From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6615 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: Constitutive Structures Date: Fri, 15 Apr 2011 09:11:20 +1000 Message-ID: References: Reply-To: Richard Garner NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1302869722 30794 80.91.229.12 (15 Apr 2011 12:15:22 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 15 Apr 2011 12:15:22 +0000 (UTC) Cc: categories To: "Ellis D. Cooper" Original-X-From: majordomo@mlist.mta.ca Fri Apr 15 14:15:18 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 1QAhvh-0000dZ-89 for gsmc-categories@m.gmane.org; Fri, 15 Apr 2011 14:15:17 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:50026) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QAhvI-0003FT-QC; Fri, 15 Apr 2011 09:14:52 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QAhvG-000269-KM for categories-list@mlist.mta.ca; Fri, 15 Apr 2011 09:14:50 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6615 Archived-At: Here's a possible answer using toposes. I don't really know enough topos theory to do this properly so I will be busking it a bit; hopefully someone more knowledgeable than I can tell me what I am up to! We define a factorisation system (E,M) on the 2-category of Grothendieck toposes, generated by the following M-maps. For each n, we take the obvious geometric morphism from the classifying topos of an object equipped with an n-ary relation to the object classifier; and we take that geometric morphism from the object classifier to the classifying topos of a monomorphism which classifies the identity map on the generic object. With any luck this generates a factorisation system on GTop; with equal luck it is a well-known one, but my knowledge of the taxonomy of classes of geometric morphisms is sufficiently hazy that I cannot say which it might be. In any case, the hope is that M-maps into the object classifier should correspond to single-sorted geometric theories. Now we work in the category of such M-maps into Set[O], and in there, there is an object which represents all the constitutive substructures of the reals. The object in question is obtained as the M-part of the (E,M) factorisation of the geometric morphism Set -> Set[O] which classifies the real numbers; it is the "complete theory of the reals", but not with respect to any particular structure, but rather with respect to all possible structures (within geometric logic) that we might impose on it. Unfortunately this would not capture, e.g., the second-order structures we might impose on the reals, but it's a start. (Of course, if we were merely interested in structures expressible by finitary algebraic theories, then we could consider the category of finitary monads on Set, and in there, the finitary coreflection of the codensity monad of the reals. That was my initial reaction to this problem, and the above is supposed to generalise this in some sense). Richard On 7 April 2011 22:50, Ellis D. Cooper wrote: > What might be the proper categorical framework to discuss, for > example, the fact that the Real Numbers have constitutive structures > such as additive abelian group, multiplicative abelian group, > topology generated by open intervals, totally ordered infinite set, and so > on? > At first one might think of forgetful functors, but then what would > be the category in which Real Numbers is one object among many? > Or, one might say take a category with exactly one object and a > functor to each of the categories of the constitutive structures. This makes > the Real Numbers look like an "element" of the "intersection" of > diverse categories. Then the Complex Numbers or the Hyperreal Numbers > which contain > the Real Numbers as sub-objects in certain ways are "elements" of > other "intersections" of categories. What am I talking about? > > Ellis D. Cooper > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]