From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3672 Path: news.gmane.org!not-for-mail From: "David Karapetyan" Newsgroups: gmane.science.mathematics.categories Subject: Re: monic epics Date: Tue, 6 Mar 2007 21:30:06 -0800 (PST) Message-ID: <200703070530.l275U6ak000843@melipona.ucdavis.edu> NNTP-Posting-Host: main.gmane.org X-Trace: ger.gmane.org 1241019447 9721 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 1HP5VJ-0001OC-FZ for categories-list@mta.ca; Wed, 07 Mar 2007 19:25:05 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 26 Original-Lines: 21 Xref: news.gmane.org gmane.science.mathematics.categories:3672 Archived-At: > Yes. Another common example of a morphism that is both a monomorphis and > an > epimorphism but not an isomorphism is the inclusion of the rational > numbers > into the real numbers in the category of topological spaces. i understand that such arrows exist and i'm trying to get an intuitive feel for why they are epic. one way i think of a surjective function is that it is a map that entirely covers the codomain so any two function that agree on all of the codomain must be the same. it is not the case with epic arrows that they cover the entire codomain as set functions but that is i think because most categories are much more structured than the category of sets so it is enough to cover certain parts of the codomain and the rest of the structure can be recovered. i think that is what happens with the inclusion of the rationals into the reals because the reals are defined as equivalence classes of sequences of rationals so if two functions agree on the rationals and they are continuous then they automatically agree on the reals.