categories: Re: Abstract Stone Duality - manifesto and draft pape r

[S.J.Vickers@open.ac.uk]

> In particular,
> 	"Subspaces in Abstract Stone Duality"
> is the successor to
> 	"Sober Spaces and Continuations",
> which I advertised on 1 September.
> 
> It shows how any category with an exponentiable object Sigma can be
> completed to one in which the adjunction Sigma^(-) -| Sigma^(-) is
monadic.

Should this be exponentiating rather than exponentiable? That was my reading
of the version of the paper that I saw - what's postulated is the ability to
form Sigma^X for every X rather than X^Sigma.

Incidentally, this points up the fact that, as an approach to topology, ASD
limits itself to locally compact spaces (or - let us focus on - locales). I
know Paul has his own justifications for this limitation.

However, some of the features Paul mentioned actually work more generally.
For instance, I have recently found that there is a monad PP on the category
Loc of all locales that agrees with Sigma^(Sigma^(-)) on the locally compact
ones (Sigma = Sierpinski locale). Then if you define Coloc ("colocales") to
be the opposite of the category of PP-algebras you get an adjunction
Sigma^(-) -| Sigma^(-) analogous to Paul's but between Loc and Coloc. PP is
the composite of the upper and lower powerlocales and the construction is
mentioned briefly in my paper with Peter Johnstone, "Preframe presentations
present".

Steve Vickers.