Re: Dualities arising via pairs of schizophrenic objects

Re: Dualities arising via pairs of schizophrenic objects

[Fred E.J. Linton]

I must say, the ideas of taking the name of the god Janus in vain
(suggested on the nLab page cited below), or using anything that 
rhymes with Bambimorphic, strike me as Bad Ideas :-) . 

But feel free, if you must ... . Cheers, -- Fred

------ Original Message ------
On Fri, 25 Nov 2011 08:28:16 AM EST, Tom Leinster <Tom.Leinster@glasgow.ac.uk>
wrote:

> There are some people, including me, who are troubled by the term
> "schizophrenic" and want to replace it. ...
> There was some discussion a while ago about what would be the best
> alternative.  I think the best candidate is "dualizing object".  See for
> instance the nLab page,
> 
> http://ncatlab.org/nlab/show/dualizing+object 
   ... [snip] ...



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Re: Dualities arising via pairs of schizophrenic objects

[Graham White]

Two possible names are "liminal" (from limen, a doorway), or
"bifrontal" (from frons, which means face: one of the titles of Janus
is Janus bifrons). I kind of like liminal, because it emphasises the
function of the twofacedness, rather than simply the fact that the
object is twofaced.

Graham

On 26/11/11 07:39, Fred E.J. Linton wrote:
> I must say, the ideas of taking the name of the god Janus in vain
> (suggested on the nLab page cited below), or using anything that
> rhymes with Bambimorphic, strike me as Bad Ideas :-) .
>
> But feel free, if you must ... . Cheers, -- Fred
>

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Re: Dualities arising via pairs of schizophrenic objects

[Jocelyn Ireson-Paine]

From the original description of this kind of object, I get the image of
two networks (the two categories) which have only one vertex in common. Or
to get more physical about it - which is a good way to generate mental
images of real-world analogues whose names we can steal - two fishing nets
or string bags tied together at a single knot. So what's a good name for
that knot? A "junction"? A "contingence"? A "taction"? It's where two
worlds touch, like a weak spot in the space-time continuum where the
threads have worn away, or the Wood Between the Worlds in the Narnia
novels. How about "crossover object"?

Jocelyn Ireson-Paine
http://www.j-paine.org
http://www.spreadsheet-parts.org
+44 (0)7768 534 091

Jocelyn's Cartoons:
http://www.j-paine.org/blog/jocelyns_cartoons/

On Sun, 27 Nov 2011, Graham White wrote:

> Two possible names are "liminal" (from limen, a doorway), or
> "bifrontal" (from frons, which means face: one of the titles of Janus
> is Janus bifrons). I kind of like liminal, because it emphasises the
> function of the twofacedness, rather than simply the fact that the
> object is twofaced.
>
> Graham

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Re: Dualities arising via pairs of schizophrenic objects

[Valeria de Paiva]

Dusko,
This time I cannot tell whether you're joking or not...

Now I must say that I totally agree with Tom Leinster (and others)
that the usage of
"schizophrenic object" is in *very* bad taste and does no good to anyone.

(this kind of stuff is acceptable for third graders. only.).

Best regards,
Valeria


On Tue, Dec 6, 2011 at 9:48 PM, Dusko Pavlovic <dusko@kestrel.edu> wrote:
> i agree that we should not use the term "schizophrenic object" in category theory.
>
> for one thing, it sounds like some sort of a metaphor. we should never use metaphors.
> for another thing, it does not sound serious. it might suggest that we are sometimes joking.
>
> i propose that we use the term *bipolar object*.
>
> for one thing, it sounds more mathematical.
> for another thing, in psychiatry they only talk about subjects, not objects, so there is no confusion.
>
> my 2c,
> -- dusko

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Re: Dualities arising via pairs of schizophrenic objects

[Fred E.J. Linton]

On Fri, 02 Dec 2011 08:35:30 AM EST, David Roberts
<david.roberts@adelaide.edu.au> wrote:
  
>  ... always thought it odd that even when one wants to accept
> category-theoretic
> foundations (e.g. ETCS or similar), then suddenly something like this
> comes along,
> where people start saying there is a thing which is an object of two
> different categories.

Ever since Eckmann-Hilton, and perhaps even before, the notion of an
object G in one category X bearing the structure of an object in some
concrete other category A (concrete via U: A -> Sets, say) has been 
clearly and unambiguously expressed as follows:

The hom functor X(-, G): X^op -> Sets is given a factorization thru' U.

If both X and A are concrete, it's perfectly plausible for an object
of X to bear the structure of an object in A, and vice versa, and a
brief peek at the example of 2 as BA w/ KT_2-space structure and as
KT_2-space with BA structure will make short work of understanding how
an object may be thought of as "inhabiting both categories at once":
indeed, it's that contravariant adjoint pair alone, between A and X,
that provides the duality in John Isbell's 1972 approach, where 
at most one of A and X need be concrete.

HTH. Cheers, -- Fred



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Re: Dualities arising via pairs of schizophrenic objects

[Jean Benabou]

Dear Fred,

I keep your notations. The concreteness of A is far from enough to
justify the definition you give, namely:

> Ever since Eckmann-Hilton, and perhaps even before, the notion of an
> object G in one category X bearing the structure of an object in some
> concrete other category A (concrete via U: A -> Sets, say) has been
> clearly and unambiguously expressed as follows:
>
> The hom functor X(-, G): X^op -> Sets is given a factorization thru'
> U.

Eckman and Hilton considered only the case when A is a category of
essentially algebraic (i.e. definable by projective limits) structures
over Sets. In more general cases, it just doesn't work. You can't even
prove, for X=A, that an object G of X bears the structure of an object
of A.
Take for A the concrete category of totally ordered sets and order-
preserving functions, with U the obvious forgetful functor.
The same is true for A=Fields, A=Topological Spaces, A=Finite Sets,
etc., to take very simple examples.

Bien amicalement,
Jean
>
> If both X and A are concrete, it's perfectly plausible for an object
> of X to bear the structure of an object in A, and vice versa, and a
> brief peek at the example of 2 as BA w/ KT_2-space structure and as
> KT_2-space with BA structure will make short work of understanding how
> an object may be thought of as "inhabiting both categories at once":
> indeed, it's that contravariant adjoint pair alone, between A and X,
> that provides the duality in John Isbell's 1972 approach, where
> at most one of A and X need be concrete.
>
> HTH. Cheers, -- Fred
>


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Re: Dualities arising via pairs of schizophrenic objects

[Fred E.J. Linton]

Anent Graham White's suggestions,

> Two possible names are "liminal" (from limen, a doorway), or
> "bifrontal" (from frons, which means face: one of the titles of Janus
> is Janus bifrons). I kind of like liminal, because it emphasises the
> function of the twofacedness, rather than simply the fact that the
> object is twofaced.

Perhaps "liminal" suggests two-facedness to some, but not to Merriam-
Webster ( http://www.merriam-webster.com/dictionary/liminal ), who believe 
it stands for

1: of or relating to a sensory threshold;
2: barely perceptible; or
3: of, relating to, or being an intermediate state, phase, or condition :
in-between, transitional <in the liminal state between life and death —
Deborah Jowitt>. 

And never mind that "two-facedness" in common parlance has to do with
a person's being (not liminal, but) deceitful, insincere, or hypocritical.

OtOH, liminal does accept prefixes nicely, as: subliminal, supraliminal :-) .

As for "bifrontal", often one is faced (sorry :-) ) with an object having  
far more "fronts" than just two. Good old 2 = {0, 1}, for example, is: 
a set, a pointed set, a bi-pointed set, a poset, a poset with top, a poset
with bottom, a compact T2 space, a pointed compact T2 space, an abelian group,
a meet-semilattice, a frame (though others will insist I should be 
saying "locale"), a Boolean Ring, a Boolean Rng, and much much more.

Does the prefix "bi-" really adequately capture the potential of having 
all that many ... umm ... hate to use this term ... personalities?

Cheers, -- Fred "it takes two to tango" Linton,
(now [re]tiring from -- sitting out the rest of -- this year's dance)



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Re: Dualities arising via pairs of schizophrenic objects

[Robert Dawson]

My own suggestion would be "pivot object".

 	Robert


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Re: Dualities arising via pairs of schizophrenic objects

[Fred E.J. Linton]

To redeploy some recent words of Tom Leinster, shingles is (as I know from 
painful first-hand experience) "a serious and often frightening condition." 

Yet I would not go so far as to insist that roofing shingles or
siding shingles be outfitted with some other name, or to urge
doctors or lawyers to refrain from speaking of "hanging out their 
shingles" when they open their practices.

I think intelligent people can be trusted to understand even potentially 
ambiguous words in a correct, mature, context-driven way.

Cheers, -- Fred Linton





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Re: Dualities arising via pairs of schizophrenic objects

[Todd Trimble]

This discussion about 'schizophrenic' mostly feels like déjà vu,
but the point about shingles, while amusing, misses the serious
point that Tom is making.

I agree that most everyone knows that the medical condition
has nothing at all to do with roofing shingles (the etymologies of
both words are interesting -- look them up). But not everyone
yet understands that schizophrenia has nothing to do with 'split
personality', as a careless folk-etymology might lead one to
suppose. But this false meaning is indeed the one reinforced
by the usage in category theory (and the coinage may have been
based on the misunderstanding).

Leaving aside this potential reinforcement of a misunderstanding,
I do agree that people experienced in category theory will
recognize the intended categorical meaning, however inapt the
coinage may be.

Todd

----- Original Message ----- 
From: "Fred E.J. Linton" <fejlinton@usa.net>
To: <categories@mta.ca>
Sent: Saturday, November 26, 2011 10:06 AM
Subject: categories: Re: Dualities arising via pairs of schizophrenic 
objects


> To redeploy some recent words of Tom Leinster, shingles is (as I know from
> painful first-hand experience) "a serious and often frightening 
> condition."
>
> Yet I would not go so far as to insist that roofing shingles or
> siding shingles be outfitted with some other name, or to urge
> doctors or lawyers to refrain from speaking of "hanging out their
> shingles" when they open their practices.
>
> I think intelligent people can be trusted to understand even potentially
> ambiguous words in a correct, mature, context-driven way.
>
> Cheers, -- Fred Linton
>
>


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Dualities arising via pairs of schizophrenic objects

[Sebastian Kerkhoff]

   Dear all,

I have a short and probably very simple question (and I apologize for it
in advance):

I believe it is a well-known fact that a potential duality arises when a
single object essentially lives in two different categories. Famous
examples for such objects and such dualities are the Gelfand-Duality
(where this object is the space of complex numbers, once as a
topological space and once as an algebraic structure) or the Stone
Duality (where this object is the two-element lattice, once as a Boolean
algebra and once as a bounded poset with discrete topology).

As far as I know (correct me if am wrong), people started to call these
objects "schizophrenic objects" after this term was introduced by Harold
Simmons in 1982. What I would like to know is the following: Could
anybody provide me with a few lines about the historical development of
this principle? I know that John Isbell is often cited as a source
(however, my impression is that people are not entirely sure), and I
have also heard that Peter Freyd was supposedly the first who studied
these kind of dual adjunctions systematically (proving that such
constructions are often essentially the only way to create dual
adjunctions between two categories).

In case you are interested, I can also provide you with the reason for
my question: I am giving a (small) course about duality theory in
Dresden, and since most of my students are very interested in universal
algebra, the course also covers the theory of natural dualities
developed by Brian Davey and his various co-authors (it is a theory that
tries to generalize the Stone duality to other algebraic structures).
However, I would like to point out to the students that the principle of
schizophrenic objects is not only a convenient ad-hoc construction for
such natural dualities, but actually a much more general principle that
gives rise to many other dualities (which will be covered in the course
in much less detail). For that, I would like to provide the students
with some historical development of this idea, which I obviously cannot
do as long as I am not at all sure about it myself. Plus, I am also
personally very interested in some background information about this
"schizophrenic" idea.

Thank you very much.

Best regards,
Sebastian Kerkhoff

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Re: Dualities arising via pairs of schizophrenic objects

[Tom Leinster]

Dear Sebastian,

There are some people, including me, who are troubled by the term
"schizophrenic" and want to replace it.  Mental health groups go to some
effort to persuade journalists not to use the word in the casual way they
sometimes do; schizophrenia is of course a serious and often frightening
condition, and it doesn't help when people use language in a way that
perpetuates an inaccurate stereotype.

There was some discussion a while ago about what would be the best
alternative.  I think the best candidate is "dualizing object".  See for
instance the nLab page,

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

Regarding the question itself, I think you'll enjoy Peter Johnstone's book
Stone Spaces, where you'll find a thorough development of the general
principle that you mention in your last paragraph.

Best wishes,
Tom


On Thu, 24 Nov 2011, Sebastian Kerkhoff wrote:

>   Dear all,
>
> I have a short and probably very simple question (and I apologize for it
> in advance):
>
> I believe it is a well-known fact that a potential duality arises when a
> single object essentially lives in two different categories. Famous
> examples for such objects and such dualities are the Gelfand-Duality
> (where this object is the space of complex numbers, once as a
> topological space and once as an algebraic structure) or the Stone
> Duality (where this object is the two-element lattice, once as a Boolean
> algebra and once as a bounded poset with discrete topology).
>
> As far as I know (correct me if am wrong), people started to call these
> objects "schizophrenic objects" after this term was introduced by Harold
> Simmons in 1982. What I would like to know is the following: Could
> anybody provide me with a few lines about the historical development of
> this principle? I know that John Isbell is often cited as a source
> (however, my impression is that people are not entirely sure), and I
> have also heard that Peter Freyd was supposedly the first who studied
> these kind of dual adjunctions systematically (proving that such
> constructions are often essentially the only way to create dual
> adjunctions between two categories).
>
> In case you are interested, I can also provide you with the reason for
> my question: I am giving a (small) course about duality theory in
> Dresden, and since most of my students are very interested in universal
> algebra, the course also covers the theory of natural dualities
> developed by Brian Davey and his various co-authors (it is a theory that
> tries to generalize the Stone duality to other algebraic structures).
> However, I would like to point out to the students that the principle of
> schizophrenic objects is not only a convenient ad-hoc construction for
> such natural dualities, but actually a much more general principle that
> gives rise to many other dualities (which will be covered in the course
> in much less detail). For that, I would like to provide the students
> with some historical development of this idea, which I obviously cannot
> do as long as I am not at all sure about it myself. Plus, I am also
> personally very interested in some background information about this
> "schizophrenic" idea.
>
> Thank you very much.
>
> Best regards,
> Sebastian Kerkhoff


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Re: Dualities arising via pairs of schizophrenic objects

[Robert Dawson]

On 24/11/2011 4:33 PM, Tom Leinster wrote:
> Dear Sebastian,
>
> There are some people, including me, who are troubled by the term
> "schizophrenic" and want to replace it. Mental health groups go to some
> effort to persuade journalists not to use the word in the casual way they
> sometimes do; schizophrenia is of course a serious and often frightening
> condition, and it doesn't help when people use language in a way that
> perpetuates an inaccurate stereotype.

 	If schizophrenia, in a medical rather than etymological sense,
actually referred to split personality, there would be some point to the
term, and the debate would have some weight on both sides - especially
as the usage is not derogatory.  But as it's inaccurate, I think a more
descriptive replacement would be in order.

 	Robert Dawson



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Re: Dualities arising via pairs of schizophrenic objects

[Eduardo J. Dubuc]

Schizophrenic seems to me a ridiculous name for a mathematical concept.
e.d.

On 25/11/11 11:38, Robert Dawson wrote:
> On 24/11/2011 4:33 PM, Tom Leinster wrote:
>> Dear Sebastian,
>>
>> There are some people, including me, who are troubled by the term
>> "schizophrenic" and want to replace it. Mental health groups go to some
>> effort to persuade journalists not to use the word in the casual way they
>> sometimes do; schizophrenia is of course a serious and often frightening
>> condition, and it doesn't help when people use language in a way that
>> perpetuates an inaccurate stereotype.

...


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Re: Dualities arising via pairs of schizophrenic objects

[Ross Street]

On 25/11/2011, at 3:06 AM, Sebastian Kerkhoff wrote:

>
> I believe it is a well-known fact that a potential duality arises  
> when a
> single object essentially lives in two different categories.

Dear Sebastian

This topic has been discussed in this forum before (e.g. Oct 2010) and I
recommend looking at the past emails. I don't think anyone mentioned the
following paper which uses the idea of an A-object in B (which is  
equally
a B-object in A) to provide examples as stated in the title.

F. Foltz, G.M. Kelly and C. Lair,
Algebraic categories with few monoidal biclosed structures or none,
J. Pure and Applied Algebra 17 (1980) 171–177.

The work of Lambek and Rattray, such as

Lambek, J.; Rattray, B. A.
A general Stone-Gelfand duality.
Trans. Amer. Math. Soc. 248 (1979), no. 1, 1–35,

should also be emphasized.

Best wishes,
Ross

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Re: Dualities arising via pairs of schizophrenic objects

[tholen]

Dear Sebastian,

Here is an elementary categorical introduction to dualities for the
"working mathematician" which you may like to have a look at for the
purpose of preparing your course:

H.-E. Porst, W. Tholen: Concrete dualities. In: Research and Exposition
in Mathematics 18 (Heldermann Verlag, Berlin 1991), pp 111-136.

Best wishes,

Walter


Quoting Sebastian Kerkhoff <Sebastian_kerkhoff@gmx.de>:

>   Dear all,
>
> I have a short and probably very simple question (and I apologize for it
> in advance):
>
> I believe it is a well-known fact that a potential duality arises when a
> single object essentially lives in two different categories. Famous
> examples for such objects and such dualities are the Gelfand-Duality
> (where this object is the space of complex numbers, once as a
> topological space and once as an algebraic structure) or the Stone
> Duality (where this object is the two-element lattice, once as a Boolean
> algebra and once as a bounded poset with discrete topology).
>
> As far as I know (correct me if am wrong), people started to call these
> objects "schizophrenic objects" after this term was introduced by Harold
> Simmons in 1982. What I would like to know is the following: Could
> anybody provide me with a few lines about the historical development of
> this principle? I know that John Isbell is often cited as a source
> (however, my impression is that people are not entirely sure), and I
> have also heard that Peter Freyd was supposedly the first who studied
> these kind of dual adjunctions systematically (proving that such
> constructions are often essentially the only way to create dual
> adjunctions between two categories).
>
> In case you are interested, I can also provide you with the reason for
> my question: I am giving a (small) course about duality theory in
> Dresden, and since most of my students are very interested in universal
> algebra, the course also covers the theory of natural dualities
> developed by Brian Davey and his various co-authors (it is a theory that
> tries to generalize the Stone duality to other algebraic structures).
> However, I would like to point out to the students that the principle of
> schizophrenic objects is not only a convenient ad-hoc construction for
> such natural dualities, but actually a much more general principle that
> gives rise to many other dualities (which will be covered in the course
> in much less detail). For that, I would like to provide the students
> with some historical development of this idea, which I obviously cannot
> do as long as I am not at all sure about it myself. Plus, I am also
> personally very interested in some background information about this
> "schizophrenic" idea.
>
> Thank you very much.
>
> Best regards,
> Sebastian Kerkhoff
>




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Re: Dualities arising via pairs of schizophrenic objects

[Vaughan Pratt]

There's something odd about when this term is used (under whatever
name).  The implication is that it's two manifestations of the "same"
object, one in each of a dual pair C, C' of categories, d in C and d' in C'.

When C is equivalent to C' (self-duality), as with FinVect, CSLat,
Chu(V,k), etc., this point of view seems mathematically appropriate.

But when not, as with the Stone duality of Boolean algebras, the duality
D: C --> C' doesn't even carry d to d', and moreover C(d,d) and
C'(d',d') typically don't even have the same number of endomorphisms.
Typically d and d' cogenerate and their respective images D(d) and
D'(d') generate.

In this case it would seem preferable to call D(d) the counterpart (up
to isomorphism) in C' of d in C, and conversely for D'(d') (writing D'
for the adjoint to D making it a duality).

What's odd is that the term seems to be used precisely when it is
mathematically inappropriate in the above sense (quite apart from
medical or sensitivity issues).

The real manifestation of the "same" entity is not with objects at all
but with homsets, namely the homset C(D'(d'), d) in C and the homset
C'(D(d), d') in C', which *do* have the same number of morphisms.

If anything deserves the epithet in question it is that homset in each
category.  The two homsets are in bijection, but their targets don't
correspond, having only in common that they are the dualizers in the
respective categories.

(I made the same point in the previous flurry on this topic a year or so
ago, hopefully more clearly this time around.)

Vaughan


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Re: Dualities arising via pairs of schizophrenic objects

[David Roberts]

Vaughan wrote:

> What's odd is that the term seems to be used precisely when it is
> mathematically inappropriate in the above sense (quite apart from
> medical or sensitivity issues).
>
> The real manifestation of the "same" entity is not with objects at all
> but with homsets, namely the homset C(D'(d'), d) in C and the homset
> C'(D(d), d') in C', which *do* have the same number of morphisms.
>
> If anything deserves the epithet in question it is that homset in each
> category.  The two homsets are in bijection, but their targets don't
> correspond, having only in common that they are the dualizers in the
> respective categories.

Yes, I always thought it odd that even when one wants to accept
category-theoretic
foundations (e.g. ETCS or similar), then suddenly something like this
comes along,
where people start saying there is a thing which is an object of two
different categories.
Such a property isn't even expressible in type theory-style
foundations, where even
elements of two different sets aren't comparable...

But since both of the categories in each pair Vaughan mentioned are
enriched over
Set, we *are* allowed to compare hom-sets, at least using some sort of roughly
canonical isomorphism. Using the formalism suggested (hom-sets), it
seems much easier
to set down a definition of these slippery objects.

And (more whimsically) regarding terminology: if someone wanted to use
the analogy
of a door, why not a window? We can have fenestral objects, by which
one can 'see'
from one category to another. And unfortunately, 'liminal' gives rise
to subliminal, which
might be a natural prefix extension mathematically but is even more
confusing than
the existing inaccurate term. :-)

David


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Re: Dualities arising via pairs of schizophrenic objects

[Dusko Pavlovic]

i agree that we should not use the term "schizophrenic object" in category theory.

for one thing, it sounds like some sort of a metaphor. we should never use metaphors.
for another thing, it does not sound serious. it might suggest that we are sometimes joking. 

i propose that we use the term *bipolar object*. 

for one thing, it sounds more mathematical.
for another thing, in psychiatry they only talk about subjects, not objects, so there is no confusion.

my 2c,
-- dusko

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Re: Dualities arising via pairs of schizophrenic objects

[Michael Barr]

When I, along with Bob Raphael and John Kennison wrote a paper on such
dualities, we called them Isbell dualities and didn't bother to name the
common object.  Of course, we still consider it meaningful to talk about
one object living in two categories.  It is not a formal notion (although
we did make an attempt to formalize it to some extent) but very useful for
intuition.  Of course for the mathematician who is strictly formal, this
is meaningless.  While I have met such people, they are thankfully rare.

See http://www.tac.mta.ca/tac/volumes/20/15/20-15.pdf for our paper.

Michael


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Re: Dualities arising via pairs of schizophrenic objects

[Jocelyn Ireson-Paine]

[Note from moderator: Someone may revive this subject again next year, but 
for this round the 48 hour rule is now in effect: nothing further will be 
posted after December 9, thanks.]

On Wed, 7 Dec 2011, Dusko Pavlovic wrote:

> I agree that we should not use the term "schizophrenic object" in
> category theory.
>
> For one thing, it sounds like some sort of a metaphor. We should never
> use metaphors. For another thing, it does not sound serious. It might
> suggest that we are sometimes joking.
>
(A) What's wrong with metaphors? In Chapter 24 of his book "Metamagical
Themas", which has some excellent chapters on analogical reasoning,
Douglas Hofstadter says:
    Don't press an analogy too far, because it will always break down. In
    that case, what good are analogies? Why bother with them? What is the
    purpose of trying to establish a mapping between two things that do
    not map onto each other in reality? The answer is surely very complex,
    but the heart of it must be that it is good for our survival (or our
    genes' survival), because we do it all the time. Analogy and reminding,
    whether they are accurate or not, guide all our thought patterns.
    Being attuned to vague resemblances is the hallmark of intelligence,
    for better or for worse.

(B) What's wrong with joking? Jokes are metaphors, so by (A), they're the
hallmark of intelligence. Besides, in my country, joking is the default
technique for talking about reality. We'd be lost without it.

> I propose that we use the term *bipolar object*.
>
> For one thing, it sounds more mathematical. For another thing, in
> psychiatry they only talk about subjects, not objects, so there is no
> confusion.
>
It's confusing if you grew up as a chemist, which I did. A bipolar object
has two ends with opposite properties. For example, detergent molecules
are bipolar. One end is hydrophilic and loves the water. The other end is
hydrophobic and loves olive oil and bacon grease. If you're lucky, it
loves them so strongly that it will wrench them away from your dirty
dishes. So to me, a bipolar object has ends, and a difference
therebetween; and therefore, it has length. Categorical objects don't.

> my 2c,
> -- dusko
>
Jocelyn Ireson-Paine


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