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 Teichmüller theory been interpreted in the language of category theory, or is that a problem that is either impossible or just trivial and unnecessary?