From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9535 Path: news.gmane.org!.POSTED!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: Topos theory for spaces of connected components Date: Tue, 6 Feb 2018 09:19:50 +0000 Message-ID: References: <244986425.357598.1517854022932.JavaMail.zimbra@math.mcgill.ca> Reply-To: Steve Vickers 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 1517922723 16935 195.159.176.226 (6 Feb 2018 13:12:03 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Tue, 6 Feb 2018 13:12:03 +0000 (UTC) Cc: categories@mta.ca To: bunge@math.mcgill.ca Original-X-From: majordomo@mlist.mta.ca Tue Feb 06 14:11:58 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 1ej32A-0002wC-3o for gsmc-categories@m.gmane.org; Tue, 06 Feb 2018 14:11:38 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:38226) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1ej33x-0000NE-81; Tue, 06 Feb 2018 09:13:29 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ej332-0004xj-Q7 for categories-list@mlist.mta.ca; Tue, 06 Feb 2018 09:12:32 -0400 In-Reply-To: <244986425.357598.1517854022932.JavaMail.zimbra@math.mcgill.ca> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9535 Archived-At: Dear Marta, Here's my thinking on connected components. For M, the paradigm example for how to get a point of MX (a cosheaf, or dist= ribution) is to take locally connected space Y with map p: Y -> X, and then t= o each sheaf U over X assign the set of connected components of p*U. This gi= ves a covariant functor from SX to Set, and it preserves colimits. If X is a= n ungeneralized space, then it suffices to do that for opens U, and the exte= nsion to sheaves follows. Your theory of complete spreads shows that that pa= radigm example is in fact general. The extreme case of p is when X is itself locally connected and we can take p= to be the identity. The corresponding cosheaf is terminal in a strong sense= : as global point of MX it provides a right adjoint to the map MX -> 1. The u= nit of the adjunction provides a unique morphism from the generic cosheaf to= the terminal one. If X is exponentiable, then (always? In favourable cases?) the cosheaf as de= scribed above can be got by taking points for a map R^X -> R, where R is (fo= llowing your notation) the object classifier. This points out Lawvere's anal= ogy with integration, where R would be the real line. Then just as Riesz pic= ks out the linear functionals as the distributions, we are interested in the= colimit-preserving ones. In the above account, the role of local connectedness is to ensure that the c= onnected components genuinely do form a set, a discrete space. What happens i= f we look for a Stone space instead? Here is my conjecture. 1. For ungeneralized X we should be looking for a Stone space of connected c= omponents of p*U for each _closed_ U. Y will need a suitable condition (stro= ngly compact?) as analogue of local compactness. (By Stone duality that coul= d also be expressed by assigning (covariantly) a Boolean algebra to each ope= n.) 2. Noting that a closed embedding is fibrewise Stone, that assignment will e= xtend to U an arbitrary fibrewise Stone (entire) bundle over X - that is to s= ay, by Stone duality and contravariantly, a sheaf of Boolean algebras. 3. For generalized X that will provide our Stone notion of cosheaf. The assi= gnment from entire bundles to Stone spaces should preserve finite colimits a= nd cofiltered limits. There's an obvious technical hurdle of how to express t= hat directly in terms of sheaves instead of entire bundles. 4. If X is exponentiable then this time, by Stone duality, we are looking fo= r maps [BA]^X -> [BA] where [BA] is the classifier for Boolean algebras. The= y must preserve filtered colimits (automatic for maps) and finite limits. NX= would exist for arbitrary X, and classify those maps.=20 Obviously there's lots to go wrong there, but do you think your coherent mon= ad fits any of those points for coherent X? By the way, although I haven't mention the effective lax descent and relativ= ely tidy maps, I am interested in them. They are connected with stable compa= ctness and Priestley duality. All the best, Steve. > On 5 Feb 2018, at 18:07, bunge@math.mcgill.ca wrote: >=20 >=20 > Dear Steve, >=20 > This is response to your message reproduced below.=20 >=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 Loc_S= , and the latter is the localic reflection for both. The upper power locale m= onad 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 effecti= ve 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 >=20 > In the case of M on BTop_S, it is the S-essential surjective geometric mor= phisms 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 l= ocales that are shown to be of lax effective descent (a result originally du= e 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 resul= t recovered from my general setting is that proper surjections of locales ar= e of effective lax descent (a result originally due to Jaapie Vermeulen). Wh= at 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 ef= fective lax descent (a result due to I. Moerdijk and J.C.C.Vermeulen).=20 >=20 > In my Pitts paper there is another consequence of the general theorem prov= ed therein and it is that coherent surjections between coherent toposes are o= f effective lax descent (a result proven by different methods and by several= people, such asM. Zawadowsky 1995, D.Ballard and W.Boshuck 1998, and I.Moe= rdijk and J.C.C.Vermeulen 1994,thus establishing a conjecture of Pitts 1985 (= in the Cambridge Conference whose slides you have requested to Andy). It is o= f interest for what we are discussing to point out that the =E2=80=9Ccoheren= t monad C=E2=80=9D that I use therein to deduce the latter from my general t= heorem 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 fu= ll subcategory of E of coherent objects with the topology of finite covering= s. This theorem is perhaps all I can get in my setting when searching for th= e still elusive N or B_U but I have not given up yet.=20 >=20 > Also in my 2015 Pitts paper there are characterizations of the algebras fo= r 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 t= erms of pointwise left Kan extensions along M-maps, and the latter in terms o= f pointwise right Kan extensions along N-maps. These notions owe much to the= work of M, Escardo, in particular to his 1998 "Properly injective spaces an= d function spaces=E2=80=9D.=20 >=20 > I will say more when i know more myself. Thanks very much for your pointer= s. I will most certainly look into them even if I do not at the moment think= they are what I need.=20 >=20 >=20 > Best regards, > Marta >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]