Rel does have products and coproducts; they coincide (by self-duality) and are just disjoint unions of sets. If's not hard to see that a relation R \subseteq A \times B is a monomorphism A \to B iff the map PA \to PB sending a subset of A to the set of all R-relatives of its members is injective; dually for epimorphisms. Rel has very few (co)limits other than (co)products; it doesn't even have splittings of all idempotents. (All symmetric idempotents have splittings, but the order-relation \leq \subseteq {0,1} \times {0,1} can't be split.) However, I don't think that the self-duality is in any sense responsible for the lack of (co)limits in Rel. The category of complete join-semilattices is self-dual, and is complete and cocomplete. Peter Johnstone On Mon, 24 Feb 2014, 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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