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

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