Post Made Community Wiki by S. Carnahan
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I was thinking about the famous question in philosophy mathematics of mathematics: "when 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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I was thinking about the famous question in philosophy mathematics of "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 "map" capture the logical structure of a proof into 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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