Re: Co-categories

Re: Co-categories

[Prof. Peter Johnstone]

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
>
>
>

Re: Co-categories

[Richard Garner]

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
>
>
>

Co-categories

[Toby Bartels]

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