After giving the matter some thought, I finally decided that the category of Hilbert spaces should have as its morphisms norm-reducing linear maps. At the very least that will ensure that an isomorphism is an isometry. And yes the category of finite dimensional Hilbert spaces is *-autonomous. The internal hom of two is the space of all linear maps and the inner product <f,g> = \sum f(u_i)g(u_i), taken over an orthonormal basis of the domain. This can be shown to be invariant under orthogonal change of basis. The norm of the identity on an n-dimensional space is sqrt(n). The dual of a space is itself, of course with the duality being adjunction (or transpose). Then the tensor product H # G = (H --o G^*)^*. Here is another approach to the same structure. Consider a pair (V,\phi) where V is a finite dimensional space and \phi is an isomorphism of V with its dual space. You have to add positive definiteness and symmetry, but that is no problem. Maps again are norm reducing. Now we can define (V,\phi) # (W,\psi) = (V # W,\phi # \psi), (V,\phi)^* = (V,\phi^{-1}), and (V,\phi) --o (W,\psi) = (V # W,\phi^{-1} # \psi). The resultant category is exactly the same as before. BTW, it is easy to see that the transpose of a norm-reducing map is norm reducing. On Sun, 12 Oct 2003, John Baez wrote:
Michael Barr wrote:
I will let others answer about the connection between closed monoidal categories and MLL, but I just wanted to say that I am not sure what you mean by the category of Hilbert spaces. If you want the inner product preserved, then only isometric injections are permitted. If you want just bounded linear maps then you are not making any real use of the inner product.
Right. I wanted to leave things flexible so different readers could interpret my question in different ways, but I also tried to hint that I think it's crucial to work with the *-category Hilb whose objects are Hilbert spaces, whose morphisms are bounded linear maps, and whose *-structure sends the bounded linear map f: H -> H' to its Hilbert space adjoint f*: H' -> H. This *-structure can be used to define concepts crucial for quantum mechanics, like "self-adjoint" and "unitary" operators, as well as "isometric injections". Isometric injections are a nice way to study subobjects in Hilb, but they're not good enough for doing full-fledged quantum mechanics, nor is ignoring the inner product altogether.
Category theorists are often a bit uncomfortable with *-categories because they prefer "adjoints" that are defined using other structure rather than put in by brute force. However, I'm convinced that we can only understand how quantum field theory exploits the analogy between differential topology and Hilbert space theory if we think about *-categories. For example, a topological quantum field theory is a symmetric monoidal functor from some *-category of cobordisms to the *-category Hilb - but the most physically realistic TQFTs are the "unitary" ones, which preserve the *-structure.
I've talked about this *-stuff and the nascent concept of "n-categories with duals" in my papers on 2-Hilbert spaces
http://math.ucr.edu/home/baez/2hilb.ps
and 2-tangles
http://math.ucr.edu/home/baez/hda4.ps
and now I want to say a bit about how it impinges on quantum logic - but to avoid reinventing the wheel, I'd like to hear anything vaguely relevant anyone knows about approaching quantum logic with an eye on category theory.
(I know a bit about quantales, but maybe there's other stuff I've never heard of.)