Re: Communes paper, schismatic objects

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

  > 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).

Good point, sorry about that.  I tend to picture all these things as
embedded in much larger categories and somehow managed to overlook the
fence that entitled both you and Peter to recycle Sierpinski's nickname.

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

Let me answer these at the end, while hastening to offer my apologies
now in case I said anything that might have appeared critical of your
point that the dualizing object in both CH and Bool has two elements,
which is perfectly true.

Earlier I had been making a different point, that in general duality
interchanges nonisomorphic objects.  When I saw your example I seized on
it as a perfect illustration of my point.  Again I'm sorry if it seemed
I was trying to replace your point with mine, I merely wanted to enlarge
on it with an additional point.

  > 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).

For any category C, nothing to do with *-autonomy, one writes G and K
for objects for which C(G,-) and C(-,K) are faithful.  In both varieties
CH and Bool, G is the (free algebra on one) generator while K is the
corresponding cogenerator, and this remains so in Stone (~ Bool^op) as a
full subcategory of CH retaining G and K.

Since the infinite parts of these categories don't bear on the
discussion, for convenience here I'll take all these names to denote
only their finite parts, making CH, Stone, and Set equivalent, and dual
to Bool.  This avoids complications with tensor products (there may be
none for all I know but this way is simple).  Pedagogically speaking
FinSet and FinBool make a great starter kit for duality, while if you're
a finite model theorist, who could ask for anything more?

In Set etc., G is the unit for the cartesian tensor, justifying calling
it I there if not for Bool ("morally" a tensor unit at least).  And K is
systematically used as the dualizing object, justifying calling it _|_
even in non-self-dual categories.

  > there is a contravariant hom
  > hom_{CH}(-, 2): CH^{op} --> Set
  > which lifts to Bool through the underlying functor Bool --> Set.

Yes, and when so lifted it interchanges I and _|_.

But being contravariant it also reverses the arrows from I to _|_, which
then end up being the arrows from I to _|_ again.

In all these categories the arrows from I to _|_ are the elements of _|_.

This is my preferred account of the dual identities of I and _|_, which
can be traced ultimately to the fact that duality does not preserve
either I or _|_, instead it preserves the arrows from I to _|_.

So when one says 2 is schizophrenic, ambimorphic, schismatic, or has a
double identity, one is really remarking on the invariance of "it's"
elements, of which there are two.  They reside neither in I nor _|_ but
in C(I,_|_), and if one takes one's frame of reference to be those
arrows then duality leaves them untouched while interchanging their
domain and codomain.

What changes with this interchange is not the elements of _|_ but those
of I, of which there are respectively 1 and 4.

Your point concerned the elements of _|_, mine the elements of I.  There
is no inconsistency, you were not confused, and you do not need to
clarify what you wrote, rather I did, which I hope I've done.

Vaughan

On 11/4/2010 11:42 PM, Todd Trimble wrote:
> 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
>

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