Re: pushouts in REL

[Vaughan Pratt <pratt@cs.stanford.edu>]

Peter Freyd gave a talk at a small conference circa 1990 (at CSLI
perhaps?) where he drew attention to the lack of (co)equalizers in Rel
as a warm-up (I forget how) to an eloquent tribute to the benefits of
toposes for intuitionistic logic.

Oddly there didn't seem to be anything about this in Cats and
Alligators.  However 1.429 therein makes the point that "if a category
has enough equalizers then all idempotents split."  So a witness to
missing equalizers is any nonsplitting idempotent, for example Dominic's
z below.

Vaughan Pratt

Dominic Hughes wrote:
> Rel doesn't have equalizers (pullbacks, pushouts, or co-equalizers).  It's
> rather surprising, and apparently not very well known.
>
> Counter-example.   Let A = { a, b }, write id for the identity A -> A, and
> define z : A -> A by adding an edge to id:
>
>     a --- a
>         /
>        /
>       /
>     b --- b
>
> Then id and z do not have an equalizer in Rel.


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