From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7565 Path: news.gmane.org!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Re: Splitting epis by wishful thinking Date: Fri, 04 Jan 2013 16:50:15 -0500 Message-ID: Reply-To: "Fred E.J. Linton" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1357344775 3639 80.91.229.3 (5 Jan 2013 00:12:55 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 5 Jan 2013 00:12:55 +0000 (UTC) Cc: "categories" To: =?ISO-8859-1?Q?Andrej=A0Bauer=A0=20?= Original-X-From: majordomo@mlist.mta.ca Sat Jan 05 01:13:12 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.32]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1TrHNq-0004Aq-E5 for gsmc-categories@m.gmane.org; Sat, 05 Jan 2013 01:13:06 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:58045) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1TrHHN-0002IS-SH; Fri, 04 Jan 2013 20:06:25 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1TrHNW-0003Uj-H8 for categories-list@mlist.mta.ca; Fri, 04 Jan 2013 20:12:46 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7565 Archived-At: On Thu, 03 Jan 2013 08:55:07 PM EST Andrej Bauer asked > Is there a construction which "freely" splits all epis in a category ..= =2E ? To the responses already received I thought it perhaps worth adding the i= dea of = freely (or generically) splitting everything, after a fashion I first hea= rd described by Bill Lawvere -- that is, freely adjoining, for each map e (epi or not), a= map f with efe =3D e (and perhaps, if you like, fef =3D f) . Note that, with e epi, efe =3D e will entail ef =3D id, i.e., f will be a= section for e. Likewise, for e mono, efe =3D e will entail fe =3D id, i.e., f will be a retraction for e. (In these two cases, of course, it will follow that fef =3D f. In general= , tho', =2E.. .) One should, of course, ask oneself whether one should really be wanting sections for = quite all epimorphisms -- in the category [R] of unital rings R, for exam= ple, should one = really ever want the trivializing homomorphisms !: R --> 1 to the termina= l ring to split, anywhere? Cheers, -- Fred = [For admin and other information see: http://www.mta.ca/~cat-dist/ ]