Alternative approach to Stone duality

Alternative approach to Stone duality

[tkenney]

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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Re: Alternative approach to Stone duality

[Graham Manuell]

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
>

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]