Dear Sam I believe that you'll find `Frobenius objects in cartesian bicategories' by Bob Walters and me, #3 in volume 20 of TAC, interesting. It is precisely this closing of Z-configurations to give an X-configuration that translates the Frobenius condition in the cartesian bicategory of profunctors. It has been known for a long time, but I think unpublished until our paper appeared, that the Frobenius objects in profunctors are groupoids. The paper by Bob W and me shows that this admits considerable generalization. Best regards Richard Wood Hello. In a category with pullbacks, say that a binary relation X <- R -> Y is "z-closed" if it satisfies the following axiom (interpreted as usual): If x R y and x' R y and x' R y' then x R y'. (The "z" in "z-closed" refers to the pattern of variables in the premise.) Z-closedness seems to be a sensible generalization of "equivalence" to relations between two different objects. (e.g. In computer science, it is common to relate the state spaces of two different systems.) Note that an endorelation is an equivalence relation if and only if it is z-closed and reflexive. Also note that, in an abelian category, every relation is z-closed. The [z-closed v. equivalence] connection seems to extend to [pullbacks v. kernel pairs]. Every span that arises from a pullback is a z-closed relation. Say that a category is "z-effective" if every z-closed relation arises as a pullback. - every abelian category is straightforwardly z-effective; - in a topos, every z-closed relation arises as a pullback span. Indeed, an extensive regular category has effective equivalence relations if and only if it is z-effective. These notions and ideas seem quite elementary, even fundamental, and I would be surprised if no-one had thought of them before. I borrowed the terminology "z-closed" from a paper by Erik de Vink and Jan Rutten (Theoret Comput Sci, 221:271-293, 1999) but I couldn't find any other references. Have I missed something? I'd be grateful for any observations or suggestions. Sam PS. I'd like to take the opportunity to acknowledge the helpful replies (public and private) to my question about W-types, a few months ago.