On Mon, 24 Feb 2014 23:36:55 +0100, Uwe.Wolter@ii.uib.no asked, among other things:
2. Is it true that there are, in general, no products and equalizer (sums and coequalizer) in Rel?
I don't know about (co-)equalizers, but (co-)products, unless my Alzheimer's is very advanced, should be available ... if by category "Rel of binary relations" you mean with sets as objects, binary relations as morphisms, and usual composition of relations as composition rule (so usual identity functions as identity maps). Indeed, let A be a family (i /in I) of sets A_i. Write C for the usual set-theoretic disjoint union of the sets A_i -- C = /join_i A_i. a) For each i /in I, write p_i: C --> A_i for the partial function which "is" the identity function on the summand A_i. It seems to me that the relations p_i serve as projections making C the product of the various A_i of the family A. b) Again, for each i /in I, write s_i :A_i --> C for the function which "is" the identity function on A_i. It seems to me that the relations s_i serve as injections making C the coproduct of the various A_i of the family A. c) In fact, much as is the case with additive categories, the compositions of these injections and projections satisfy p_i s_k = /empty (i /ne k), p_i s_i = id_(A_i), s_i p_i = the partial function on C which is identity on summand A_i, and /join(s_i p_i) = id_C. Of course, Wolters' own observation, in 4 (below), serves as link between a) and b); and c) is, as it were, the observation that "coproduct and product both serve as biproduct":
4. ... the fact that the formation of converse relations establishes an isomorphism between Rel and its opposite ...
Indeed, p_i and s_i are mutually converse. Do let me know, please, if I've just been talking through my hat -- that'll be a sign it's time for me to retire for real :-) . Cheers, -- Fred
Any reply or reference is well-come.
Best regards
Uwe Wolter
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