From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10346 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: Graham Manuell Newsgroups: gmane.science.mathematics.categories Subject: Re: Alternative approach to Stone duality Date: Sat, 12 Dec 2020 10:49:52 +0200 Message-ID: References: Reply-To: Graham Manuell Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="22911"; mail-complaints-to="usenet@ciao.gmane.io" Cc: categories@mta.ca To: tkenney Original-X-From: majordomo@rr.mta.ca Mon Dec 14 04:22:19 2020 Return-path: Envelope-to: gsmc-categories@m.gmane-mx.org Original-Received: from smtp2.mta.ca ([198.164.44.74]) by ciao.gmane.io with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.92) (envelope-from ) id 1koeRG-0005r3-As for gsmc-categories@m.gmane-mx.org; Mon, 14 Dec 2020 04:22:18 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:46120) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1koeKo-00035X-Au; Sun, 13 Dec 2020 23:15:38 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1koeOq-0003Qi-1c for categories-list@rr.mta.ca; Sun, 13 Dec 2020 23:19:48 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10346 Archived-At: Dear Toby, What you call 'drunk spaces' are usually called T_D spaces. The idea of using them in a duality as an alternative to sober spaces is briefly discussed in the first chapter of the book "Frames and Locales" by Jorge Picado and Ale=C5=A1 Pultr and I believe some references can be found there= . I don't believe that they discuss the morphisms much though. As an aside, I think that the 'fibration' you mentioned can also be understood in terms of the Skula topology (though I must say, I'm not convinced it is actually a fibration: how do you lift a morphism where the domain coframe doesn't have enough points?). Recall that the Skula modification of a topological space X is a new topological space with the same points as X and with subbasic opens given by both the open and the closed sets of X. (This shows up in the characterisation of epimorphisms of T_0 spaces, amongst other places.) The Skula modification of X is discrete if and only if X is T_D. The lattice of closed sets of the Skula modification of X equipped with the sublattice of closed sets of X is enough to recover a T_0 space X completely. The fibres of your functor are then given by the different possible Skula topologies for a space with a given lattice of opens. I studied a pointfree variant of this situation in my paper "Strictly zero-dimensional biframes and a characterisation of congruence frames" published in Applied Categorical Structures (arXiv link ) and also in my MSc thesis "Congruence frames of frames and =CE=BA-frames" at the University of Cape Town. (In thi= s analogue we do not have a fibration, but we do have a semitopological/solid functor.) Best regards, Graham On Sat, 12 Dec 2020 at 04:15, tkenney wrote: > Hi. > > Does anyone know if the following perspective on topology has bee= n > studied before (and if so, is there a good reference)? Apologies if I'm > missing something very basic here. > > Let T_0 be the category of T_0 topological spaces and continuous > homomorphisms. We have the usual functor (T_0)^op ---> Coframe (this is > all 1-dimensional, so you can call it Frame if you prefer) sending a > topological space to the coframe of closed sets. This is a faithful > fibration. (It can be extended to arbitrary topological spaces, but isn't > faithful.) Furthermore, all the non-empty fibres are posets with top > elements. These top elements are the sober spaces, and the restriction of > the functor to them is full and has an adjoint, which is the usual > equivalence between sober spaces and spatial locales. > > On the other hand, for a large class of coframes (coframes in > which every element is a sup of elements which are not _equal_ to the > sup of a set of strictly smaller elements), the fibres are complete > boolean algebras. Thus the fibres have bottom elements. These > are topological spaces where for any point x, x is open in the subspace > topology on its closure. Since these spaces are at the bottom of the > boolean algebra with sober spaces at the top, they should presumably be > called "drunk spaces", though this does lead to there being a large class > of spaces which are both sober and drunk. All T_1 spaces are drunk. When > restricted to drunk spaces, the functor is not full. However, its image i= s > a subcategory of Coframe (I think the morphisms in the image are complete > co-Heyting homomorphisms). When we restrict to this subcategory, we get > an equivalence between drunk topological spaces and > completely indecomposable-generated coframes with complete co-Heyting > algebra homomorphisms. > > Does anyone know if this duality between "drunk" spaces and > indecomposably-generated coframes has been studied? > > The motivation here is that the fibration extends to a fibration > from closure spaces to Inf-lattices, and the usual top element adjoint in > this extension is not very interesting, and is on the wrong side for my > purposes, but the restricted equivalence above looks like it covers more > of the cases of interest. > > Regards, > > Toby Kenney > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]