From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4399 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Has anyone seen this condition? Date: Fri, 16 May 2008 17:04:54 -0400 (EDT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1241019922 13124 80.91.229.2 (29 Apr 2009 15:45:22 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:45:22 +0000 (UTC) To: Categories list Original-X-From: rrosebru@mta.ca Fri May 16 20:30:46 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 16 May 2008 20:30:46 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Jx9Ls-0000ZH-Tz for categories-list@mta.ca; Fri, 16 May 2008 20:28:40 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 22 Original-Lines: 21 Xref: news.gmane.org gmane.science.mathematics.categories:4399 Archived-At: In our (John Kennison, Bob Raphael, and I) work, the following condition has arisen. Has anyone seen or named it? Say that an object E of a category is a ???? if it is a cogenerator and if whenever f: A ---> B is not an epimorphism and g: B ---> C is a regular monomorphism, then there are two maps h,k: C ---> E s.t. hg is unequal to kg, while hgf = kgf. This is related to the questions Paul Taylor and I have raised recently. Theorem. If E satisfies ????, then whenever A has an extremal monomorphism into a power of E, then A ---> TA ===> T^2A is an equalizer where T is the triple from the adjoint pair Hom(-,E) and E^{(-)}. What's interesting is that while it is obvious that any injective cogenerator satisfies ????, it is also the case that any cogenerator that contains an injective cogenerator also satisfies ????. Thus, in completely regular spaces, the interval is a cogenerator and both it and the real line (and many, many other spaces) also satisfy ????. Michael