Re: Co-categories

["Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

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
>
>
>