Many thanks to all who replied to my message, in private and publicly. I notice that these concerns also arose on this mailing list back in 1992: http://www.mta.ca/~cat-dist/archive/1992/92-06.txt At that time, Michael Barr was asking, amongst other things, about the exactness property that "every Mal'cev [=difunctional=z-closed] relation is a pullback", which holds both in toposes and in abelian categories (as I mentioned below). I wonder if anything more came out of that. He mentioned a possible connection with "effective unions", but I haven't been able to get anything to work there. By the way, following the comments about Mal'cev operators, and Peter Freyd's "Mal'cev allegories", I note that (exact) categories in which every relation is difunctional have been called "Mal'cev categories" [see e.g. the book by Bourn and Borceux on the topic (pointed out by Peter Lumsdaine), or Carboni, Lambek, Pedicchio, Diagram chasing in Mal'cev categories, JPAA 69]. Sam On 29 May 2008, at 10:01, Sam Staton wrote:
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.