diagrams From: Robin Cockett <robin@ucalgary.ca> To: Steve Lack <steve.lack@mq.edu.au> Cc: <categories@mta.ca> Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Sender: categories@mta.ca Precedence: bulk Reply-To: Robin Cockett <robin@ucalgary.ca> 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 :=3D 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=E2=80=99m travelling at the moment, and can=E2=80=99t 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=E2=80=94> X @ B, whe= re @ 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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