[Title changed, and wording of question tweaked, by YC, because the original title asked a question which seems different from the one people want to answer.]
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.
Sorry for cross posting to MSE. However, somebody there suggested I posted it here to get better response (well, more than two replies). I'm surprised that most books on category theory have very little mention of Analysis. Especially since Grothendieck originally studied functional analysis.