What is category theory?
It’s an area of mathematics… and also an area of philosophy.
How is it mathematical?
It gives one framework to recognise patterns common to many areas of mathematics. It provides a “language” — or theory — with which to describe and manipulate these patterns.
It can feel like mathematics is made up of lots of different approaches. Geometry explores ideas of space and measurement. Algebra explores operations like addition and multiplication. Probability explores randomness and chance. Each approach has a set of rules guiding its behaviour. We can call these rules the structure of that approach.
A category is a structure of a particular shape.
That’s it.
Maybe this sounds very basic, or very restrictive. In fact, this basic shape turns up all the time!
Mathematicians have found ways of moving between these different approaches. (And they continue to find even more ways!) For example, there are recipes for turning geometric problems into algebraic ones. Category theory gives a way to express how these recipes take structure into account.
You could say category theory is a way to describe different kinds of structures and how they relate. (And how their relations relate. And how the relations of their relations relate. And so on.)
How is it philosophical?
How is it not?