Re: dualities

[Michael Barr <mbarr@math.mcgill.ca>]

First, let me say I have avoided contributing to this thread because I
don't understand what Vaughan is asking.  He knows, as well as anyone,
since he put them on the map, about Chu categories.  He knows about
*-autonomous categories as well.  So what is the question, really?

The simplest answer to Mamuka's question is the duality between vector
spaces (over any field, including the 2 element field) and linearly
compact vector spaces.  In the case of a finite field, linear compactness
is the same as the ordinary topological kind.  One proof of this fact is
that the category of finite dimensional spaces is self-dual and if two
categories are dual, the inductive completion of one is dual to the
projective completion of the other.  For finite dimensional vector spaces,
the inductive completion is vector spaces and the projective completion is
linearly compact ones.  Another example is the obvious duality between
finite sets and finite boolean algebras that gives Stone duality on one
hand and the duality between Set and CABA on the other, depending which
one you complete which way.

Most examples I am aware of of self-dualities are Chu categories (or chu
categories).  And if V_k is the category of k-vector spaces, then
Chu(V_k,k) (an object is a pair of spaces and a bilinear pairing into k)
is *-autonomous, as is chu(V_k,k) of separated extensional pairs.  Peter's
example is one of the very few *-autonomous categories I cannot relate to
Chu.  Complete (say inf) semi-lattices is another.

Michael