Re: Communes paper, schismatic objects

[Todd Trimble <trimble1@optonline.net>]

Thanks, Vaughan, for clearing that up. I've only been consistently tuning in
to this list recently, and so I may have missed the last time you made this
point.

I'll take this opportunity to re-iterate a point I was trying to make which
may have gotten lost in this side discussion: that IMO the words
"schizophrenic" (and "schismatic" for that matter) are not really all that
apt, even if we put aside Tom Leinster's concern about perpetuating popular
misconceptions about a psychiatric term. I'll only say this one more time,
because I understand that many readers are tired of terminological debates
(I usually quickly get tired of them too).

In the case of classical Stone duality, we have a span of functors CH <--
BoolCH --> Bool, where 2 is a Boolean algebra object in the category of
compact Hausdorff spaces, or equivalently a "compact Hausdorff space object"
in the category of Boolean algebras (where a compact Hausdorff object can be
defined algebraically in any category with small products as a
product-preserving functor from the large infinitary Lawvere theory whose
operations are parametrized by ultrafilters).

I proposed "ambimorphic" to describe such an object where we have two
commutatively interacting structures, and here it is immediate from
algebraic theory nonsense that the ambimorphic object 2 induces the two
sides hom(-, 2)^{op}: CH --> Bool^{op}  and  hom(-, 2): Bool^{op} --> CH  of
an adjoint pair leading up to Stone duality. The natural "home" of 2 from
this point of view is in Bool(CH) = CH(Bool). (Not in CH or Bool, because
these two senses of 2 do not match up under the equivalence StoneSpace^{op}
~ Bool, as Vaughan has pointed out.)

Of course I understand where the expression "schizophrenic" comes from: in
our running example we can push 2 down either to CH or to Bool, and from
that point of view 2 is considered as having a kind of "split personality"
(sorry, Tom). But that's sort of a funny way of thinking about it: those
personalities are perfectly and harmoniously united in the home Bool(CH).
It's as if we were to think of the left arm of the span as split from the
right arm, but it's a little odd to contemplate two arms as "split" from
each other if there's a body in the middle connecting them and the two work
together.

For this reason I consider "ambimorphic" as a far more apt term for the
general situation (and it seems to pass some of Eduardo's criteria as well).
If you (the plural "you", not you Vaughan) don't want to use it, fine, or if
you think battling against "schizophrenia" is a losing battle, that's
obviously your prerogative. I for my part will continue using "ambimorphic",
and obviously would be pleased if others began to adopt that term as well.

Todd

----- Original Message -----
From: "Vaughan Pratt" <pratt@cs.stanford.edu>
To: "Categories list" <categories@mta.ca>
Sent: Friday, November 05, 2010 4:00 PM
Subject: categories: Re: Communes paper, schismatic objects


> > 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.
>

...


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