From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8996 Path: news.gmane.org!.POSTED!not-for-mail From: Thomas Streicher Newsgroups: gmane.science.mathematics.categories Subject: Re: Grothendieck toposes Date: Tue, 1 Nov 2016 11:30:55 +0100 Message-ID: References: Reply-To: Thomas Streicher NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: blaine.gmane.org 1478045322 16377 195.159.176.226 (2 Nov 2016 00:08:42 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Wed, 2 Nov 2016 00:08:42 +0000 (UTC) Cc: Marta Bunge , "categories@mta.ca" To: Steve Vickers Original-X-From: majordomo@mlist.mta.ca Wed Nov 02 01:08:38 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 1c1j6M-0001Uw-Sq for gsmc-categories@m.gmane.org; Wed, 02 Nov 2016 01:08:23 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:60419) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1c1j5w-0006hC-TY; Tue, 01 Nov 2016 21:07:56 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1c1j5y-0004R8-So for categories-list@mlist.mta.ca; Tue, 01 Nov 2016 21:07:58 -0300 Content-Disposition: inline In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8996 Archived-At: As to the question brought up by Steve I want to remark that Grothendieck always spoke about U-topos for some (Grothendieck) universe U. To write simply Set instead of U is covenient but slightly misleading if one takes logical foundations seriously since what after all is this Set? One shouldn't forget that ZFC has many models even when one adds the axiom that every set is element of a Grothendieck universe (it has a countable model by downward Loewenheim-Skolem). If one uses an extended set-theoretic foundation as Grothendieck did Set is just a name for a generic Grothendieck universe. With the advent of elementary topos theory one wanted to forget about set theory since one thought that category theory provide its own foundation via elementary toposes. This certainly makes sense but what then is Set? Well, one may choose some (unspecified) base topos SS and consider categories relative to SS as Grothendieck fibrations over SS. The role of "Set" is then taken by the fundamental ("codomain") fibration P_SS = cod : SS^2 -> SS (where 2 is the ordinal 2). From this relative point of view Grothendieck toposes over SS correspond to bounded geometric morphisms to SS as worked out in detail in Johnstone's 1977 book. But, of course, there may be many non isomorphic g.m.s from EE to SS. However, in Top/SS there is a (kind of) terminal object, the identity g.m. on SS. As explained in Moens's 1982 Thesis g.m.s to SS correspond to cocomplete locally small fibrations of toposes over SS (he assumed that the internal sums were stable and disjoint which 6 years later was shown by Jibladze to be the case for all cocomplete fibered toposes. But if one has a Grothendieck topos EE over SS the internal language of SS doesn't allow one to speak about EE in all relevant respects. In particular, one cannot quantify over the objects of EE within the internal language of SS. However, one may "blow up" SS so that one can. It is an old observation by Benabou that *split* fibrations over SS correspond to categories internal to presheaves over SS (for a large enough "Set"). This, however, is not possible for non-split fibrations like P_SS. Different ways of overcoming this problem have been found by Awodey, Butz, Simpson and myself "Relating first-order set theories, toposes and categories of classes" (APAL 2014) and in an unpublished paper by Mike Shulman arXiv:1004.3802. The restrictions of the internal language of the base topos w.r.t. speaking about a fibration over it can be overcome when one admits universes in the base topos. These universes are less set-theoretic than Grothendieck's ones but play a similar role. Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]