Hi all, ==Background== I have finally read Jean Benabou's Louvain lectures on distributors, which he was kind enough to send me earlier this year. In them he describes the following bicategories of distributors (some paraphrasing/simplification may occur, and modernisation of terms - all errors are mine): Dist objects: categories C,D,... arrows: functors C^op x D--> Set (equiv. opfibrations H --> C^op x D) 2-arrows: natural transformations (equiv. cartesian functors over C^op x D) Dist(V), for V a symmetric closed monoidal category objects: V-enriched categories C,D,... arrows: V-functors C^op x D--> V 2-arrows: V-natural transformations Dist(E), for E a regular category objects: categories internal to E C,D,... arrows: internal opfibrations H --> C^op x D 2-arrows: internal (cartesian) functors over C^op x D Dist(K), for K a bicategory objects: monads in K . . . . And here it seems to me the pattern breaks down, as taking as input the bicategories Cat, V-Cat and Cat(E), we do not arrive at any of the examples on the previous list. I understand the motivation behind Dist(K), namely that one considers the process E |--> Span(E) |--> Dist(Span(E)) = Dist(E) (E regular category) as Cat(E) = Monads(Span(E)). I'm not worried about that too much. ==Question== 1) Has anyone done any work on distributor-like constructions for bicategories that recover the processes Cat |--> Dist, V-Cat |--> Dist(V), Cat(E) |--> Dist(E)? Something like universally adding adjoints to all 1-arrows in a bicategory, I would imagine. I'm asking this in the context of the equivalence between representable distributors and anafunctors, so I suppose a secondary question is: 2) Given 1) above, is there a notion of 'representable 1-arrow' in this universal construction? Thanks, David ------------------------------ David Roberts david.roberts@adelaide.edu.au University of Adelaide [For admin and other information see: http://www.mta.ca/~cat-dist/ ]