Disclaimer: I posted this question on MSE only a few days ago; and received very few comments. I know that the etiquette is to wait a bit more than that before moving a post from MSE to MO, but I figured that posting it on MO would be an actual improvement because there would be some actual researchers in category theory on this site, willing to give details about what it is they do, what's interesting about it, etc, whereas there may be less of them on MSE. If this isn't appropriate, I'll remove this post, and if it's the case I'm sorry for the disturbance.
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 (more than just giving a common language) ?