Re: Communes paper, schismatic objects

[Todd Trimble <trimble1@optonline.net>]

Dear Vaughan,

I don't understand why you mailed that explanation. Did you
think I was confused?  Do I need to clarify what I wrote?

The two-element set carries a Boolean algebra structure and
a compact Hausdorff structure, and the two structures commute.
The two-element set equipped with those two structures is what
I was calling 2 in my prior post.

You know the relevant material perfectly well, but a suitable
reference for what I was referring to is Johnstone's Stone
Spaces. The original Stone duality takes this very structure 2
as a 'schizophrenic' object, as discussed on p. 260, example
(e). Peter also calls it "2". I didn't think anyone here would
find that notation confusing.

For example, it seemed unlikely to me that anyone here would
confuse this with the Sierpinski space 2 (which isn't compact
Hausdorff after all).

The underlying Boolean algebra of this structure is, strangely
enough, conventionally called 2, and there is a contravariant hom

hom_{Bool}(-, 2): Bool^{op} --> Set

which lifts to CH through the underlying functor CH --> Set,
according to the well-known Stone duality (where the lift factors
through the full subcategory of Stone spaces).

The underlying compact Hausdorff space of this structure is
again, strangely enough, also conventionally called 2, and there
is a contravariant hom

hom_{CH}(-, 2): CH^{op} --> Set

which lifts to Bool through the underlying functor Bool --> Set.

The notations I and _|_ which you brought into this discussion
are perhaps best understood in the context of *-autonomous
categories, for example Chu(Set, 2). (That last mention of 2
refers to a 2-element set. Throughout this discussion, wherever
I wrote "2", it refers to a 2-element set, possibly with extra
structure as appropriate.) You seemed to think I was guilty of
confusing I and _|_, but of course I didn't even mention them,
and actually I do understand the difference between the units
I and _|_ for the tensor and par in a *-autonomous category.
I hope you will take my word for that.

Best regards,

Todd

----- Original Message -----
From: "Vaughan Pratt" <pratt@cs.stanford.edu>
To: "Categories list" <categories@mta.ca>
Sent: Wednesday, November 03, 2010 6:35 PM
Subject: categories: Re: Communes paper, schismatic objects


>
> 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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