From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6074 Path: news.gmane.org!not-for-mail From: Peter Freyd Newsgroups: gmane.science.mathematics.categories Subject: even more on inclusion maps Date: Sun, 29 Aug 2010 17:21:26 -0400 Message-ID: Reply-To: Peter Freyd NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1283129692 31399 80.91.229.12 (30 Aug 2010 00:54:52 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 30 Aug 2010 00:54:52 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Mon Aug 30 02:54:50 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Opse9-00068z-4q for gsmc-categories@m.gmane.org; Mon, 30 Aug 2010 02:54:49 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:47587) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1OpsdK-00042G-T0; Sun, 29 Aug 2010 21:53:58 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1OpsdI-00019M-UX for categories-list@mlist.mta.ca; Sun, 29 Aug 2010 21:53:57 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6074 Archived-At: The second construction I gave in my last note ("more on inclusion maps") allowed a choice-free equivalence functor to one with an inclusion-map structure. A simple argument shows that one could use such not to find an inclusion structure on the original category (which we know is not always possible) but at least to find a choice nof monics, one in each set of names for a given subobject. And that of course would imply the axiom of choice (see below). Yikes. In case one needs a proof, given a family of sets construct a category as follows: for each S in the family let A_S B_S be names of objects. The home-sets of the form (B_S, B_S) each have only one element; S itself will be the hom-set (A_S, B_S), the complete group of permutations of S will be the home-set (A_S, A_S), all other hom-sets are empty. The composition of the endomorphism of A_S with the maps to B_S is just what you would expect. All elements of (A_S, B_S) name the same subject. Hence a choice of a monic from each set-of-subobject-names yields a choice function for the non-empty sets in the given family. The second construction doesn't work. The much easier first construction -- fortunately -- does work. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]