Re: Dualities arising via pairs of schizophrenic objects

[tholen@mathstat.yorku.ca]

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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