Tensor products as isomorphic functors in category theory - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T22:07:59Z http://mathoverflow.net/feeds/question/13617 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/13617/tensor-products-as-isomorphic-functors-in-category-theory Tensor products as isomorphic functors in category theory Ian Durham 2010-02-01T03:05:19Z 2010-02-01T03:05:19Z <p>An <a href="http://mathoverflow.net/questions/13581/quantum-channels-as-categories-question-1" rel="nofollow">earlier question that I posed</a> sought to define a category with a set of quantum channels as arrows and the C<code>$^{*}$</code>-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<code>$^{*}$</code>-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.</p> <p>Now suppose we have one such category whose channels have dimension 2 that I'll call <strong>Chan</strong>(2). Suppose we also have a category whose channels have dimension 4 that I'll call <strong>Chan</strong>(4). A tensor product of the channels in <strong>Chan</strong>(2) with themselves produces a channel that is in <strong>Chan</strong>(4), i.e. it is a mapping from <strong>Chan</strong>(2) to <strong>Chan</strong>(4). The tensor product is known to be a functor so this isn't unexpected.</p> <p>But here's my question: is there a way to define an isomorphism between <strong>Chan</strong>(2) and <strong>Chan</strong>(4)? In other words, can the action of the tensor product be "undone," i.e. if the tensor product is the functor going from <strong>Chan</strong>(2) to <strong>Chan</strong>(4), is there a functor going from <strong>Chan</strong>(4) to <strong>Chan</strong>(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?</p>