From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9731 Path: news.gmane.org!.POSTED!not-for-mail From: Christopher Townsend Newsgroups: gmane.science.mathematics.categories Subject: Re: Re: sheaves on localic groupoids Date: Sat, 20 Oct 2018 18:21:44 +0100 Message-ID: References: Reply-To: Christopher Townsend NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 (1.0) Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1540142507 27480 195.159.176.226 (21 Oct 2018 17:21:47 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Sun, 21 Oct 2018 17:21:47 +0000 (UTC) Cc: John Baez , categories To: "Eduardo J. Dubuc" Original-X-From: majordomo@mlist.mta.ca Sun Oct 21 19:21:42 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 1gEHQ2-0006xS-Ks for gsmc-categories@m.gmane.org; Sun, 21 Oct 2018 19:21:38 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:56512) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1gEHRH-0000FH-Rh; Sun, 21 Oct 2018 14:22:55 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1gEHQD-0003Pr-LS for categories-list@mlist.mta.ca; Sun, 21 Oct 2018 14:21:49 -0300 In-Reply-To: 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:9731 Archived-At: Hi Eduardo & all If F is a Grothendieck topos then there is a bounded geometric morphism p:F-= >Set. Let B be a bound for p, then there is a locale of surjections [N->>B],= where N is the natural numbers. This is a locale over F that can be mapped t= o a locale over Set (apply the direct image of p to the frame of opens of [N= ->>B]). You get the object locale of the localic groupoid that =E2=80=98repr= esents=E2=80=99 F via Joyal and Tierney. The morphism locale is the image of= [N->>B]x[N->>B].=20 Not sure if that sheds much light on what the localic groupoid really is, bu= t it is one way of constructing it.=20 I=E2=80=99d like to add that any geometric morphism p:F->E, we know, gives r= ise to a =E2=80=98localic=E2=80=99 adjunction between locales over F and lo= cales over E; the right adjoint being effectively pullback in the category o= f toposes. If you can find a locale W over F such that W->1 is an effective d= escent morphism and the slice of this localic adjunction at W is an equivale= nce then in fact the adjunction is a connected components adjunction of a lo= calic groupoid in E; this can be shown by application of Janelidze=E2=80=99s= categorical Galois Theorem (the result holds at the generality of cartesian= categories - it=E2=80=99s quite straightforward). If you know that the loca= lic adjunction is a connected components adjunction it is easy to get the Jo= yal and Tierney result by restricting to local homeomorphisms. But of course you should complain that it must be hard to show that any loca= lic adjunction, sliced at W, is an equivalence. In fact even at the general l= evel of cartesian categories it is not that hard once we recall that the loc= alic adjunction is stably Frobenius (something that is in the original Joyal= and Tierney paper but not dwelt on). For a Frobenius adjunction to be an eq= uivalence it only needs to have its left adjoint preserve 1 and its unit to b= e a regular monomorphism. At the slice, 1 is automatically preserved, so we a= re just left checking that the (sliced) unit is a regular monomorphism. It t= urns out that this is so for W=3D[N->>B] precisely when B is a bound, effect= ively completing a proof of the Joyal and Tierney representation theorem. I hope it is OK that I=E2=80=99ve used this thread to effectively advertise s= ome work(*) that I did a few years ago. Kind regards, Christopher (*) A localic proof of the localic groupoid representation of Grothendieck t= oposes Proc. Amer. Math. Soc. 142 (2014), 859-866 > 19 Oct 2018, at 19:57, Eduardo J. Dubuc wrote: >=20 >=20 > A time ago when I was working on the subject I was also very curious > about the same (or related) questions of Johon Baez. > More concretely, if you take a pointless connected atomic topos (we know > they are): >=20 > How or which is the localic point that Joyal-Tierney using change of > base take out of the hat ? >=20 > How or which is the localic groupoid of Joyal-Tierney ? >=20 >> On 17/10/18 23:12, John Baez wrote: >> Dear Categorists - >>=20 >> 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 th= is >> 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 b= e >> seen as the category of continuous actions of a localic group on (discret= e) >> sets. >>=20 >> I'm curious about how these three conditions combine to get the job done.= >> So suppose G is a localic groupoid. >>=20 >> Under which conditions is the category of sheaves on G a connected >> Grothendieck topos? >>=20 >> Under which conditions is the category of sheaves on G an atomic >> Grothendieck topos? >>=20 >> Under which conditions is the category of sheaves on G a Grothendieck top= os >> with a point? >>=20 >> (Maybe we should interpret "with a point" as an extra structure on G rath= er >> than a mere extra property; I don't know how much this matters.) >>=20 >> Best, >> jb >>=20 >>=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]