Re: Alternative approach to Stone duality

[Graham Manuell <grmanuell@gmail.com>]

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 Aleš 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
<https://arxiv.org/abs/1710.10894>) and also in my MSc thesis "Congruence
frames of frames and κ-frames" at the University of Cape Town. (In this
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 <tkenney@mathstat.dal.ca> wrote:

> Hi.
>
>         Does anyone know if the following perspective on topology has been
> 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 is
> 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
>

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