Dualities arising via pairs of schizophrenic objects

[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/ ]