From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4663 Path: news.gmane.org!not-for-mail From: Gaucher Philippe Newsgroups: gmane.science.mathematics.categories Subject: Re: right adjoint by forgetting relational symbols Date: Thu, 16 Oct 2008 12:19:39 +0200 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241020089 14278 80.91.229.2 (29 Apr 2009 15:48:09 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:48:09 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Oct 16 08:51:39 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 16 Oct 2008 08:51:39 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KqRId-00043g-Hm for categories-list@mta.ca; Thu, 16 Oct 2008 08:45:51 -0300 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 20 Original-Lines: 45 Xref: news.gmane.org gmane.science.mathematics.categories:4663 Archived-At: Le Wednesday 15 October 2008 12:03:23, vous avez =E9crit=A0: > Dear all, > > I need some references for this problem. Suppose we have a locally > presentable category C axiomatized by a limits theory T, so C=3DMod(T). > Let us forget some relational symbols in T and all axioms containing these > relational symbols. One obtains a theory T'. There is a forgetful functor > Mod(T) --> Mod(T'). Does this functor have always a right adjoint ? If no= t, > what conditions must we add ? > > Thanks in advance. pg. Dear all,=20 Thank you for all answers. But they do not give what I want. I am looking f= or=20 a -R-I-G-H-T- adjoint, and by comparing the example I have and the limit=20 theory axiomatizing the category of small categories (for which the forgetf= ul=20 functor Mod(T)-->Set does not have any right adjoint since it is not=20 colimit-preserving), i found (maybe) the following sufficient condition:=20 If T is a limit theory without equality symbol before the implication signs= ,=20 then any forgetful functor Mod(T) --> Mod(T') has a right adjoint. Indeed, all sentences of T are of the form (Ax)(F(x)=3D>((E!y)G(x,y)) where= F(x)=20 and G(x,y) are conjunctions of atomic formulas. By hypothesis, F does not=20 contain the symbol =3D. So the forgetful functor Mod(T) --> Mod(T') is coli= mit=20 preserving. Since Mod(T) is locally presentable, it is cocomplete,=20 cowellpowered and has a strong generator. So by SAFT, the forgetful functor= =20 Mod(T) --> Mod(T') has a right adjoint. Does it sound good ? Thanks in advance. pg.