Happy New Year! Peter wrote: Consider the following two categories:
(a) the category of finite dimensional complex vector spaces and linear maps, and (b) the category of finite dimensional Hilbert spaces and linear maps.
Specifically, take Toby's proposal, and consider two different objects A,B of (b) such that both A and B are two-dimensional Hilbert spaces. Let u:A->B be some non-unitary isomorphism.
Using "u" to stand for a non-unitary morphism! Reminds me of the joke: Teacher: Suppose p is a prime number... Student: But what if it's not? Teacher: Well then it wouldn't be called "p", now, would it!
Then you can easily find an equivalence of categories which identifies both A and B with the two-dimensional vector space C^2, and which identifies u with the identity morphism on C^2. At this point, you have not equipped the category (a) with anything useful, because it does not induce a notion of unitary map on C^2.
Okay, that's a nice argument. I'm pretty sure Lurie gave me some similar argument: take a dagger-category, try to transport the structure along an equivalence of categories, and get something unacceptable. So it seems that, to define the extra structure of Hilbert spaces (on top of
vector spaces), one needs at least one "evil" concept, be it that of unitary maps or the dagger structure.
If this is really true (and I think it is), we're pushed towards Mark Weber's idea: dagger-categories are not best thought of as categories but rather something new, based on graphs-with-involution instead of graphs. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]