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 on understanding the role of renormalization in quantum field theory in these terms (specifically in terms of homotoptic geometry 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 any good references that try to capture the non-perturbative aspects of QFT from this perspective?

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    $\begingroup$ Well, the starting point for that would be the nLab page on renormalization, wouldn't it? There's a load of references to various formalizations of renormalization there. $\endgroup$ – ACuriousMind Oct 18 '16 at 13:37
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    $\begingroup$ I updated the question to be slightly more specific -- I'm wondering if there are any references using something along the lines of homotopical geometry to understand either of these aspects of QFT. I didn't see anything on the nLab page seemed to do so (granted, that probably means that there aren't any references, otherwise I would definitely expect them to be on there...). $\endgroup$ – OperaticDreamland Oct 18 '16 at 13:50
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    $\begingroup$ Dennis Sullivan has been asking for a homotopic approach to lattice QFT for years, but as far as I know, no one has even done the 1d case. $\endgroup$ – user1504 Oct 18 '16 at 14:50
  • $\begingroup$ That's unfortunate, but... I suppose that just means there is still something to work on. ;) $\endgroup$ – OperaticDreamland Oct 18 '16 at 14:51
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    $\begingroup$ If you want to try an easy case, there's an exact lattice representation of 2d Yang-Mills (due to Migdal, I think). $\endgroup$ – user1504 Oct 18 '16 at 15:27

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