From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9732 Path: news.gmane.org!.POSTED!not-for-mail From: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= Newsgroups: gmane.science.mathematics.categories Subject: RE: sheaves on localic groupoids Date: Sat, 20 Oct 2018 21:24:58 +0000 Message-ID: References: Reply-To: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1540142561 31591 195.159.176.226 (21 Oct 2018 17:22:41 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Sun, 21 Oct 2018 17:22:41 +0000 (UTC) To: John Baez , categories Original-X-From: majordomo@mlist.mta.ca Sun Oct 21 19:22:36 2018 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from [198.164.44.40] (helo=smtp2.mta.ca) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1gEHQx-00085P-A8 for gsmc-categories@m.gmane.org; Sun, 21 Oct 2018 19:22:35 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:56532) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1gEHT5-0000N3-WB; Sun, 21 Oct 2018 14:24:48 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1gEHRx-0003R4-9x for categories-list@mlist.mta.ca; Sun, 21 Oct 2018 14:23:37 -0300 In-Reply-To: Accept-Language: en-US, en-CA Content-Language: en-US Precedence: bulk X-Broken-Reverse-DNS: no host name found for IP address 198.164.44.40 Xref: news.gmane.org gmane.science.mathematics.categories:9732 Archived-At: Dear John, Eduardo and Simon,=0A= =0A= A few comments and questions.=0A= =0A= If G is a connected localic group, then every continuous action of G on a s= et is trivial.=0A= Hence the topos of G-sets contains no information about G.=0A= This is annoying.=0A= Of course, we could replace G-sets by G-locales (=3D actions of G on locale= s). =0A= But the category of G-locales is not a topos in general.=0A= Should we consider the gross-topos of sheaves on the category of G-locales = ?=0A= What should be the Grothendieck topology?=0A= What are the applications ?=0A= (the applications may guide the developement of a theory).=0A= =0A= I do not have a satisfactory answer to these questions .=0A= =0A= Remark: There are plenty of connected Lie groups.=0A= Equivariant homotopy theory is an important branch of topology.=0A= =0A= Best wishes,=0A= Andr=E9=0A= =0A= =0A= =0A= =0A= =0A= =0A= =0A= =0A= =0A= You are considering the topos T of equivariant G-sheaves on a localic group= oid G.=0A= In the case where G is a group, T is the topos of continuous G-sets.=0A= =0A= =0A= =0A= =0A= =0A= =0A= Under which conditions is the category of sheaves on G a connected Grothend= ieck topos?=0A= =0A= =0A= =0A= =0A= =0A= ________________________________________=0A= From: John Baez [baez@math.ucr.edu]=0A= Sent: Wednesday, October 17, 2018 10:12 PM=0A= To: categories=0A= Subject: categories: sheaves on localic groupoids=0A= =0A= Dear Categorists -=0A= =0A= Joyal and Tierney proved that any Grothendieck topos is equivalent to the= =0A= category of sheaves on a localic groupoid. I gather that we can take this= =0A= localic groupoid to have a single object iff the Grothendieck topos is=0A= connected, atomic, and has a point. In this case the topos can also be= =0A= seen as the category of continuous actions of a localic group on (discrete)= =0A= sets.=0A= =0A= I'm curious about how these three conditions combine to get the job done.= =0A= So suppose G is a localic groupoid.=0A= =0A= Under which conditions is the category of sheaves on G a connected=0A= Grothendieck topos?=0A= =0A= Under which conditions is the category of sheaves on G an atomic=0A= Grothendieck topos?=0A= =0A= Under which conditions is the category of sheaves on G a Grothendieck topos= =0A= with a point?=0A= =0A= (Maybe we should interpret "with a point" as an extra structure on G rather= =0A= than a mere extra property; I don't know how much this matters.)=0A= =0A= Best,=0A= jb=0A= =0A= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]