An addition to Steve's comment: Going even further back in history these ideas were used in Richard Wood's thesis: there he started with a bicategory and created a new bicategory by setting the new 1-cells (f,X): A --> B := f: A -> X @ B as Steve suggested. Composition then uses the associativity isomorphism ... and the resulting 1-cell composition is certainly bicategorical. One can, of course, also do the I-cell dual construction and, indeed your construction seems to amalgamate these two constructions. One can also always extract a category from a bicategory by identifying isomorphic 1-cells ... -robin (Robin Cockett) On Thu, Sep 25, 2014 at 1:52 PM, Steve Lack <steve.lack@mq.edu.au> wrote:
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.
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