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In my limited understanding, one can always find a category that captures the data of a deformation problem. But for a given deformation problem with a topologized deformation space, is there any advantage of reinterpreting it in the language of category theory (stacks? fibered categories? descent theory? Grothendieck topologies?)?

I guess, I am wondering if the answer might be "Hey, if you have a topological space already, why do you need a deformation category?", or if there is a kind of formalism that does help to answer the deformation problem more wholly or more naturally than the approach using tools from general topology.

Has Teichm├╝ller theory been interpreted in the language of category theory, or is that a problem that is either impossible or just trivial and unnecessary?

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Yes, Teichmüller theory has a category-theoretic interpretation, just like any moduli problem. This was initiated by Grothendieck in 1961 or so in one of his Bourbaki seminar "Techniques of construction" papers. –  S. Carnahan Apr 25 '12 at 6:23
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