I’m a mathematics student in abstract algebra and algebraic geometry. Most of my books cover a great deal of category theory and it is an essential tool in understanding these two subjects.
Recently, I started taking some functional analysis courses and I discovered that there is almost no category theory done in these courses. But since most of the spaces studied in functional analysis are objects in categories (e.g. the normed spaces form a category), I find it rather strange that the books leave the category theory out.
Is there a reason for this?
Let me ask you a dual question: I am a mathematics student in set theory, why don’t category theory students do set theory? I find it strange that most books on category theory have only a naive handling of set theory.
Now let me answer your question. Category theory is an impressive tool for abstraction, but analysis is not always in need for abstraction – it looks for concrete solutions and ideas. In that aspect categories are not too useful. In fact, if you try to insist on concreteness, categories can become a burden when you insist to carry the category around instead of just talking about functions and spaces.
On the other hand, abstract algebra is very fitting for category theory. We focus on “all” groups or “all” modules over a certain ring. Analysis focuses on particular spaces, continuous/differentiable/analytical functions over C, for example. The situation is similar with set theory, while a very useful tool for formalizing arguments in the better parts of mathematics – it is often neglected and cast aside as a non-issue.
The best tip to remember (as a set theory student) is that while a screwdriver is a very useful tool to carry, you don’t really need it if you are making a sandwich.