# What's there to do in category theory?

I'm sure anyone who's heard of categories has also heard the classical "Well obviously there aren't any real theorems in category theory, it's much too general", or something in the likes of it.

Now obviously this argument is invalid (although its conclusion may be correct) because the same could be said of set theory, but there are clearly many really important theorems and results in set theory (I guess I don't have to justify that's it a huge field of research).

Now these theorems come from the fact that, when we do set theory, we don't just look at $\in$, we look at "derived stuff", like transitive sets, well-ordered sets, models of certain things, filters, etc. (I'm just giving a few examples to explain what I mean, I perfectly know that there's much much more to set theory than just those).

So the same thing should apply to category theory : of course we're not going to prove of we just stand there with our arrows and objects; you have to consider interesting ones, with more properties etc.

My question is about these (sorry for the lengthy intrduction). I know that a big part of category theory (although I don't really know in what proportion) is devoted to studying topoi(/ses ?) and for instance cartesian closed categories.

But I'm also guessing that there's much more than that to category theory; and my problem is that I don't know much about what is currently studied, what the major subfields of category theory are, or for that matter what subfields there are; so that when I want to refute the argument given at the very beginning I'm a bit stuck because I feel like I'm reducing category theory to topos theory and abelian categories.

Here's the actual question (after the too wordy introduction) : could you give some examples of subfields of research (if possible, currently, or previously very active fields) in category theory, paradigmatic questions or theorems in those subfields; how they're interesting in themselves and for some, how they can be interesting for other areas in maths ?

(I hope this is appropriate for MSE, otherwise please tell me so)

EDIT : I also posted the question on MO (https://mathoverflow.net/q/287334), the question has received some interesting answers !

• At the beginning of Set theory for category theory, Mike Shulman states Freyd's Special Adjoint Functor Theorem is "one of the first results [he] quote[s] when people ask [him] 'are there any real theorems in category theory?'" – Derek Elkins Nov 22 '17 at 22:29
• Another example of an extremely useful theorem: the Freyd-Mitchell embedding theorem which allows you to do element theoretic proofs even if your category is not concrete. – Jim Nov 22 '17 at 22:36
• Nowadays a lot of people are interested in not just category theory but higher category theory (involving morphisms between morphisms and so forth) and there's a lot to do here. I don't have a great one-stop-shop introduction here but you can try the very beginning of HTT: math.harvard.edu/~lurie/papers/highertopoi.pdf – Qiaochu Yuan Nov 22 '17 at 22:36
• @Jim I, personally, would include all of the theory of internal logics/internal languages (of which I consider Freyd-Mitchell embedding at least a close cousin) and related to those things like monadicity and its relations to universal algebra. I'm never really sure what makes a theorem "real" or when a result ceases to be "category theory" in questions like this though. – Derek Elkins Nov 22 '17 at 22:46
• @DerekElkins : I agree that the Freyd adjoint functor theorem is what I'd call a "real theorem", but it's only a theorem, not exactly a subfield of category theory. – Max Nov 23 '17 at 22:29