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What I mean by higher level before this gets closed is functional analysis, complex analysis and harmonic analysis?

I've read looked at the examples in most category theory books and it normally has little Analysis. Which, is strange as I've even seen lattice theory be used to motivate a whole book on category theory.

I was wondering is there a nice application of category theory to functional analysis?

It's weird as read that higher category theory is used in Quantum mechanics as it foundation, yet QM has heavy use of Hilbert spaces.

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    You might repost this question to MO as it may attract more replies there. Also there has been some mention of categories of Banach spaces, Banach bundles etc on discussions on the nForum and entries in the nLab, so have a look over there as well.2011-12-13

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This was cross-posted to MO, where it got changed slightly, and it received 13 answers. (just posting this so the question doesn't sit with 0 answers)