From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9029 Path: news.gmane.org!.POSTED!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: Re: Grothendieck toposes Date: Thu, 10 Nov 2016 16:34:32 +0000 Message-ID: References: Reply-To: Steve Vickers NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 (1.0) Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1478960950 2431 195.159.176.226 (12 Nov 2016 14:29:10 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Sat, 12 Nov 2016 14:29:10 +0000 (UTC) Cc: categories@mta.ca To: wlawvere =0A= Original-X-From: majordomo@mlist.mta.ca Sat Nov 12 15:29:06 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 1c5ZIh-0007hR-E2 for gsmc-categories@m.gmane.org; Sat, 12 Nov 2016 15:28:59 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:46636) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1c5ZIH-0007p0-DH; Sat, 12 Nov 2016 10:28:33 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1c5ZIH-0004LA-UK for categories-list@mlist.mta.ca; Sat, 12 Nov 2016 10:28:33 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9029 Archived-At: Dear Bill, I'd really like to understand these issues about gros and petit toposes bett= er. My own current direction with arithmetic universes is - I believe - a reason= able one to try in respect to geometric theories and classifying toposes, bu= t it loses sight of gros toposes and synthetic approaches, which, after all,= were part of the founding ideas of toposes. I need to know better what it i= s that I risk losing. However, my attempts to understand gros toposes run into difficulties with g= eometricity. This is illustrated by the Zariski topos for algebras over a fi= eld k. If k is R or C then the geometric methodology expects it and its alge= bras to be locales (and non-discrete), not sets. But does the polynomial rin= g R[x] exist localically? The degree of a polynomial looks like being part o= f the geometric structure, and removing leading zeros (e.g. after a subtract= ion) is not continuous. Do you know if anyone has investigated a localic form of these methods in al= gebraic geometry? All the best, Steve. p.s. I like to think I'm following Grothendieck's insights, but the truth is= I understand only a tiny fraction of them. > On 6 Nov 2016, at 15:41, wlawvere =3D0A=3D wrote: >=20 >=20 > Dear friends and colleagues, >=20 > In Spring 1981, near a lavender field in Southern France, > Alexander Grothendieck greeted me at the door of his home. > He wasted no time and immediately put the question: >=20 > 'What is the relationship between the two uses of the term 'topos'?' >=20 > This led to a very interesting discussion.The first thing that > was established as a basis was that SGA4 never defined 'topos', > but rather spoke always of 'U-topos', where U was a certain > kind of model of set theory. All the categories so arising have > common features, such as cartesian closure, and the U itself can > be construed as such a category. (TAC Reprints no. 11). >=20 > Thus we arrived at the notion of 'U-topos' as a special geometric > morphism E =E2=86=92U of 'elementary' toposes. Grothendieck's > general method of relativization suggests the usefulness > of a general topos as a codomain or base U. (see Giraud, SLN 274). > But to focus more specifically on the original case, various special > properties of the base U could also be considered: > Booleanness (note for example, that Booleanness distinguishes > algebraic points among algebraic figures) > Axiom of choice; > Lack of measurable cardinals; et cetera. >=20 > One of the many topics we discussed was the > 'Medaille de Chocolat' exercise in SGA4, and its basic importance > for understanding applications of topos theory: the gros and > petit sheaves of an object point out that there should be a > qualitative distinction between a topos of SPACES and a topos > of set-valued sheaves on a generalized space. I believe that > considerable progress is now being made on the characterization > of 'gros' toposes under the name of Cohesion. Grothendieck made > a big step towards the characterization of 'petit' under the name > of 'etendu' (sometimes known as 'locally localic'). Concerning > Grothendieck's most famous contribution, the 'petit etale' topos, > what are it's distinguishing properties as a topos? >=20 > We also discussed the Grauert direct image theorem as a > relativization of the Cartan-Serre theorem. It is important to > note that Grothendieck's work was not limited to the Weil > conjectures but, for example, involved around 1960 several > categories related to complex analysis which were perhaps > part of his inspiration for the notion of topos. >=20 >=20 > Separation? > Actually, separation has been one of the main sources of confusion. > I wish that someone with internet confidence would correct the > Wikipedia article that claims that pre-1970 toposes were about > geometry, but that post-1970 toposes were about logic. Certainly, > that discourages students from studying either. > Omitted was the fact that logic has always been used to sharpen the > study of geometry; in the last 50 years we have been able to make > this relation more explicit, with the help of categories. >=20 > Of course, separating a certain kind of object from a certain kind > of map would be basic 'grammar'. > But we cannot separate the legacy of Grothendieck from the > inspiration it gives to the continuing development of topos theory. >=20 > Best wishes [For admin and other information see: http://www.mta.ca/~cat-dist/ ]