From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6647 Path: news.gmane.org!not-for-mail From: =?ISO-8859-1?Q?Mattias_Wikstr=F6m?= Newsgroups: gmane.science.mathematics.categories Subject: Re: Explanations Date: Wed, 27 Apr 2011 10:16:37 +0200 Message-ID: Reply-To: =?ISO-8859-1?Q?Mattias_Wikstr=F6m?= NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1303992003 11293 80.91.229.12 (28 Apr 2011 12:00:03 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 28 Apr 2011 12:00:03 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Thu Apr 28 13:59:59 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1QFPt1-00048y-CV for gsmc-categories@m.gmane.org; Thu, 28 Apr 2011 13:59:59 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:47407) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QFPpT-0001lt-0Z; Thu, 28 Apr 2011 08:56:19 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QFPpP-0004SS-B3 for categories-list@mlist.mta.ca; Thu, 28 Apr 2011 08:56:15 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6647 Archived-At: > On 26/04/2011 06:55, Timothy Porter wrote: > In category theory, many proofs are transparent and of the form: what do > we know about the situation, just one fact, so we have to use that.... > it works. (I am thinking of classical Yoneda lemma type situations, > since the only elements in hom-sets that we can be sure exist are the > identities.) [...] The question is: "What facts and what objects do we have?", "What is given?". In general we need to work with what is directly given in order to arrive at some indirectly given thing that we are seeking (what we are seeking has to be given in some sense, or else the problem cannot be solved). In category-theoretic terms we can think of what is given as a subobject A of some larger object B in an allegory (where B represents things that "exist" but which we may or may not be able to refer to). A subobject of B is then given just in case it factors through A. (For this to work well the allegory we are working with should contain lots of different objects so that "subobject of X" and "part of X" become practically synonymous.) We may also view things in terms of symmetry and invariants. We are directly given certain subobjects A1, A2, ..., An of some object B in an allegory and we are indirectly given any subobject of B which stays invariant as we apply isomorphisms to B that fix A1, A2, ..., An. (The earlier object A would be the smallest subobject of B containing A1, A2, ..., An as parts.) Finally, we can view what we are given as a theory (axiom system), and the question is what can be defined/specified/referred to in that theory. Of course, mathematics inevitably involves axiom systems, but any theorem which starts "for all ..." can be thought of as involving an axiom system of its own. Mattias [For admin and other information see: http://www.mta.ca/~cat-dist/ ]