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.)