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* Re:  Communes paper, schismatic objects
@ 2010-11-07 20:05 Fred E.J. Linton
  0 siblings, 0 replies; 7+ messages in thread
From: Fred E.J. Linton @ 2010-11-07 20:05 UTC (permalink / raw)
  To: Todd Trimble, Vaughan Pratt; +Cc: Categories list

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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^ permalink raw reply	[flat|nested] 7+ messages in thread
* Re:  Communes paper, schismatic objects
@ 2010-11-01 23:52 Todd Trimble
  2010-11-03 22:35 ` Vaughan Pratt
  0 siblings, 1 reply; 7+ messages in thread
From: Todd Trimble @ 2010-11-01 23:52 UTC (permalink / raw)
  To: Categories list

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


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


^ permalink raw reply	[flat|nested] 7+ messages in thread
* Communes paper, schismatic objects
@ 2010-11-01 17:44 Vaughan Pratt
  0 siblings, 0 replies; 7+ messages in thread
From: Vaughan Pratt @ 2010-11-01 17:44 UTC (permalink / raw)
  To: categories list

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


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


^ permalink raw reply	[flat|nested] 7+ messages in thread

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Thread overview: 7+ messages (download: mbox.gz / follow: Atom feed)
-- links below jump to the message on this page --
2010-11-07 20:05 Communes paper, schismatic objects Fred E.J. Linton
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2010-11-01 23:52 Todd Trimble
2010-11-03 22:35 ` Vaughan Pratt
2010-11-05  6:42   ` Todd Trimble
     [not found]   ` <3827E0D667BC4DDB9CA346B29AF2DD9B@PC162032150465>
2010-11-05 20:00     ` Vaughan Pratt
2010-11-06 14:12   ` Todd Trimble
2010-11-01 17:44 Vaughan Pratt

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