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