This post is about two very different areas of mathematics that I’m pretty interested in (although I know a lot more about one of them…) and the relationships between them. It’s well-known that category theory grew out of investigations in algebra and topology, and that its insights are today still most useful in those and closely related subjects. (Algebraic geometry, I’m looking at you.) In particular, the famous “slum of topology” epithet notwithstanding, combinatorics is pretty far-removed from both the above areas, and indeed, I don’t know of an introductory textbook in combinatorics that gives category theory a second (or first!) glance.

And why should it? Doubtless many have tried, but no one’s (as far as I know) managed to give combinatorics a Grothendieck- or MacLane-type makeover. On one level, this makes sense; combinatorics is a huge area that encompasses problems that look completely different. It can almost be described as the trash-heap of mathematics; combinatorics is the study of problems that don’t really fit in anywhere else. To abstract from such a huge generality would seem sort of… onanistic, “abstraction for abstraction’s sake.” But of course this doesn’t mean that category theory has no place in combinatorics!

In this post, I plan to discuss two (and a half) different applications of category theory to combinatorics. (more…)