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