From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9550 Path: news.gmane.org!.POSTED!not-for-mail From: Matias M Newsgroups: gmane.science.mathematics.categories Subject: Re: Reflection to 0-dimensional locales Date: Wed, 14 Feb 2018 16:06:45 -0300 Message-ID: References: <5D815D7C26A24888833B8478A002DE64@ACERi3> Reply-To: Matias M 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 1518648464 9931 195.159.176.226 (14 Feb 2018 22:47:44 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Wed, 14 Feb 2018 22:47:44 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Wed Feb 14 23:47:40 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 1em5ps-0001oP-5t for gsmc-categories@m.gmane.org; Wed, 14 Feb 2018 23:47:32 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:42499) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1em5rG-0002Ib-7w; Wed, 14 Feb 2018 18:48:58 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1em5qG-0004oA-Mr for categories-list@mlist.mta.ca; Wed, 14 Feb 2018 18:47:56 -0400 In-Reply-To: <51180F2A7C24424DAB19B751068688C5@ACERi3> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9550 Archived-At: Dear colleagues, this is to clarify my references to one of George Janelidze's messages in this thread and to add a couple of comments about preservation of finite products by pizero. 2018-02-11 18:38 GMT-03:00 George Janelidze : > My question above is mentioned (among many other things) in the message > of Matias Menni posted on February 8, although it is not clear to me > whether > or not Matias already knew then that the answer to it is negative. I did not know. 2018-02-11 18:38 GMT-03:00 George Janelidze : > I also don't understand what exactly does Matias mean by asking whether o= r > not the > inclusion functor > 0-Dimensional locales--->Locales > "is the result of a variant of Bill's construction (using an exponentiati= ng > object and a `good' factorization system)". Apologies, I inserted my question in the wrong place. What I meant to ask is the following. Let L : Locales ---> Locales be the monad determined by the inclusion 0-Dimensional locales--->Locales. Is it the case that the unit X ---> L X is the first component of a factorization X ---> L X ---> 2^(2^X) of the canonical map to a double dualization, where 2 is a suitable exponentiating object in Locales? 2018-02-11 18:38 GMT-03:00 George Janelidze : > Note that Matias speaks of > preservation of finite products while the reflection > Locales--->0-Dimensional locales > does not preserve them. > I did not mean to suggest that a positive answer to my question would entail an answer to George's question. I simply wanted to understand the construction of L : Locales ---> Locales in terms that I find more familiar. (The finite-product preservation result I mentioned is not applicable to Locales.) 2018-02-11 18:38 GMT-03:00 George Janelidze : > Note also the big (and well known) difference between semi-left-exactness > and preservation of finite products: for every locally connected topos E > with coproducts, the functor > Pizero : E--->Sets > is a semi-left-exact reflection - but if it were always finite product > preserving, then, say, homotopy theory would not exist (all fundamental > groups of 'good' spaces would be trivial)... > Concerning local connectedness, finite-product preservation of pizero and even some homotopy theory, let me add a couple of related remarks. The reason I looked for a finite-product preserving pizero in the Tbilisi paper comes from "Axiom 1" in Lawvere's "Categories of spaces ..." paper, TAC Reprint 9, which postulates an essential geometric morphism Gamma: E ---> S such that the leftmost adjoint Gamma_! : E ---> S preserves finite products. Then Lawvere says: "The axiom is necessary for the naive construction of the homotopic passage from quantity to quality; namely, it insures that (not only Gamma_* but also) Gamma_! is a closed functor thus inducing a second way of associating an S-enriched category to each E-enriched category [...] For example, E itself as an E-enriched category gives rise to a homotopy category in which [E](X, Y )=3DGamma_!(Y^X)." Axiomatic Cohesion (TAC 2007) continued the work in "Categories of spaces ..." introducing, among other things, the axiom: (Nullstellensatz) The canonical Gamma_* ---> Gamma_! is epic which captures the intuition that every piece has a point. The above is one source of inspiration for Johnstone's TAC 2011 paper where he studies locally connected geometric morphisms p: E ---> S such that the leftmost adjoint p_! : E ---> S preserves finite products. Among other things he shows that: if p: E ---> S is lc, hyperconnected and local then p_! preserves finite products. It also follows from Johnstone's results that: if p: E ---> S is lc and local then, p is hyperconnected iff the Nullstellensatz holds. A rough conclusion that one may arrive at is that: if pizero does not preserve finite products then there is some connected space without points. Apologies again for the lack of clarity in my previous message. I look forward to see how the different aspects of the 'pizero idea' evolve and I am glad that Steve Vickers' message brought them up. All the best, Mat=C3=ADas 2018-02-11 18:38 GMT-03:00 George Janelidze : > Dear Colleagues, > > Concerning Steve's messages started with "Topos theory for spaces of > connected components" sent on February 4 and comments to them, I am not > sure > I understand what was the end of the story, but I would like to comment o= n > a > part of the story related to my question, in the 'chronological' order: > > 1. I think on February 6 I have written three messages, the first of whic= h > was not posted (which is reasonable since my second message contained its > copy). In the third message, whose subject was "Reflection to 0-dimension= al > locales", I wrote that the answer to my question > > "Is the reflection > Locales--->0-Dimensional locales > semi-left-exact?" > > is NO. I also wrote that I know this from Graham Manuell, who explained t= o > me that this follows from the existence of a counter-example, due to I. > Kriz, presented in the book "Frames and Locales" by J. Picado and A. Pult= r > (Pages 260-266). And I asked if it is possible to construct a simpler > counter-example. > > 2. My question above is mentioned (among many other things) in the messag= e > of Matias Menni posted on February 8, although it is not clear to me > whether > or not Matias already knew then that the answer to it is negative. I also > don't understand what exactly does Matias mean by asking whether or not t= he > inclusion functor > > 0-Dimensional locales--->Locales > > "is the result of a variant of Bill's construction (using an exponentiati= ng > object and a `good' factorization system)". Note that Matias speaks of > preservation of finite products while the reflection > > Locales--->0-Dimensional locales > > does not preserve them. > > Note also the big (and well known) difference between semi-left-exactness > and preservation of finite products: for every locally connected topos E > with coproducts, the functor > > Pizero : E--->Sets > > is a semi-left-exact reflection - but if it were always finite product > preserving, then, say, homotopy theory would not exist (all fundamental > groups of 'good' spaces would be trivial)... > > 3. Andrej Bauer, in his message of February 9, also mentions my question > and > says: > > Can Example 1 in >> >> https://dml.cz/bitstream/handle/10338.dmlcz/119250/Commentat >> MathUnivCarolRetro_42-2001-2_13.pdf >> >> be put to some use to answer the question negatively? It shows that >> the zero-dimensional reflection in topological spaces does not >> preserve finite products. The example uses fairly nice subspaces of R >> and R^2. >> > > I think the topological version does not help; note also that the > non-semi-left-exactness there was known for a very long time. > > Summarizing, I thank again Graham for his help, and Matias and Andrej for > their comments, but let me insist: the counter-example of Kriz is so > complicated... can someone construct an easier one? > > George Janelidze > > Disclaimer - University of Cape Town This email is subject to UCT policie= s > and email disclaimer published on our website at > http://www.uct.ac.za/main/email-disclaimer or obtainable from +27 21 650 > 9111. If this email is not related to the business of UCT, it is sent by > the sender in an individual capacity. Please report security incidents or > abuse via https://csirt.uct.ac.za/page/report-an-incident.php. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]