Co-categories

[Toby Bartels <toby+categories@ugcs.caltech.edu>]

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