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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