# Tensor products as isomorphic functors in category theory

An earlier question that I posed sought to define a category with a set of quantum channels as arrows and the C$^{*}$-algebra that these channels map from and to as the object. So, for example, my channels might be the set of all channels with 2 x 2 Kraus operators mapping some C$^{*}$-algebra to itself. Since the channels are always $d \times d$ square matrices, I'll refer to them as having dimension $d$. So far the answers to that question seem to have indicated that my formulation is correct.

Now suppose we have one such category whose channels have dimension 2 that I'll call Chan(2). Suppose we also have a category whose channels have dimension 4 that I'll call Chan(4). A tensor product of the channels in Chan(2) with themselves produces a channel that is in Chan(4), i.e. it is a mapping from Chan(2) to Chan(4). The tensor product is known to be a functor so this isn't unexpected.

But here's my question: is there a way to define an isomorphism between Chan(2) and Chan(4)? In other words, can the action of the tensor product be "undone," i.e. if the tensor product is the functor going from Chan(2) to Chan(4), is there a functor going from Chan(4) to Chan(2) and, can the two together define an isomorphism? If I make a larger category out of all these little categories, it seems like I could make the tensor product and anything that undoes it, arrows in the larger category and then I'd have my isomorphism. Can I do this and, if so, how does one undo a tensor product?

• Are you familiar with multicategories, because they seem relevant to what you're looking for. Feb 1, 2010 at 3:11
• Also, about your question, the tensor product cannot be undone unless you tensor over Z, because this is the direct sum of commutative rings. By construction, the tensor product "cuts out" too much stuff to recover everything about the rings. Feb 1, 2010 at 3:14
• What exactly is the use of category theory adding to the problem here?
– j.c.
Feb 1, 2010 at 4:00
• Harry: I am not familiar with multicategories. Is there a reference you could point me to or are they found somewhere in Saunders and MacLane? jc: This is a smaller part of a larger problem I'm working on related to a quantum extension of Birkhoff's theorem. A quantum extension already exists for 2-d and reversible channels. By wrapping a channel up inside a category (while still trying to maintain the structure somehow), I hope to be able to exploit the category theoretic definition of isomorphism which will eventually allow me to represent n channels by a combination of unitaries. Feb 1, 2010 at 12:43
• Ian: there's an OK wikipedia article on multicategries and, um, a reference there that you can follow up. They first appeared in a paper of Lambek in the late 1960s. They're not in Mac Lane (and incidentally, Saunders was Mac Lane's first name -- they're not two separate people). Feb 2, 2010 at 21:54