Communes paper, schismatic objects

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

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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