Re: Complete atomic Boolean algebra: Reference?

Re: Complete atomic Boolean algebra: Reference?

[jvoosten]

This is of course folklore. I believe
there is a proof (essentially) in
Johnstone's Stone Spaces.

I have written out a proof for students in
chapter 1 of my "Basic Category Theory"
notes; see

http://www.math.uu.nl/people/jvoosten/onderwijs.html

Jaap van Oosten

Re: Complete atomic Boolean algebra: Reference?

[Dusko Pavlovic]

isn't the correspondence of sets and caBa the "discrete part" of the
stone duality?

the other monad prof wyler mentions appears, i think, in e. manes'
thesis, and in his 1976 book "algebraic theories", perhaps as a step
towards deriving the monad for compact hausdorff spaces.

all the best,
-- dusko pavlovic

jvoosten@math.uu.nl wrote:

> This is of course folklore. I believe
> there is a proof (essentially) in
> Johnstone's Stone Spaces.
>
> I have written out a proof for students in
> chapter 1 of my "Basic Category Theory"
> notes; see
>
> http://www.math.uu.nl/people/jvoosten/onderwijs.html
>
> Jaap van Oosten

Complete atomic Boolean algebra: Reference?

[Oswald Wyler]

Every reader of this post probably knows that the algebras for the monad
on sets induced by the self-adjoint contravariant powerset functor are
the complete atomic Boolean algebras, with maps preserving all infima
and all suprema as morphisms.  I know how to prove this without much
trouble, but I have not been able to find a proof of this fact, or even
a good reference to such a proof, in the literature available to me
at my present location (which is essentially what I have at home).
If you know such a reference, please e-mail it to me at owyler@nqi.net.
Related question.  The functor on sets which assigns to every set X the
set of increasing subsets of PX is the functor part of a monad, with
completely distributive complete lattices as algebras.  Again, I have
a proof but no references.