Dualities arising via pairs of schizophrenic objects
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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] ...
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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) [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
My own suggestion would be "pivot object". Robert [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Monday, November 28, 2011 at 21:04 , Jocelyn Ireson-Paine wrote:
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"?
Given how much category theory tends to borrow from philosophy, "pineal object" ("pineal gland" is where, according to Descartes, material and ideal worlds intersect, http://plato.stanford.edu/entries/pineal-gland/) might be a candidate. Nikita.
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Jean, thanks for your remarks (below). You are of course quite right: there is no reason whatsoever to expect that an object of X should 'bear the structure of an X-object' in the Eckmann-Hilton sense, even when X is concrete. If I ever gave the impression that it should, please believe me when I say that I had absolutely no intention to do so, and absolutely no expectation that it should be so. Quite to the contrary, even a group (qua object of Grps) will not bear the structure of a Grps object unless it's an Abelian group, as we know since Eckmann-Hilton (or before) as well. Cheers, -- Fred --- [Original message] --- ------ Original Message ------ Received: Sun, 04 Dec 2011 09:25:38 AM EST From: Jean Benabou <jean.benabou@wanadoo.fr> To: Fred Linton <fejlinton@usa.net>, Categories <categories@mta.ca> Subject: Re: categories: Re: Dualities arising via pairs of schizophrenic objects
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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.
... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
[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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (17)
-
David Roberts -
Dusko Pavlovic -
Eduardo J. Dubuc -
Fred E.J. Linton -
Graham White -
Jean Benabou -
Jocelyn Ireson-Paine -
Michael Barr -
Nikita Danilov -
Robert Dawson -
Ross Street -
Sebastian Kerkhoff -
tholen＠mathstat.yorku.ca -
Todd Trimble -
Tom Leinster -
Valeria de Paiva -
Vaughan Pratt