Re: Dualities arising via pairs of schizophrenic objects

[Tom Leinster <Tom.Leinster@glasgow.ac.uk>]

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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