of mathematics.

What is the problem, Sir?