From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6329 Path: news.gmane.org!not-for-mail From: "George Janelidze" Newsgroups: gmane.science.mathematics.categories Subject: Re: property_vs_structure Date: Thu, 21 Oct 2010 02:14:41 +0200 Message-ID: References: Reply-To: "George Janelidze" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1287620730 6010 80.91.229.12 (21 Oct 2010 00:25:30 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 21 Oct 2010 00:25:30 +0000 (UTC) To: , "Marta Bunge" Original-X-From: majordomo@mlist.mta.ca Thu Oct 21 02:25:28 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P8iyG-0006XL-4z for gsmc-categories@m.gmane.org; Thu, 21 Oct 2010 02:25:28 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:49734) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P8ixf-0005YP-7O; Wed, 20 Oct 2010 21:24:51 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P8ixb-0006fW-6c for categories-list@mlist.mta.ca; Wed, 20 Oct 2010 21:24:47 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6329 Archived-At: Dear Marta, Many thanks, and apologizing for the delay, I am now answering: As you know better than I do, Topos Theory has many aspects and great impact (using these days' expression) on many areas of mathematics. But in this message let me consider only one of its aspects, namely that the (2-)category TOP of toposes can be considered as one of the candidates for 'the right geometric/topological category'. The long list of other possible candidates includes Fam(A) = the category of families of objects of a category A, such that Fam(A) has pullbacks (e.g. every (cocomplete) locally connected topos is such); Loc = the category of locales, Top = the category of topological spaces, CHTop = the category of compact Hausdorff spaces, LaxAlg(T,V) = the categories of lax (T,V)-algebras in the sense of M. M. Clementino, D. Hofmann, and W. Tholen (if T is the ultrafilter monad of Sets, and V = {0,1}, then LaxAlg(T,V) = Top by a theorem of M. Barr), Schemes = the category of schemes in algebraic geometry, CR^o = the opposite category of commutative rings, and many others (I listed only those that will be mentioned below). Each of them certainly has various (subcategories with various) Galois structures with interesting covering morphisms. But essentially only in the cases of CR^o, CHTop, and Fam(A) I have a feeling that the Galois structure I am using (which is just the adjunction with Stone spaces for CR^o and for CHTop, and with sets for Fam(A)) I am using is THE right one. In particular the case of TOP seems to be very interesting, and probably "the answer" would give an answer for Loc, while a "good answer" for Loc might suggest something for TOP. I am not sure I fully understood what you say about unramified coverings versus locally constant coverings. Are you even saying that you found a Galois structure on TOP, or on any subcategory of TOP, whose coverings are exactly the unramified coverings (and the situation is non-trivial in the sense that unramified coverings are not the same as locally constant coverings? That would be wonderful! Independently of that your work on study and comparing what you call local homeomorphisms, complete spreads, and unramified coverings in TOP is absolutely very interesting! And there should be a connection to be understood between it and the work of Maria Manuel Clementino and Dirk Hofmann on similar concepts in LaxAlg(T,V). You say "The notions of discrete fibration and discrete opfibration are lifted from categories to geometric morphisms of toposes (in M. Bunge and J. Funk, Singular Coverings of Toposes, LNM 1890, Springer, 2006, Chapter 9) relative to the symmetric KZ-monad called M therein for "measures" (M.Bunge and A.Carboni, JPAA 105 (1995) 233-249). They are, respectively, the local homeomorphisms and the complete spreads (singular coverings)..." And this is to be compared with the following: Just as for categories, there are discrete fibrations and discrete opfibrations of preorders, and coverings are exactly those that are discrete fibrations and discrete opfibrations at the same time. On the other hand finite preorders are the same as finite topological spaces, and discrete fibrations of finite preorders are the same as the local homeomorphisms of finite topological spaces. This generalizes to the infinite case as follows: Using the ultrafilter convergence, one can define discrete fibrations of T-preorders (=lax T-algebras = T-categories), where T is the ultrafilter monad on the category of sets; and it turns out that: (i) the class of discrete fibrations of T-preorders is not pullback stable; (ii) the pullback stable discrete fibrations of T-preorders are the same as local homeomorphisms of general topological spaces (see [M. M. Clementino, D. Hofmann, and G. Janelidze, Local Homeomorphisms via Ultrafilter Convergence, Proc. AMS 133, 3, 2004, 917-922]). Similarly to the case of T-preorders one can define discrete fibrations and discrete opfibrations of lax (T,V)-algebras (for arbitrary T and V), and this is what I would like to compare with your local homeomorphisms and complete spreads. Of course the categories TOP and LaxAlg(T,V) are so different that the only way to make such a comparison, would be to find appropriate categorical definitions. I don't know how to define a local homeomorphism categorically, but maybe something similar to the story of separability (see [A. Carboni and G. Janelidze, Decidable (=separable) objects and morphisms in lextensive categories, Journal of Pure and Applied Algebra 110, 1996, 219-240] and [G. Janelidze and W. Tholen, Strongly separable morphisms in general categories, Theory and Applications of Categories 23, 5, 2010, 136-149]) can be done. It is also interesting that you say: "...Under hypotheses of the locally simply connected kind, unramified coverings are locally constant..." while what is done in my paper with Aurelio almost suggests to use the path lifting property to make a separable morphism a covering morphism (the occurrence of the path-lifting property is familiar of course, but the fact that it is "almost suggested" categorically a kind of new). All these problems, as well as the absence (so far!) of good Galois structures in TOP, Loc, Top, and Schemes, are related to each other, and more purely-categorical concepts are needed to understand these relationships better. Best wishes, George [For admin and other information see: http://www.mta.ca/~cat-dist/ ]