Re: Limits in REL

["René Guitart" <rene.guitart@orange.fr>]

It is known that REL (also denoted as COR) has biproducts, even products which are also sums for small infinite indexation.
We can give examples that  some kernel, some cokernel, some image or some coimage do not exist
(see :  Davar-Panah, Thèse de 3ème cycle, Paris, 1968 ;   Guitart, Thèse de 3ème cycle, Paris, 1970).
For the lack of splitting idempotents :  
the completion of REL with respect to splitting idempotents is  the category of complete completely  distributive lattices, with sup compatible maps (R. Guitart and J. Riguet, Enveloppe karoubienne de catégories de Kleisli, CTGDC, XXXIII-3 (1992), p. 261-266). 
The case of {0,1} it is clear because a preorder is splittable if and only if it is an equivalence relation (Prop. 5 in Guitart-Riguet). It is   also  indicated exactly when an idempotents split in REL. 
In fact the construction in Guitart-Riguet works for any Kleisli category i.e. category of free algebras, and if the monad (T, u,m) on C is with T an injective map, then the completion of Kl(T) is the full subcategory of EM(T) with objects U_T-projective algebras.  So  idempotents split in Kl(T) if and only if every projective algebra is free.
This analysis works also for the description of the splitting of idempotent of the category of continuous relations between compact spaces: cf. my talk at the PSSL 51, Valenciennes, 13-14 février 1993 (An unpublished paper available on my page, in the section preprint).
Best regards,
René Guitart

Le 4 juil. 2014 à 09:45, Peter Johnstone a écrit :

> REL has very few limits other than biproducts: it doesn't even have
> splittings for all idempotents (so no equalizers or coequalizers).
> The simplest non-splittable idempotent is the usual order relation
> on {0,1}, and the same example works in REL(C) for any regular C
> where the disjoint coproduct 1+1 exists.
> 
> Peter Johnstone
> 
> On Thu, 3 Jul 2014, Ondrej Rypacek wrote:
> 
>> Hi all
>> 
>> What is known about limits in REL , the (bi)category of sets and relations?
>> I know there are biproducts; are there equalisers?
>> 
>> And what about SPAN(C) or REL(C), spans and relations over a suitable
>> category  C ?
>> 
>> Thanks a lot in advance,
>> Ondrej
> 



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