From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3677 Path: news.gmane.org!not-for-mail From: David Karapetyan Newsgroups: gmane.science.mathematics.categories Subject: Re: monic epics Date: Wed, 07 Mar 2007 12:23:25 -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 9735 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 19:34:13 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 07 Mar 2007 19:34:13 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1HP5dK-0005aa-QK for categories-list@mta.ca; Wed, 07 Mar 2007 19:33:23 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 31 Original-Lines: 54 Xref: news.gmane.org gmane.science.mathematics.categories:3677 Archived-At: Steve Vickers wrote: > Dear David, > > There are more complicated examples. Here's one. > > Take A to be the semigroup {0,a,b,c} in which x0=0x=0 for all x, aa=a, > cc=c, ab = bc = b, and all other products are 0. (This of this as > being derived from a category with two objects and three morphisms, so > a and c represent the two identities and b is a morphism between the > two objects. 0 takes care of products of non-composable pairs.) > > Take B = A u {d}, with cd = da = d, bd=a, db=c and all other binary > products involving d give 0. (Think of adjoining an inverse to b in > the category.) > > Now the inclusion A -> B is a semigroup epi. To see this, suppose f: A > -> C is a semigroup homomorphism, and x in C satisfies > > xf(a) = f(c)x = x > f(b)x = f(a) > xf(b) = f(c) > > Then x is the unique such. For if x' is another then > > x' = x'f(a) = x'f(b)x = f(c)x = x > > If g: B -> C is a semigroup homomorphism agreeing with f on A, then > g(d) does satisfy those equations for x, and so any two such g's must > be equal. > > This semigroup example can be easily turned into monoids by adjoining > a unit element. > > There is still the same idea that B is got by adjoining inverses to > elements of A, but they are not inverses in the monoid sense and it is > not clear to me in general how one would formalize the idea that they > are inverses when embedded in some category. > > There is a related epi in rings: the inclusion of upper triangular 2x2 > matrices (over any ring) into all 2x2 matrices. > > Regards, > > Steve Vickers. > i like the example of the semigroups. i think in some sense the addition of d does not add enough information to our monoid so if we have two semigroup homomorphisms that agree on A then by using the properties of semigroup homomorphisms we are forced to define how they behave on d in only one way. i'm still trying to incorporate the inclusion of the 2x2 upper triangular matrices into all 2x2 matrices and Michael Barr's example into this framework.