Dear Jason, I’m travelling at the moment, and can’t look up details, but similar things have been done in the past, in particular by Bob Walters. If you start with a monoidal category, then you can define a bicategory with the same objects, and in which a morphism from A to B consists of an object X and a morphism A—> X @ B, where @ denotes the tensor product. There is a dual version in which the original category in which morphisms have the form A@X->B. Your version is a combination of both of these. (Once again, you get a bicategory rather than a category, unless for some reason + is strictly associative.) Another closely related notion is that of Elgot automaton, studied by Walters and various collaborators. Regards, Steve Lack. On 25 Sep 2014, at 4:49 am, Jason Erbele <erbele@math.ucr.edu> wrote:
Ah, it figures I would leave something out in my first post here. My apologies to all of you that were scratching your heads over the missing rules for composition. At least five of you have responded with essentially the same question, so this will be sort of a blanket response.
Composition of (f,g,h): A --> B through X and (f',g',h'): B --> C through Y will go through the biproduct X (+) Y with (f'f, [f'g, g'], [h, h'(f+gh)]^T): A --> C. That is, the first component composes in the usual way, the second component is a row matrix, and the third component is a column matrix. In the original category, the various morphisms in the composed triple correspond to A --> B --> C, directly; X --> B --> C and Y --> C; and A --> X and A --> B --> Y, where this A --> B is f+gh rather than f.
Identity morphisms in the new category are those with f=id and the zero object for X, which uniquely determines g and h. The embedding I mentioned of the original category into the new category is a functor, after all.
Best, Jason
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]