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I'm (vaguely) aware of certain uses of higher category theory in attempts to mathematically understand quantum field theories -- for example, Lurie's work on eTQFTs, the recent-ish book by Paugam, and a bunch of work by people like Urs Schreiber.
What I'm wondering is: what work has been done at understanding the role of renormalization in quantum field theory in these terms (specifically in terms of homotoptic geoemtry or something along those lines)? And since much of the work I've seen along these lines tends to focus on perturbative QFT (with good reason, of course), are there are good references which try to capture the non-perturbative aspects of QFT from this perspective?
The recent book by Costello on factorization algebras is nonperturbative in the sense that it delivers asymptotic series in $\hbar$ rather than in a coupling constant.
Without a small parameter, a coupling or $\hbar$ or the inverse of the number of sites of a discretization, only exactly solvable problems (which, in dimension 4 means free theories) can be treated nonperturbatively in a very strict sense, in the absence of a rigorous mathematical foundation.
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