June 19th, 2008 the combinatorial nature of space

If there is one area of mathematics I do not enjoy, it is combinatorics. Whenever people start saying things like “n choose k, but divide by two because you double counted,” it goes right over my head. I can’t quite put my finger on it, but there is something really hard about counting arguments for me.

I would have dismissed combinatorics outright as being petty and stupid (perhaps unfairly) if it had not been for my Topology seminar last semester. In studying simplicial homology, I learned that at the heart of space, there is something fundamentally combinatorial about it. You see, simplices (which are in essence n-dimensional generalizations of triangles) are defined and dealt with in a purely combinatorial fashion. And yet, this reveals something deep and fundamental about the nature of space.

The moral of the story is that combinatorics is very fundamental. It seems to crop up in nearly every branch of mathematics. And yet, it seems to be the only area that category theory has not swept under the rug as being a “trivial application” of category theory.

I wonder if there is a way to formalize and abstract counting arguments in a category-theoretic fashion? (I think Yoneda’s lemma is already hinting at this.)

P.S. Sorry about the exceptionally nerdy post. Not every post is like this, I promise.