Profunctors and spans of sets

Profunctors and spans of sets

[Mike Stay]

A span of sets X <-f- S -g-> Y can be viewed as a profunctor between
discrete categories by assigning to each pair (x,y) its preimage under
(f,g) o Delta.  Similarly, a map of spans of sets can be seen as a
natural transformation between profunctors, while a span of spans of
sets can be seen as a profunctor between the corresponding collages.

Where has this been discussed in the literature?

In the bicategory of categories, profunctors, and natural
transformations, a profunctor from C to D with a right adjoint is (up
to some details around Cauchy completion) just a functor from C to D.
Is there a nice characterization of profunctors with right adjoints
when the 2-cells are profunctors between collages?
-- 
Mike Stay - metaweta@gmail.com
http://www.cs.auckland.ac.nz/~mike
http://reperiendi.wordpress.com


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Re: Profunctors and spans of sets

[Aleks Kissinger]

I'm not sure about spans of spans, but the relationship between
profunctors and spans (or cospans) of categories was pointed out by
Benabou. I first saw this in a talk, which, fortuitously, Jeff Morton
was also at, and wrote a blog post all about it:
https://theoreticalatlas.wordpress.com/2011/05/18/benabou-spans-distributors/

a

On 25 August 2012 16:19, Mike Stay <metaweta@gmail.com> wrote:
> A span of sets X <-f- S -g-> Y can be viewed as a profunctor between
> discrete categories by assigning to each pair (x,y) its preimage under
> (f,g) o Delta.  Similarly, a map of spans of sets can be seen as a
> natural transformation between profunctors, while a span of spans of
> sets can be seen as a profunctor between the corresponding collages.
>
> Where has this been discussed in the literature?
>
> In the bicategory of categories, profunctors, and natural
> transformations, a profunctor from C to D with a right adjoint is (up
> to some details around Cauchy completion) just a functor from C to D.
> Is there a nice characterization of profunctors with right adjoints
> when the 2-cells are profunctors between collages?
> --
> Mike Stay - metaweta@gmail.com
> http://www.cs.auckland.ac.nz/~mike
> http://reperiendi.wordpress.com
>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]