From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8193 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Ren=E9_Guitart?= Newsgroups: gmane.science.mathematics.categories Subject: Re: Limits in REL Date: Sat, 5 Jul 2014 09:11:20 +0200 Message-ID: References: Reply-To: =?iso-8859-1?Q?Ren=E9_Guitart?= NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Apple Message framework v1085) Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1404591219 31372 80.91.229.3 (5 Jul 2014 20:13:39 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 5 Jul 2014 20:13:39 +0000 (UTC) Cc: Ondrej Rypacek , "categories@mta.ca" To: Peter Johnstone Original-X-From: majordomo@mlist.mta.ca Sat Jul 05 22:13:32 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1X3WKs-00039U-4W for gsmc-categories@m.gmane.org; Sat, 05 Jul 2014 22:13:26 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:49550) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1X3WJs-0000pi-Fm; Sat, 05 Jul 2014 17:12:24 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1X3WJL-0003lM-M0 for categories-list@mlist.mta.ca; Sat, 05 Jul 2014 17:11:56 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8193 Archived-At: 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, Th=E8se de 3=E8me cycle, Paris, 1968 ; Guitart, = Th=E8se de 3=E8me cycle, Paris, 1970). For the lack of splitting idempotents : =20 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 = cat=E9gories de Kleisli, CTGDC, XXXIII-3 (1992), p. 261-266).=20 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.=20 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 f=E9vrier 1993 = (An unpublished paper available on my page, in the section preprint). Best regards, Ren=E9 Guitart Le 4 juil. 2014 =E0 09:45, Peter Johnstone a =E9crit : > 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. >=20 > Peter Johnstone >=20 > On Thu, 3 Jul 2014, Ondrej Rypacek wrote: >=20 >> Hi all >>=20 >> What is known about limits in REL , the (bi)category of sets and = relations? >> I know there are biproducts; are there equalisers? >>=20 >> And what about SPAN(C) or REL(C), spans and relations over a suitable >> category C ? >>=20 >> Thanks a lot in advance, >> Ondrej >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]