Re: Reference search: new categories by replacing morphisms with diagrams

[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 := 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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