From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5502 Path: news.gmane.org!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: equality is beautiful Date: Sun, 10 Jan 2010 17:17:02 +0000 Message-ID: References: Reply-To: Steve Vickers NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v753.1) Content-Type: text/plain; charset=ISO-8859-1; delsp=yes; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1263149019 9994 80.91.229.12 (10 Jan 2010 18:43:39 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 10 Jan 2010 18:43:39 +0000 (UTC) To: Joyal@mx4.nbpei-ecn.ca, =?ISO-8859-1?Q?Andr=E9?= , Original-X-From: categories@mta.ca Sun Jan 10 19:43:32 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 4.50) id 1NU2kg-0003rf-V2 for gsmc-categories@m.gmane.org; Sun, 10 Jan 2010 19:43:03 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NU2Ic-00001v-Gu for categories-list@mta.ca; Sun, 10 Jan 2010 14:14:02 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5502 Archived-At: On 9 Jan 2010, at 03:29, Joyal, Andr=E9 wrote: > Dear All > > Many people seem to distrust the equality > relation between the objects of a (large) category. > Is this a philosophical conundrum or a mathematical problem? > > Can we define a notion of (large) category without supposing > that its (large) set of objects has a diagonal Dear Andre, Can I explore this with regard to topologies? Suppose we compare FinSet with Set, defining FinSet very small with N =20= for its object space. The object diagonal N -> NxN is an open inclusion. Now look at Set. The natural topology on the class of sets, as the =20 Ind-completion of FinSet, is the one whose sheaves are given by the =20 object classifier. Thus continuous maps from it are functorial and =20 preserve filtered colimits (the categorical analogue of Scott =20 continuity). (This introduces a confusing issue. Functors from the category Set =20 are already determined by their object maps. But it is a special =20 category in which the morphism space is the comma object got from two =20= copies of the identity map on the object space - we are using the =20 fact that the category of spaces is actually a 2-category, using the =20 specialization morphisms. FinSet was certainly not of this kind.) Now the object diagonal is not even an inclusion, since it is not =20 full. I would speculate, by analogy with what I know for ideal =20 completions, that it is essential but not locally connected. I don't really know what to make of this, but it does seem that there =20= are topological distinctions to be made between the two categories =20 based on object equality. Now let me wonder about classifying toposes. I love using them (as I did above), but they always seem slightly =20 fuzzy because they are really defined only up to equivalence. So I =20 certainly would distrust the object equality. I think the discussion =20 becomes slightly sharper in terms of arithmetic universes (as =20 mentioned by Paul Taylor) instead of toposes. Given a base AU A0, there are two obvious places to look for an =20 object classifier (representing the class of sets) a la Grothendieck =20 topos theory. First, there is A0[U], the AU freely generated over A0 =20 by an object U. This can be constructed by universal algebra, and =20 then is a classifying A0-AU (for the theory with one sort and no =20 predicates, functions or axioms) characterized up to isomorphism with =20= respect to strict A0-AU homomorphisms. However, its object equality =20 depends rather delicately on the precise structure used to =20 characterized AUs. For example, I suspect it will differ according as =20= AUs are taken to have canonical pullbacks or canonical binary =20 products and equalizers. On the other hand, sheaf theory would suggest using the category Presh=20= (FinSet^op) of internal diagrams over the internal FinSet in A0. I =20 conjecture that this (is an AU and) is equivalent but not isomorphic =20 to A0[U]. Hence there are issues of object equality when one compares =20= them. (Milly Maietti and I are looking at "subspaces" in this AU =20 setting, and again are having to be rather careful about equality and =20= the distinction between strict and non-strict AU homomorphisms.) A further issue, seen in AUs but not toposes, is that A0[U] can be =20 internalized in A0, as the internal AU in A0 freely generated by one =20 object. The comparison with the external A0[U] will presumably raise =20 truth v. proof issues similar to those that you have already =20 investigated for the initial AU and Goedel's Theorem. To summarize: (1) Granted the existence of the object diagonals, they may still =20 have different topological characteristics for different kinds of =20 categories. (2) A universal characterization (such as for classifying toposes) =20 that is only up to equivalence will make it difficult to rely on =20 object equality. (3) In arithmetic universes we can perhaps (allowing for the fact =20 that the theory is not fully developed yet) see situations where the =20 object equality definitely exists, but is sensitive to inessential =20 differences. Best regards, Steve Vickers. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]