As Peter J. is saying, categories of relations have poor (co)limits. 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. oSee [GP] 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 On 24 Feb 2014, at 23:36, 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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