I was expecting Peter Lumsdaine to reply to this, but perhaps he's away. In discussions with Steve Awodey and myself, Peter recently established the fact that every co-category in a pretopos is a co-equivalence relation; more specifically, the "co-domain" and "co-codomain" maps (sorry, but I can't see any other way to describe them) are the cokernel pair of a (unique) monomorphism (namely, their equalizer). Peter Johnstone On Tue, 12 Aug 2008, Toby Bartels wrote:
I've been thinking idly about a concept dual to categories in much the same way that co-algebras are dual to algebras, and I've decided that I'd like to more about it. To be precise, if V is a monoidal category, then a category enriched over V has maps [A,B] (x) [B,C] -> [A,C], while a cocategory enriched over V has maps [A,C] -> [A,B] (x) [B,C]. (You can fill in the rest of the definition for yourself.)
Searching Google, this concept appears to be known (under this name) in the case where V is Abelian, but I'm not so interested in that. I'm more interested in the case where V is a pretopos (like Set) equipped with the coproduct (disjoint union) as the monoidal structure (x). My motivation is that this concept is important in constructive analysis when V is a Heyting algebra equipped with disjunction as (x). (This defines a V-valued apartness relation on the set of objects; but I'm stating even this fact in more generality than I've ever seen.)
So if anyone has heard of this concept where V is not assumed abelian, or even knows of a good introduction where V is assumed abelian, then I would be interested in references.
--Toby