From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9533 Path: news.gmane.org!.POSTED!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories Subject: Re: Topos theory for spaces of connected components Date: Mon, 5 Feb 2018 13:07:02 -0500 (EST) Message-ID: Reply-To: Marta Bunge NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1517881238 27562 195.159.176.226 (6 Feb 2018 01:40:38 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Tue, 6 Feb 2018 01:40:38 +0000 (UTC) To: categories@mta.ca, s.j.vickers@cs.bham.ac.uk Original-X-From: majordomo@mlist.mta.ca Tue Feb 06 02:40:34 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 1eisFJ-0006fF-AK for gsmc-categories@m.gmane.org; Tue, 06 Feb 2018 02:40:29 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:38044) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1eisGx-0006om-LG; Mon, 05 Feb 2018 21:42:11 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1eisG3-0002DE-HU for categories-list@mlist.mta.ca; Mon, 05 Feb 2018 21:41:15 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9533 Archived-At: Dear Steve, This is response to your message reproduced below.=20 I am aware of Johnstone=E2=80=99s results on the lower bagdomain. However, = both the symmetric monad M and the lower bagdomain monad B_L on BTop_S are = =E2=80=9Con the same side=E2=80=9D as the lower power locale monad P_L on L= oc_S, and the latter is the localic reflection for both. The upper power lo= cale monad P_U on Loc_S is =E2=80=9Con the other side=E2=80=9D, in a sense = that is explained in my =E2=80=98Pitts monads paper=E2=80=9D.In it I deduce= effective lax descent theorems in a general setting of what I call "Pitts = KZ-monads" and "Pitts co-KZ-monads" on a =E2=80=9C2-category of spaces=E2= =80=9D.=20 In the case of M on BTop_S, it is the S-essential surjective geometric morp= hisms that are shown to be of lax effective descent (a result originally du= e to Andy Pitts). In the case of P_L on Loc-S it is the open surjections of= locales that are shown to be of lax effective descent (a result originally= due to Joyal-Tierney). Both M and P_L are instances of "Pitts KZ-monads". = Now, P_U on Loc_S is instead an instance of a" Pitts co-KZ-monad" and the = result recovered from my general setting is that proper surjections of loca= les are of effective lax descent (a result originally due to Jaapie Vermeul= en). What I seek is a co-KZ-monad N (or perhaps B_U) on BTop_S for which my= general theorem would give me that relatively tidy surjections of toposes = are of effective lax descent (a result due to I. Moerdijk and J.C.C.Vermeul= en).=20 In my Pitts paper there is another consequence of the general theorem prove= d therein and it is that coherent surjections between coherent toposes are = of effective lax descent (a result proven by different methods and by sever= al people, such asM. Zawadowsky 1995, D.Ballard and W.Boshuck 1998, and I.= Moerdijk and J.C.C.Vermeulen 1994,thus establishing a conjecture of Pitts 1= 985 (in the Cambridge Conference whose slides you have requested to Andy). = It is of interest for what we are discussing to point out that the =E2=80= =9Ccoherent monad C=E2=80=9D that I use therein to deduce the latter from m= y general theorem is a Pitts co-KZ-monad, hence on the =E2=80=9Csame side= =E2=80=9D as P_U for Loc_S. For a coherent topos E, the coherent monad C(E= ) applied to it classifies pretopos morphisms E_{coh} =E2=80=94> S. where E= _{coh} is the full subcategory of E of coherent objects with the topology o= f finite coverings. This theorem is perhaps all I can get in my setting whe= n searching for the still elusive N or B_U but I have not given up yet.=20 Also in my 2015 Pitts paper there are characterizations of the algebras for= a Pitts KZ-monad M (dually for a Pitts co-KZ-monad N) as the "stably M-com= plete objects" ("stably N-complete objects"), where the former is stated in= terms of pointwise left Kan extensions along M-maps, and the latter in ter= ms of pointwise right Kan extensions along N-maps. These notions owe much t= o the work of M, Escardo, in particular to his 1998 "Properly injective spa= ces and function spaces=E2=80=9D.=20 I will say more when i know more myself. Thanks very much for your pointers= . I will most certainly look into them even if I do not at the moment think= they are what I need.=20 Best regards, Marta ________________________________________________ =EE=9C=92 From: Steve Vickers Sent: February 5, 2018 9:03 AM To: martabunge@hotmail.com Cc: categories@mta.ca Subject: Re: categories: Topos theory for spaces of connected components =20 Dear Marta, Johnstone showed that B_L(X) is a partial product of X against the "generic= local homeomorphism", a geometric morphism p from the classifier of pointe= d objects to the object classifier. A point of B_L(X) is a family of points= of X, indexed by elements of a set. He also proposed other partial products, for example those against the gene= ric entire map, which goes to the classifier for Boolean algebras from the = classifier of Boolean algebras equipped with prime filter. Wouldn't that be= your B_U? A point would be a family of points of X, indexed by points of a= Stone space. Steve. 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