Classification Problems - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T11:47:54Z http://mathoverflow.net/feeds/question/43623 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43623/classification-problems Classification Problems Manuel Rivera 2010-10-26T03:06:10Z 2010-11-09T05:22:14Z <p>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 <a href="http://mathoverflow.net/questions/3776/when-are-two-proofs-of-the-same-theorem-really-different-proofs" rel="nofollow">this</a> interesting post which contained some similar ideas.</p> <p>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.</p> http://mathoverflow.net/questions/43623/classification-problems/43630#43630 Answer by David Roberts for Classification Problems David Roberts 2010-10-26T04:27:52Z 2010-10-26T04:27:52Z <p>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...</p>