From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3669 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: monic epics Date: Wed, 7 Mar 2007 07:40:40 -0500 (EST) Message-ID: References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1241019445 9708 80.91.229.2 (29 Apr 2009 15:37:25 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:37:25 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Mar 7 19:29:32 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 07 Mar 2007 19:29:32 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1HP5Xe-0002aE-72 for categories-list@mta.ca; Wed, 07 Mar 2007 19:27:30 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 23 Original-Lines: 33 Xref: news.gmane.org gmane.science.mathematics.categories:3669 Archived-At: 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. On Tue, 6 Mar 2007, David Karapetyan wrote: > Hi, I've been trying to learn some category theory and I came upon the > example of a monic, epic in the category of monoids given by the > inclusion function of (N,0,+) into (Z,0,+). I know that in monoids every > monic arrow is also an injective function but the inclusion function of > N into Z provides a counterexample of every epic arrow being a > surjective function. I noticed that N is just a "folded" version of Z, > where by "folded" I mean take Z and throw away all the inverses of the > natural numbers. So does every monic, epic arrow determine such a > "folding" or are there monic, epics that can't be characterized in such > a way? > > -- Any society that would give up a little liberty to gain a little security will deserve neither and lose both. Benjamin Franklin