From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9726 Path: news.gmane.org!.POSTED!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: sheaves on localic groupoids Date: Wed, 17 Oct 2018 19:12:23 -0700 Message-ID: Reply-To: John Baez NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" X-Trace: blaine.gmane.org 1539972504 8225 195.159.176.226 (19 Oct 2018 18:08:24 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Fri, 19 Oct 2018 18:08:24 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Fri Oct 19 20:08:20 2018 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1gDZC3-0001zG-KE for gsmc-categories@m.gmane.org; Fri, 19 Oct 2018 20:08:15 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:56053) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1gDZE7-0002kC-Pe; Fri, 19 Oct 2018 15:10:23 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1gDZD5-00073H-D9 for categories-list@mlist.mta.ca; Fri, 19 Oct 2018 15:09:19 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9726 Archived-At: Dear Categorists - Joyal and Tierney proved that any Grothendieck topos is equivalent to the category of sheaves on a localic groupoid. I gather that we can take this localic groupoid to have a single object iff the Grothendieck topos is connected, atomic, and has a point. In this case the topos can also be seen as the category of continuous actions of a localic group on (discrete) sets. I'm curious about how these three conditions combine to get the job done. So suppose G is a localic groupoid. Under which conditions is the category of sheaves on G a connected Grothendieck topos? Under which conditions is the category of sheaves on G an atomic Grothendieck topos? Under which conditions is the category of sheaves on G a Grothendieck topos with a point? (Maybe we should interpret "with a point" as an extra structure on G rather than a mere extra property; I don't know how much this matters.) Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]