From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8994 Path: news.gmane.org!.POSTED!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: Grothendieck toposes Date: Mon, 31 Oct 2016 11:27:57 +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=us-ascii Content-Transfer-Encoding: 7bit X-Trace: blaine.gmane.org 1477941789 8028 195.159.176.226 (31 Oct 2016 19:23:09 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Mon, 31 Oct 2016 19:23:09 +0000 (UTC) Cc: "categories@mta.ca" To: Marta Bunge Original-X-From: majordomo@mlist.mta.ca Mon Oct 31 20:23:05 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 1c1IAR-0007o6-RC for gsmc-categories@m.gmane.org; Mon, 31 Oct 2016 20:22:47 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:57425) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1c1IA2-0003Go-JF; Mon, 31 Oct 2016 16:22:22 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1c1IA3-0007Te-Te for categories-list@mlist.mta.ca; Mon, 31 Oct 2016 16:22:23 -0300 In-Reply-To: Original-Content-Transfer-Encoding: quoted-printable Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8994 Archived-At: Dear Marta, Thanks for your reply. My question was a survey of usage, so what follows is= not meant disputatiously, but I thought your reasoning raised some interest= ing issues. As I see it you give two reasons here for taking Grothendieck toposes as bei= ng over Set =3D ZFC, both pertinent: (1) what Grothendieck meant, and (2) th= e (essential) uniqueness of geometric morphisms to Set. (1) is delicate, given Grothendieck's underlying implicit definition of topo= s as "that of which topology is the study". My understanding of this (you may know more - my knowledge of Grothendieck's= work is almost all second-hand) is that there are two ways of viewing it, s= omewhat akin (respectively) to algebraic and general topology. The first is that he meant a topos to be a category with which to do sheaf c= ohomology, by forming an (exact) injective resolution of Abelian groups, tak= ing global sections (becoming non-exact), and then extracting the ker/im coh= omology groups. This is perhaps an algebraic topologist's idea of what topol= ogy means. Classicality of Set seems then to be needed in order to get injec= tive hulls in the category of sheaves over a site.=20 On the other hand, there is also the idea that global points can be recovere= d as sections of the geometric morphism to Set, and the topos simultaneously= embodies both the points and the topology on them. Is this also part of wha= t Grothendieck meant? It is closer to general topology, points and their con= tinuous transformations. This idea then generalizes well to elementary toposes, replacing Set by S. E= lementary toposes are not generalized spaces in themselves, but bounded geom= etric morphisms between them are, and many topological properties are redefi= ned for geometric morphisms. (2) raises the question of why Set should have its privileged property. Tech= nically, it is that every object of Set is a colimit of copies of 1, and tha= t is preserved up to unique isomorphism by the inverse image part of any geo= metric morphism. But is that not because ZFC provides a 2-level structure of= sets and classes, and we are implicitly using ZFC classes for our toposes? A= s we explore the foundational options then we should expect this uniqueness p= roperty for Set =3D ZFC to evaporate. All the best, Steve. > On 30 Oct 2016, at 20:17, Marta Bunge wrote: >=20 > Dear Steve, >=20 > When an elementary (base) topos S is specified, I use "S-bounded topos" to= mean the pair (E, e), with E an elementary topos and e: E--> S a (bounded) g= eometric morphism. When S =3D Set (a model of ZFC) and E an arbitrary elemen= tary topos, then there is a most one geometric morphism e:E---> Set, so in t= hat case the latter need not be specified. I therefore use "E is a Grothendi= eck topos" to mean "E is an elementary topos bounded over Sets". The latte= r has been shown to be equivalent to what Grothendieck meant by it. =20 >=20 > Best regards, > Marta=20 > ************************************************ > Marta Bunge > Professor Emerita > Dept of Mathematics and Statistics=20 > McGill University=20 > Montreal, QC, Canada H3A 2K6 > Home: (514) 935-3618 > marta.bunge@mcgill.ca=20 > http://www.math.mcgill.ca/people/bunge > ************************************************ > From: Steve Vickers > Sent: October 27, 2016 7:07:52 AM > To: Categories > Subject: categories: Grothendieck toposes > =20 > For some years now, I have been using the phrase "Grothendieck topos" - > category of sheaves over a site - to allow the site to be in an > arbitrary base elementary topos S (often assumed to have nno). Hence > "Grothendieck topos" means "bounded S-topos". The whole study of > Grothendieck toposes, as of geometric logic, is parametrized by choice of S= . >=20 > That's presumably not how Grothendieck understood it, and I know some of > his results assumed S =3D Set, some classical category of sets. Moreover, > the Elephant defines "Grothendieck topos" that way. >=20 > On the other hand, if a topos is a generalized space, with a classifying > topos being the space of models of a geometric theory, then that surely > meant Grothendieck topos; and there are various reasons for wanting to > vary S. For example, using Sh(X) as S gives us a generalized topology of > bundles, fibrewise over X. >=20 > I'm coming to suspect my usage may confuse. >=20 > How do people actually understand the phase "Grothendieck topos"? Do > they hear potential for varying an implicit base S, or do they hear a > firm implication that S is classical? >=20 > Steve Vickers. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]