From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1953 Path: news.gmane.org!not-for-mail From: Jiri Adamek Newsgroups: gmane.science.mathematics.categories Subject: connected functors Date: Mon, 7 May 2001 14:36:35 +0200 (MET DST) Message-ID: <200105071236.OAA09189@lisa.iti.cs.tu-bs.de> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018228 1371 80.91.229.2 (29 Apr 2009 15:17:08 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:17:08 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue May 8 09:01:33 2001 -0300 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f48BGUI18048 for categories-list; Tue, 8 May 2001 08:16:30 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: ELM [version 2.5 PL2] Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 21 Original-Lines: 39 Xref: news.gmane.org gmane.science.mathematics.categories:1953 Archived-At: Connected Functors Peter Johnstone has asked during the last PSSL for a characterization of functors p: E -> B which are "connected" in the sense that the functor Set^B -> Set^E of composition with p is fully faithful. We have found two necesary and sufficient conditions; in the following E and B are arbitrary small categories. Theorem 1. A functor p: E -> B is connected iff every object X of B is an absolute limit of the diagram of all arrows X -> p(Z) for Z ranging through E . Theorem 2. A functor p: E -> B is connected iff for every morphism x: X -> X' of B the category of all factorizations of x through objects of p[E] is connected. More precisely, in Thm 1 we form the diagram of all arrows X -> p(Z) and all E-morphisms whose p-image forms a commutative triangle in B. Then X is equipped with a canonical cone of that diagram; this cone is requested to be an absolute limit. In Thm 2 we consider the category of all triples (Z,q,m) where Z is an object of E and m,q are morphisms of B with x = q.m (and morphisms between these triples are the E-morphisms whose p-images form two commutative triangles in B ). Connectedness of that category has been, for the case of x = id , observed as a necessary condition by Peter. J. Adamek, R. El Bashir, M. Sobral and J. Velebil xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx alternative e-mail address (in case reply key does not work): J.Adamek@tu-bs.de xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx alternative e-mail address (in case reply key does not work): J.Adamek@tu-bs.de xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx