General notions of equivalence and exactness

[Sam Staton <ss368@cam.ac.uk>]

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.