From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3671 Path: news.gmane.org!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: monic epics Date: Wed, 07 Mar 2007 11:47:06 +0000 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 1241019447 9716 80.91.229.2 (29 Apr 2009 15:37:27 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:37:27 +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 1HP5X0-0002Bu-L9 for categories-list@mta.ca; Wed, 07 Mar 2007 19:26:51 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 25 Original-Lines: 60 Xref: news.gmane.org gmane.science.mathematics.categories:3671 Archived-At: 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. 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? > >