From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9004 Path: news.gmane.org!.POSTED!not-for-mail From: majordomo@mlist.mta.ca Newsgroups: gmane.science.mathematics.categories Subject: Re: Grothendieck toposes Date: Wed, 2 Nov 2016 17:50:08 +0000 Message-ID: <8C57894C7413F04A98DDF5629FEC90B138BD447A@Pli.gst.uqam.ca> References: Reply-To: Bob Rosebrugh NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1478194842 2791 195.159.176.226 (3 Nov 2016 17:40:42 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Thu, 3 Nov 2016 17:40:42 +0000 (UTC) To: Marta Bunge , "categories@mta.ca" Original-X-From: majordomo@mlist.mta.ca Thu Nov 03 18:40:37 2016 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.7.28]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1c2Lzf-000579-ND for gsmc-categories@m.gmane.org; Thu, 03 Nov 2016 18:40:03 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:33951) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1c2LzH-00050T-2V; Thu, 03 Nov 2016 14:39:39 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1c2LzI-0002nQ-4a for categories-list@mlist.mta.ca; Thu, 03 Nov 2016 14:39:40 -0300 Thread-Topic: categories: Re: Grothendieck toposes Thread-Index: AQHSMuqQbPGTFq4B80awlyPIslE5nKDEOGQNgAGIWQCAABIcgIAAG2R6 In-Reply-To: Accept-Language: en-US, en-CA Content-Language: en-US Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9004 Archived-At: Dear Marta, Mathematics and science are very often regarded as the pure product of human rationality. I can agree with the importance of rationality, except that humanity is as much the product of nature as it is of rational choices. You will agree that the natural world is only partially explained by scienc= e. The rest is a big mystery. Not that the mystery is absolutly impenetrable. I feel compelled to recognize the presence of mysteries even in mathematics. The history of complex numbers, from the discovery by Cardano to their applications in quantum physics is bewilderin= g. They belong to this universe as much as the electron and the human mind. The fact that we human can understand complex numbers may have a metaphysical meaning. What is it? Best, Andr=E9 ________________________________ From: Marta Bunge [martabunge@hotmail.com] Sent: Wednesday, November 02, 2016 7:18 AM To: categories@mta.ca Cc: Steve Vickers; Patrik Eklund; Joyal, Andr=E9 Subject: Re: categories: Re: Grothendieck toposes Dear all, > It is marvelous that the two notions should be so related. > But it is be better to keep them appart before uniting them. > Otherwise the miracle disappear in confusion. The above is a quotation from a recent posting by Andre Joyal. To the risk= of boring everyone I offer the following comment on it here. There is no n= eed to talk about miracles in mathematics, not even as some sort of analogy= . Why not instead give credit to the very important insight of an elementar= y topos as embodying both the logic and the geometry? There are two notions= of morphism between elementary toposes, not a preferred one - the geometri= c and the logical. One structure - to wit that of an elementary topos, can = be seen in two different ways depending on what the mathematical uses one w= ants to give it. There is no confusion here - just richness. Let me be mor= e specific. Thinking of an elementary topos S as the chosen "set theory", a Grothendiec= k topos (including any category of the form Sh(X) for X a locale in S, but = more generally as a category of sheaves on a site in S) can be recovered as= a pair (E, e) where E is another elementary topos and e: E -> S a bounded = geometric morphism. Thinking of elementary toposes from the logical point o= f view, and so of logical morphisms between them, there are other ideas and= constructions that profit from this point of view - for instance a formula= tion and proof of realizability by means of Artin-Wraith glueing. Both the geometric and the logical are sides of the same coin. The notion o= f an elementary topos (or "topos" for short) is simple yet powerful and unt= il now it has served most of the mathematical purposes for which it was int= ended and more. Best wishes, Marta [For admin and other information see: http://www.mta.ca/~cat-dist/ ]