From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5603 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: abstraction of notation from sets. Date: Sun, 28 Feb 2010 13:30:46 -0800 Message-ID: References: Reply-To: Vaughan Pratt NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1267448850 7691 80.91.229.12 (1 Mar 2010 13:07:30 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 1 Mar 2010 13:07:30 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Mon Mar 01 14:07:26 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Nm5LI-0002KN-6M for gsmc-categories@m.gmane.org; Mon, 01 Mar 2010 14:07:24 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Nm4iW-0004q9-H7 for categories-list@mta.ca; Mon, 01 Mar 2010 08:27:20 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5603 Archived-At: Besides Peter Easthope's question on the appropriateness of considering an object of C as an element of C there is also (in fact more generally as I'll mention near the end) the question of whether the notion of "element" is well established in a presheaf category C = Set^{J^op} on J, made a topos by choice of a final object 1 and a subobject classifier O, and (by Yoneda) with the further choice of a full embedding of J in C. With that arrangement, and with the idea that distinct provenances for sets give rise to distinct notions of "element," it seems to me that at least four natural kinds of set arise in a presheaf topos in ordinary mathematical practice. First kind. As a homset from 1. This notion makes the connection with set theory that toposes were developed for, with the caveat that it should be understood in the light of the third and fourth kinds so as not to overstate its applicability. Second kind. As a morphism to O. This represents a subobject of the domain of the morphism (the subobject itself being in general a proper class and therefore not a fit entity for ordinary mathematics). For example in the topos Set with natural numbers object N it is natural to write {2,3,5,7} for the set of 1-digit primes, understood as (the subobject of N represented by) its characteristic function as a morphism from N to O. Third kind. As a homset to O. This is the power *set* C(X,O) of subobjects of the domain X of the homset, which by the cartesian closed structure of C is in a natural bijection with the power *object* O^X when considered as a set C(1,O^X) of the first kind. Fourth kind. As a homset from the image Y(j) under the embedding Y: J --> C of some object j of J. For example if J has objects V and E making each object of the topos a graph G then we think of G as formed from two sets, and refer to the morphisms to G from Y(V) and Y(E) as respectively the vertices and edges of G. In this example the set of vertices of G also happens to be a set of the first kind, but not the set of edges, pointing up the need I mentioned earlier not to overstate the significance of sets of the first kind while also addressing Peter's original question in a roundabout way in terms of the underlying graph of a category. A morphism of a presheaf category is properly understood as an ob(J)-indexed family of functions mapping sets of the fourth kind to their counterparts in the codomain of the morphism. Among these the monics as families of injections furnish the notion of subobject in a presheaf topos with its intuitive meaning complementary to that of the second kind of set, the remark about proper classes notwithstanding. Of course any homset of a topos is a set, but it seems to me that the above four kinds deserve special recognition as sets commonly encountered in mathematical practice having readily distinguishable provenances. Vaughan Pratt [For admin and other information see: http://www.mta.ca/~cat-dist/ ]