Dear Toby, What you call a cocategory enriched over V can also be described as a category enriched over V^op. These have been studied by Paddy McCrudden in his thesis under the name "coalgebroids" (= many-object coalgebras). The main results are on a generalised notion of Tannakian duality; and on transfer of extra structure across this duality. See respectively: [1] Paddy McCrudden, Categories of Representations of Coalgebroids, Advances in Mathematics Volume 154, Issue 2, Pages 299-332 [2] Paddy McCrudden, Balanced Coalgebroids Theory and Applications of Categories, Vol. 7, pp 71-147. Richard --On 12 August 2008 20:45 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