Tensor products as isomorphic functors in category theory

2010-02-01T03:05:19Z

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?

Tensor products as isomorphic functors in category theory - MathOverflow

Tensor products as isomorphic functors in category theory