Re: Communes paper, schismatic objects

[Vaughan Pratt <pratt@cs.stanford.edu>]

On 11/1/2010 4:52 PM, Todd Trimble wrote:
> It is both at once: a Boolean algebra
> object in the category of compact Hausdorff spaces, and we need both
> forms in the same body so that we can say hom_{CH}(-, 2) is a Boolean
> algebra valued functor.

Your example perfectly illustrates my point about I and _|_ being
distinct but dual objects.  In CH, I = 1 and _|_ = 1+1 (I'm assuming by
2 you mean 1+1 rather than the Sierpinski space).  The contravariant
Boolean algebra valued functor hom_{CH}(-, 1+1): CH^op --> Bool sends I
and _|_ in CH to respectively _|_ and I in Bool.  In both categories I
is the free object on one generator and as such a generator and the
tensor unit, while its dual _|_ is cofree, a cogenerator, and the unit
for par (to the extent tensor and par are defined in each category --
they become fully defined in a common self-dual unification that
covariantly embeds both categories, namely Chu(Set,2)).

Understood via the above functor as Boolean algebra objects, in CH I = 1
and _|_ = 1+1 are respectively the 2-element and 4-element Boolean
algebras, while in Bool these are interchanged: I has 4 elements (the
free Boolean algebra on one generator) and _|_ has 2.

In both categories _|_ is the dualizing object.  I would not say that
the 2-element and 4-element Boolean algebras are the same.  In my book
they are distinct.

Vaughan


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