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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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depends on how you want to study qm- although this probably does injustice to classify work this way, there seems to be two ways to do qm rigorously: constructive quantum field theory and algebraic quantum field theory. I've only heard about category theory in the context of AQFT. I recommend looking at a book called "Deep Beauty" edited by Hans Halvorson. –  r.g. Dec 13 '11 at 1:05
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@simplicity: As you know, Grothendieck started his professional life in functional analysis... –  Zhen Lin Dec 13 '11 at 2:51
    
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. –  Tim Porter Dec 13 '11 at 13:06

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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)

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