Question

I was thinking about the famous question in philosophy of mathematics: "When are two proofs the same?" and I was wondering if we could somehow "classify" proofs by establishing some sort of functorial relationship between proofs and other mathematical objects which we can classify (like for example, surfaces; my initial idea was to somehow capture the logical structure of a proof in a graph and then classify graphs by their topological structure). I searched MO and found this interesting post which contained some similar ideas.

However, I was wondering if we can come up with a list of examples of classification problems in mathematics which have been answered using category theoretic tools by functorially "translating" the original problem into a different category in which we can classify the corresponding objects... and everything works in a nice way. The natural place to start is obviously algebraic topology.

Answer 1

I suppose the first one on a number of people's minds is moduli spaces. More specifically, we can in Top form a moduli space (of curves, say), but not in the category of schemes. Thus stacks were born...