From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9047 Path: news.gmane.org!.POSTED!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories Subject: Re: Fibred 2-category of Grothendieck toposes? Date: Fri, 02 Dec 2016 08:30:46 -0500 Message-ID: References: <8135_1480682187_58416ACB_8135_508_1_E1cCn2y-0006W1-1X@mlist.mta.ca> Reply-To: Marta Bunge NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 (1.0) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1480984738 28356 195.159.176.226 (6 Dec 2016 00:38:58 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Tue, 6 Dec 2016 00:38:58 +0000 (UTC) To: Steve Vickers , Categories Original-X-From: majordomo@mlist.mta.ca Tue Dec 06 01:38:53 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 1cE3mW-0006RV-D8 for gsmc-categories@m.gmane.org; Tue, 06 Dec 2016 01:38:52 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:57587) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1cE3m9-00067V-Io; Mon, 05 Dec 2016 20:38:29 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1cE3mB-0004L2-I2 for categories-list@mlist.mta.ca; Mon, 05 Dec 2016 20:38:31 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9047 Archived-At: Dear Steve, The setting of a 2-category GTop bounded over an elementary topos S has been= extensibly worked out in practice in (for instance) the book by Marta Bunge= and Jonathon Funk, Singular Coverings of Toposes, LNM 1890, Springer 2006, a= nd in several papers by myself or with collaborators which you can look up i= n my Research Gate page. The terminology that I have used for it everywhere (= including lectures) is Top_S, by which it is not meant Top/S but the sub 2-c= ategory of it whose objects are bounded geometric morphisms between elementa= ry toposes, with codomain S. In particular It often becomes necessary to con= sider change of base. The terminology is well adapted to the consideration o= f certain distinguished sub 2-categories of Top_S - for instance LTop_S deno= tes that whose objects are geometric morphisms with codomain S and a locally= connected elementary topos. I hope this is useful to you.=20 Cordial regards, Marta > On Dec 2, 2016, at 5:20 AM, Steve Vickers wrot= e: >=20 > If "Grothendieck topos" means bounded geometric morphism into a given > base S, then by allowing the base to vary we can get a 2-category GTop > of Grothendieck toposes, fibred over some form of ETop (elementary > toposes). This is because pseudopullbacks of bounded geometric morphisms > along arbitrary geometric morphisms exist and are still bounded. (I say > "some form of" ETop because it may be better to restrict the 2-cells > downstairs to be isos, even though we certainly don't want to do the > same upstairs. Also nnos are needed if classifying toposes are to exist.) >=20 > Has anyone worked on that particular combination of bounded and unbounded?= >=20 > Steve Vickers. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]