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From: "John Baez" <baez@math.ucr.edu>
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Subject: Re: dualities
Date: Tue, 2 May 2006 15:05:24 -0700 (PDT)
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Hi -

> First, let me say I have avoided contributing to this thread because I
> don't understand what Vaughan is asking.

He asked if dualities categorify.  I guess he meant something
like this:

There are lots of interesting examples of a pair of categories C,D
together with an object c in C and an object d in D such that

hom(-,c): C -> D

and

hom(-,d): D -> C

are part of an equivalence of categories.  In the nicest examples,
c and d are in some sense the same mathematical entity regarded
as living in two different categories - a "schizophrenic object",
in the words of Harold Simmons.

So, can we find equally nice examples where C and D are instead
2-categories?  In particular, can we find examples where C and D
are 2-categorical generalizations of the 1-categorical examples
we already know?

In particular, he suggested taking the example where C is the
category of finite distributive lattices and finding an analogous
example where C is the 2-category of (maybe finite, in some sense?)
distributive categories.

For more on "schizophrenic objects", Peter Johnstone's review
of Clark and Davies' "Natural dualities for the working algebraist"
makes good reading:

http://north.ecc.edu/alsani/ct99-00(8-12)/msg00116.html