Re: pushouts in REL

[Michael Barr <barr@math.mcgill.ca>]

On Thu, 20 Aug 2009, soloviev@irit.fr wrote:

> To the list:
>
> I am actually travelling (in Russia) and I need more or
> less  urgently a reference concerning pullbacks and pushouts
> in the category of sets and relations - do they always exist
> etc - I would not adress it to the list if it would be
> not urgent and I would not have some difficulty with
> search from here -
>
> Best to all -
>
> Sergei Soloviev
>

You should check this, but it seems right.  Rel is self dual and each
object is too.  So limits are colimits (of the dual diagram).  Rel has
arbitrary sums and products--they are disjoint unions.  The empty set is
initial and terminal.  So to have pullbacks you need equalizers.  So let A
and B be sets and R,S \inc A x B.  Let A_0 be the subset of A consisting
of all a such that (a,b) \in R iff (a,b) \in S.  Then it seems to me that
the inclusion function of A_0 into A is the equalizer of R and S.


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