Re: Limits in REL

[Marco Grandis <grandis@dima.unige.it>]

There was a similar question "Limits and colimits in Rel?" by  UweWolter, on 24 Feb 2014, which had various replies.
I copy my own (with an addition in [[...]]).
Regards   MG

----COPY----

As Peter J. is saying, categories of relations have poor (co)limits. [[ Eg no equalisers nor coequalisers.]]
For abelian groups, Rel(Ab) does not even have products (sums).

However, if you insert the 2-category Rel  into the double category RRel of sets, mappings and relations [GP1]
you have a double category with all double limits and colimits.
For instance: the obvious cartesian product  a x b: XxY --> X' x Y'  (resp. sum  a + b: X+Y --> X' + Y')  of two relations a, b
is indeed a product (resp. a sum) in the double category.
See [GP1] for definitions and discussion of these aspects.

Similarly, many bicategories of spans, cospans, relations, profunctors... have poor (co)limits, but can be usefully embedded in
weak double categories (with the same objects, "strict morphisms", "same morphisms", suitable double cells) that have all limits
and colimits.

Also adjoints work well in the extended settings: see [GP2].

Best regards

Marco

[GP1] M. Grandis - R. Paré, Limits in double categories, Cah. Topol. Géom. Différ. Catég. 40 (1999), 162-220.
[GP2] M. Grandis - R. Paré, Adjoint for double categories,  Cah. Topol. Géom. Différ. Catég. 45 (2004), 193-240.
   both downloadable at:   http://ehres.pagesperso-orange.fr/Cahiers/Ctgdc.htm



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