Re: Communes paper, schismatic objects

Re: Communes paper, schismatic objects

[Fred E.J. Linton]

Todd Trimble writes, inter alia,

> ... The natural "home" of 2 from
> this point of view is in Bool(CH) = CH(Bool). 

I quite agree that Bool(CH) and CH(Bool) are quite equivalent.
But they are far from *equal*. Each serves as a 'natural "home"'
for 2, and the object 2 in its one home has a radically different
personality -- and interacts with radically different colleagues
-- than in its other. 

Moreover, as shown in Stone Spaces, 2 has rather more natural
homes -- and personalities -- than merely these two :-) , whence
my inclination to speak of its *multiple* personalities (pace Tom L.).

Cheers, -- Fred



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Re: Communes paper, schismatic objects

[Todd Trimble]

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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Re: Communes paper, schismatic objects

[Vaughan Pratt]

  > 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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Re: Communes paper, schismatic objects

[Todd Trimble]

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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Re: Communes paper, schismatic objects

[Vaughan Pratt]

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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Re: Communes paper, schismatic objects

[Todd Trimble]

A couple of things related to recent comments on "schizophrenic".

Vaughan Pratt wrote, with regard to possible alternatives to "schizophrenic"

"So I followed Tom's pointer

     http://ncatlab.org/nlab/show/dualizing+object

linking to a discussion of alternatives, which seemed inconclusive.  Sam
(Staton?) made the point however that even if schizophrenia is not the
appropriate word, schizo is the appropriate prefix, having derived from
the Greek 'split'. "

Although the discussion at the nLab might appear inconclusive, in actual
fact a number of people at the nLab and n-Category Cafe seem to have
provisionally adopted "ambimorphic", which I coined with the intended
meaning, "having both forms". I actually feel that is very appropriate in
practice; for example, in classical Stone duality, it is not enough to say 
the
dualizing object 2 is "split" between being seen as a compact Hausdorff
space and as a Boolean algebra. 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.

With regard to Dusko's recent comments: it's quite understandable that
"political correctness" and endless debates over terminology can become
tiresome. But I'm not sure "political correctness" is quite the angle from
which Tom's objection comes. At the Cafe he brought it up here:

http://golem.ph.utexas.edu/category/2007/01/more_on_duality.html#c007089

(where you can also see the consequent discussion of suggested alternatives)
and the sense I get is that it's not so much about  "protecting the weak"
as it is about wishing not to perpetuate pop misconceptions. But putting
all that aside, perhaps the emphasis on being "split" is not quite accurate
in the first place, or at least should be reconsidered, as I argue above.

Todd

----- Original Message ----- 
From: "Vaughan Pratt" <pratt@cs.stanford.edu>
To: "categories list" <categories@mta.ca>
Sent: Monday, November 01, 2010 1:44 PM
Subject: categories: Communes paper, schismatic objects


A couple of things.  First, I neglected to mention that "Communes via
Yoneda, from an Elementary Perspective," Fundamenta Informaticae 123
(2010) 1–16, DOI 10.3233/FI-2010-315 is about to appear and won't be
findable by Google just yet.  Those interested in seeing it sooner can
find it on my site at

http://boole.stanford.edu/pub/CommunesFundInf2010.pdf

Second, as I said I wasn't passing judgment on the wisdom of avoiding
the term "schzophrenic" but merely pointing out the associated cost,
which needs to be balanced against the harm of any given word.

So I followed Tom's pointer

     http://ncatlab.org/nlab/show/dualizing+object

linking to a discussion of alternatives, which seemed inconclusive.  Sam
(Staton?) made the point however that even if schizophrenia is not the
appropriate word, schizo is the appropriate prefix, having derived from
the Greek "split."

So it is the medical condition that is inappropriately named, namely as
"split madness," with phrenitis and frenzy having a common origin.

With that in mind it occurred to me that "schismatic" might be a
suitable alternative, as providing better continuity with the older
terminology by coming from the same root schizo, but more honestly so
than schizophrenia since in this case there really is a multiple
personality, and moreover there's nothing insane about it.  (And it's a
syllable shorter to boot.)

Third, while it is true that the schismatic object (to give the term a
trial run) is usually observed manifesting its split personality in
different categories, this is not the case in *-autonomous categories
where I and _|_ are the Jekyll and Hyde of the same category.  (I
apologize to readers of this list with either of those surnames.)

In all the examples I'm aware of, the two categories in which the
schismatic object occurs (once in each) admit a common completion to a
*-autonomous category which embeds one object as I and the other as _|_.
    Considering them to be the "same" object found in two categories
misses the contravariance between them, which is brought out more
clearly by this joint completion, where they are clearly not the same
object but a pair of dual objects.

My paper accounts for C.I. Lewis's qualia by viewing them as morphisms
running from I to _|_.  If I and _|_ are rigid (|C(x,x)|=1) as for
Chu(Set,K)), the presence of a morphism from _|_ to I is inconsistent in
the sense that it collapses Hom(I,_|_) to a singleton, since I and _|_
respectively generate and cogenerate.  So in order to have more than one
quale (in the Chu setting) there cannot be any morphism from _|_ to I.
(That was mainly in the nature of background on the neighborhood of I
and _|_.)

Vaughan


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Communes paper, schismatic objects

[Vaughan Pratt]

A couple of things.  First, I neglected to mention that "Communes via 
Yoneda, from an Elementary Perspective," Fundamenta Informaticae 123 
(2010) 1–16, DOI 10.3233/FI-2010-315 is about to appear and won't be 
findable by Google just yet.  Those interested in seeing it sooner can 
find it on my site at

http://boole.stanford.edu/pub/CommunesFundInf2010.pdf

Second, as I said I wasn't passing judgment on the wisdom of avoiding 
the term "schzophrenic" but merely pointing out the associated cost, 
which needs to be balanced against the harm of any given word.

So I followed Tom's pointer

    http://ncatlab.org/nlab/show/dualizing+object

linking to a discussion of alternatives, which seemed inconclusive.  Sam 
(Staton?) made the point however that even if schizophrenia is not the 
appropriate word, schizo is the appropriate prefix, having derived from 
the Greek "split."

So it is the medical condition that is inappropriately named, namely as 
"split madness," with phrenitis and frenzy having a common origin.

With that in mind it occurred to me that "schismatic" might be a 
suitable alternative, as providing better continuity with the older 
terminology by coming from the same root schizo, but more honestly so 
than schizophrenia since in this case there really is a multiple 
personality, and moreover there's nothing insane about it.  (And it's a 
syllable shorter to boot.)

Third, while it is true that the schismatic object (to give the term a 
trial run) is usually observed manifesting its split personality in 
different categories, this is not the case in *-autonomous categories 
where I and _|_ are the Jekyll and Hyde of the same category.  (I 
apologize to readers of this list with either of those surnames.)

In all the examples I'm aware of, the two categories in which the 
schismatic object occurs (once in each) admit a common completion to a 
*-autonomous category which embeds one object as I and the other as _|_. 
   Considering them to be the "same" object found in two categories 
misses the contravariance between them, which is brought out more 
clearly by this joint completion, where they are clearly not the same 
object but a pair of dual objects.

My paper accounts for C.I. Lewis's qualia by viewing them as morphisms 
running from I to _|_.  If I and _|_ are rigid (|C(x,x)|=1) as for 
Chu(Set,K)), the presence of a morphism from _|_ to I is inconsistent in 
the sense that it collapses Hom(I,_|_) to a singleton, since I and _|_ 
respectively generate and cogenerate.  So in order to have more than one 
quale (in the Chu setting) there cannot be any morphism from _|_ to I. 
(That was mainly in the nature of background on the neighborhood of I 
and _|_.)

Vaughan


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