From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3678 Path: news.gmane.org!not-for-mail From: David Karapetyan Newsgroups: gmane.science.mathematics.categories Subject: Re: monic epics Date: Wed, 07 Mar 2007 12:53:00 -0800 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019450 9741 80.91.229.2 (29 Apr 2009 15:37:30 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:37:30 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Mar 7 19:35:23 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 07 Mar 2007 19:35:23 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1HP5eV-0006A2-Ol for categories-list@mta.ca; Wed, 07 Mar 2007 19:34:35 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 32 Original-Lines: 20 Xref: news.gmane.org gmane.science.mathematics.categories:3678 Archived-At: Michael Barr wrote: > Category theory is too abstract for any such statement to be true (or > even make sense). For example, in the category denoted . ---> . (with > two objects and one non-identity map), that map is monic and epic for > want of any test maps. More concretely, the inclusion of Z into R is > both in the category of commtative rings. In fact the following a > characterization of monic/epics in commutative rings is this: a > subring R \inc S is epic iff every element of of S can be written s = > vAw where for some n, v is an n-dimensional row vector, w is an > n-dimensional column vector and A is an n x n matrix of elements of S > such that the entries of A, vA, and Aw all belong to R. In general, > very little can be said about monic/epics. > ok i got it. in all the examples given the subobjects given by the monics are "generators" for the object, where by "generators" i mean the elements of the subobject in some way determine the elements of the bigger object. so how about this then: any time we have the situation described above the monic arrow will also be epic.