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/ ]