There seems to be some confusion relating terminology on bimodule-equivalent structures and their actual technical definitions, so perhaps the following overview of their relationships is relevant: 1) profunctor = distributeur = bivariant functor on categories into Set, F: A^op x B -> Set 2) bimodule = two-sided discrete fibration = fibrational span <d,c>: M -> A x B (discrete fibres + cartesian liftings for d, cocartesian liftings for c + interchange) (sadly, enriched categorists, lacking generally the span notion in the V-cat setting, use (bi)modules to mean actually bivariant presheaves as in 1) 3) cospans [l,r]: A + B -> G satisfying a) l and r full and faithful b) [l,r] bijective on objects c) There are no morphisms in G from r(B) to l(A) A cospan satfisfying a-c is what might conceivably be called a `bipartite category´ (I ignore Prof. Pratt´s incarnation of this concept, but this is what it ought to be if it actually is the `same as a bimodule´). I haven´t come across this notion before, so I don´t know of any established terminology. The equivalence between 1 and 2 is a two-sided version of the Grothendieck correspondence, which as far as I know, was first made explicit in Street´s "Cosmoi of Internal Categories". The paper exhibits the passage 1 -> 2 as a weighted colimit. The converse requires, as usual, a pro-choice stance. The equivlance between 2 and 3 results from the following adjunction between Spn(A,B) and CoSpn(A,B): Ker: CoSpn(A,B) -> Spn(A,B) sends a cospan [l,r]: A + B -> G to the comma-category l/r with its two canonical projections to A and B (this is is a bimodule) Coll: Spn(A,B) -> CoSpn(A,B) sends a span <d,c>: M -> A x B to its lax colimit: it consists of functors l:A -> G and r:B -> G and a 2-cell \alpha: ld => rc, universal among such (with the evident 2-dimensional property). When <d,c> is a bimodule, G is what (I think) Bob Walters calls the collage of the bimodule. It´s rather evident how to describe G by generators and relations (as 3) above makes clear). Equally easily, Coll is left adjoint to Ker. Bimodules are recovered from their collages, M \iso l/r, and a cospan is a collage of its kernel precisely when it is a bipartite category. Incidentally, the correspondence of bimodules and bipartite categories is analogoous to the fairly useful one between bimodules bearing a monad structure and the bijective-on-object functors, obtained via the Kleisli construction on the bimodule monad (which subsumes the ordinary Kleisli construction on a monad in Cat as a special case). Claudio Hermida 19-Oct-2004 17:24:01 -0300,4986;000000000000-00000000