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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.

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And I wonder whether this should be CW... –  David Roberts Oct 26 '10 at 4:28
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I think this is a too broad question. There are too many examples for this paradigm of translating classifications (lie groups -> lie algebras, complex manifolds -> complex varieties, field extensions -> groups, etc.), I don't know if such a big list will be an enrichment. –  Martin Brandenburg Oct 26 '10 at 9:02
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closed as too broad by Stefan Kohl, Andrey Rekalo, Chris Godsil, j.c., Carlo Beenakker Nov 6 '13 at 13:06

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.If this question can be reworded to fit the rules in the help center, please edit the question.

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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...

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