[Note from moderator: Apologies for forgetting how recently this was asked and extensivly answered. Some short responses are not being posted.] Ondrej Rypacek writes:
Hi all
What is known about limits in REL , the (bi)category of sets and relations? I know there are biproducts; are there equalisers?
Professor Johnstone answered this question a few months ago as below. Since REL is the category of *free algebras* for the powerset monad, there is very little chance of a limit of such algebras being free again. To get decent limits, you need to move to the Eilenberg-Moore category of the powerset monad, viz., the category of complete semilattices. 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. Cheers, Uday [For admin and other information see: http://www.mta.ca/~cat-dist/ ]