Re: Limits and colimits in Rel?

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

Rel does have products and coproducts; they coincide (by self-duality)
and are just disjoint unions of sets. If's not hard to see that
a relation R \subseteq A \times B is a monomorphism A \to B iff the
map PA \to PB sending a subset of A to the set of all R-relatives
of its members is injective; dually for epimorphisms. Rel has very few
(co)limits other than (co)products; it doesn't even have splittings of
all idempotents. (All symmetric idempotents have splittings, but the
order-relation \leq \subseteq {0,1} \times {0,1} can't be split.)

However, I don't think that the self-duality is in any sense
responsible for the lack of (co)limits in Rel. The category of
complete join-semilattices is self-dual, and is complete and cocomplete.

Peter Johnstone

On Mon, 24 Feb 2014, Uwe.Wolter@ii.uib.no wrote:

> Dear all,
>
> I remember that there was some time ago a discussion on this list
> about limits and colimits in the category Rel of binary relations.
> Unfortunately, I can not remember or trace the final answer. But, if I
> remember right there are, besides initial and terminal objects, in
> general no limits or colimits in Rel.
>
> So my questions are:
>
> 1. Is there a characterization of monomorphisms and epimorphisms in Rel?
> 2. Is it true that there are, in general, no products and equalizer
> (sums and coequalizer) in Rel?
> 3. Are there some general results about what limits/colimits exist or
> don't exist?
> 4. Is the presumable non-existence related to the fact that the
> formation of converse relations establishes an isomorphism between Rel
> and its opposite Rel^op?
>
> Any reply or reference is well-come.
>
> Best regards
>
> Uwe Wolter

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