From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9018 Path: news.gmane.org!.POSTED!not-for-mail From: wlawvere =0A= Newsgroups: gmane.science.mathematics.categories Subject: Re: Grothendieck toposes Date: Sun, 6 Nov 2016 15:41:05 +0000 Message-ID: Reply-To: wlawvere =0A= NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-2022-jp" Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1478569211 20045 195.159.176.226 (8 Nov 2016 01:40:11 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Tue, 8 Nov 2016 01:40:11 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Tue Nov 08 02:40:06 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 1c3vOG-0003fl-Tm for gsmc-categories@m.gmane.org; Tue, 08 Nov 2016 02:39:57 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:40474) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1c3vNw-0001jG-Ln; Mon, 07 Nov 2016 21:39:36 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1c3vNw-0003oq-FN for categories-list@mlist.mta.ca; Mon, 07 Nov 2016 21:39:36 -0400 Thread-Topic: Categories: Re: Grothendieck toposes Accept-Language: en-CA, en-US Content-Language: en-CA Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9018 Archived-At: =0A= Dear friends and colleagues,=0A= =0A= In Spring 1981, near a lavender field in Southern France,=0A= Alexander Grothendieck greeted me at the door of his home.=0A= He wasted no time and immediately put the question:=0A= =0A= 'What is the relationship between the two uses of the term 'topos'?'=0A= =0A= This led to a very interesting discussion.The first thing that=0A= was established as a basis was that SGA4 never defined 'topos',=0A= but rather spoke always of 'U-topos', where U was a certain=0A= kind of model of set theory. All the categories so arising have=0A= common features, such as cartesian closure, and the U itself can=0A= be construed as such a category. (TAC Reprints no. 11).=0A= =0A= Thus we arrived at the notion of 'U-topos' as a special geometric=0A= morphism E =1B$B"*=1B(BU of 'elementary' toposes. Grothendieck's=0A= general method of relativization suggests the usefulness=0A= of a general topos as a codomain or base U. (see Giraud, SLN 274).=0A= But to focus more specifically on the original case, various special=0A= properties of the base U could also be considered:=0A= Booleanness (note for example, that Booleanness distinguishes=0A= algebraic points among algebraic figures)=0A= Axiom of choice;=0A= Lack of measurable cardinals; et cetera.=0A= =0A= One of the many topics we discussed was the=0A= 'Medaille de Chocolat' exercise in SGA4, and its basic importance=0A= for understanding applications of topos theory: the gros and=0A= petit sheaves of an object point out that there should be a=0A= qualitative distinction between a topos of SPACES and a topos=0A= of set-valued sheaves on a generalized space. I believe that=0A= considerable progress is now being made on the characterization=0A= of 'gros' toposes under the name of Cohesion. Grothendieck made=0A= a big step towards the characterization of 'petit' under the name=0A= of 'etendu' (sometimes known as 'locally localic'). Concerning=0A= Grothendieck's most famous contribution, the 'petit etale' topos,=0A= what are it's distinguishing properties as a topos?=0A= =0A= We also discussed the Grauert direct image theorem as a=0A= relativization of the Cartan-Serre theorem. It is important to=0A= note that Grothendieck's work was not limited to the Weil=0A= conjectures but, for example, involved around 1960 several=0A= categories related to complex analysis which were perhaps=0A= part of his inspiration for the notion of topos.=0A= =0A= =0A= Separation?=0A= Actually, separation has been one of the main sources of confusion.=0A= I wish that someone with internet confidence would correct the=0A= Wikipedia article that claims that pre-1970 toposes were about=0A= geometry, but that post-1970 toposes were about logic. Certainly,=0A= that discourages students from studying either.=0A= Omitted was the fact that logic has always been used to sharpen the=0A= study of geometry; in the last 50 years we have been able to make=0A= this relation more explicit, with the help of categories.=0A= =0A= Of course, separating a certain kind of object from a certain kind=0A= of map would be basic 'grammar'.=0A= But we cannot separate the legacy of Grothendieck from the=0A= inspiration it gives to the continuing development of topos theory.=0A= =0A= Best wishes=0A= Bill Lawvere=0A= =0A= =0A= =0A= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]