From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2290 Path: news.gmane.org!not-for-mail From: Robin Cockett Newsgroups: gmane.science.mathematics.categories Subject: Re: connected categories and epimorphisms. Date: Sun, 18 May 2003 11:45:37 -0600 (MDT) Message-ID: <0HF3009TMFEZKY@l-daemon> References: <20030516162342.M13454-100000@mx1.mat.unb.br> 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 1241018552 3406 80.91.229.2 (29 Apr 2009 15:22:32 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:22:32 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon May 19 14:52:01 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 19 May 2003 14:52:01 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19HohT-0006Yn-00 for categories-list@mta.ca; Mon, 19 May 2003 14:45:27 -0300 In-reply-to: <20030516162342.M13454-100000@mx1.mat.unb.br> Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 26 Original-Lines: 41 Xref: news.gmane.org gmane.science.mathematics.categories:2290 Archived-At: Of course it depends rather heavily on what you mean by connected: (1) if you mean that there is a way to get between any two objects via arrows -- and one is allowed to go backwards along arrows -- then this is not true. Any category with products is necessarily connected in this manner and the category of Sets provides a counter-example. Any projection p_0: A x 0 -> A where 0 is the empty set and A is non-empty is non-epic. (2) if you mean that given any objects A and B there is always an arrow f: A -> B (differs from (1) in that you are not allowed to go backwards along arrows) -- that is homsets are non-empty -- then this IS true. This is because every projection in such a category has a section as the composite <1_A,f> p_0 A --------> A x B --------> A is the identity. This makes the projection a retraction and thus epic. (2) if you mean (stretching a bit) that every object has a (regular) epic onto the final object (all objects have global support) then all you need in addition is that the product functors _ x A preserves these epics. This will be the case, for example, if the category is cartesian closed ... however, such a category better not have an initial object! -robin On 16 May, Flavio Leonardo Cavalcanti de Moura wrote: > Hi, > > How can I show that, in a connected category, projections (of the > product) are epimorphisms? > > Thank you, > > Flavio Leonardo.