converse relations and distributors

converse relations and distributors

[John Stell]

The notion of a "distributor between posets" is used as a motivating example in the document
Distributors at Work (www.mathematik.tu-darmstadt.de/~streicher/). This gives one way to
see what a relation between posets should be.

For posets A and B, what is considered are (ordinary) relations R \subseteq B \times A such that
b2 \leq b1 and a1 \leq a2 and  b1 R a1 implies b2 R a2.

What I'm interested in is the converse of such a relation between posets. Clearly we cannot simply
take the usual converse of R. I can see what to do for the context in which I need this, but I'd
be interested to know if this issue appears in the literature. Is there some well-known construction
on distributors that tells us what the converse 'should' be?

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Re: converse relations and distributors

[Steve Lack]

Dear John,

Let Dist be the category of posets and distributors (also known as
profunctors, modules, ...) between them. Dist is monoidal via the
categorical product of posets. This is not the product in Dist, but I will
still write AxB for the product of A and B.

Then the opposite poset of A (reverse the direction of inequalities) is
dual to A, in the sense of monoidal categories: there are distributors
1--/--> A*xA and AxA*--/-->1 satisfying the triangle equations for an
adjunction.

It follows that any distributor A--/-->B induces a distributor B*--/-->A*,
which one could call the converse (it is the converse as a relation).

All this works for categories rather than posets, but then Dist has to be
regarded as a monoidal bicategory rather than a monoidal category, and the
notion of dual has to be suitably adapted. More generally it holds for
enriched or internal categories, provided that the base for enrichment or
internalization has enough structure.

Regards,

Steve Lack.

On 26/03/10 12:38 AM, "John Stell" <J.G.Stell@leeds.ac.uk> wrote:

> The notion of a "distributor between posets" is used as a motivating example
> in the document
> Distributors at Work (www.mathematik.tu-darmstadt.de/~streicher/). This gives
> one way to
> see what a relation between posets should be.
>
> For posets A and B, what is considered are (ordinary) relations R \subseteq B
> \times A such that
> b2 \leq b1 and a1 \leq a2 and  b1 R a1 implies b2 R a2.
>
> What I'm interested in is the converse of such a relation between posets.
> Clearly we cannot simply
> take the usual converse of R. I can see what to do for the context in which I
> need this, but I'd
> be interested to know if this issue appears in the literature. Is there some
> well-known construction
> on distributors that tells us what the converse 'should' be?
>



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]