Re: dualities

[Vaughan Pratt <pratt@cs.stanford.edu>]

Michael Barr wrote:
> 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?

Duality is of necessity between categories, and involves associating an
object (say an algebra or space) of one category with its dual in
another, or in the same category in the self-dual case.  Downstairs,
i.e. in 2-CAT.  By "categorifying duality" I meant a duality of
2-categories in which one associates an object (this time a category
rather than an algebra) of one 2-category with its dual in another, with
the functors being reversed (op) as opposed to the natural
transformations (co).  Upstairs, i.e. in 3-CAT.

Regarding Chu, I was going to respond that the Chu construction works
downstairs with categories (from my usual V=Set perspective), or at most
V-categories, categories enriched in V, as objects of the 2-category
V-CAT.  However if the enriched Chu construction can be organized to
allow V to be a 2-category, with Chu(V,k) then being a 3-category, maybe
Mike is on to a promising approach (though it's not clear that's what he
actually meant).  It's an aspect of Chu spaces I know next to nothing
about however.

My first guess would be that it (moving Chu up into 3-CAT) ought to work
fine, with the caveat that the simple notion of Stone topology as a
totally disconnected compact Hausdorff topology would turn into the
proverbial thousand flowers---there's far more room for such stuff in
3-CAT than 2-CAT.  (Actually there's also a lot of unexplored such
territory even just in ordinary Chu(Set,3).)  Along those lines, Peter
Johnstone's mention of quasi-injective toposes dual to continuous
categories in his 1982 paper with Joyal is surely just scratching the
surface of the possible permutations and combinations up there in 3-CAT.

   Peter's
> example is one of the very few *-autonomous categories I cannot relate to
> Chu.  Complete (say inf) semi-lattices is another.

Oh, but complete inf semilattices are one of the most elegant
self-dualities of chupology.  They embed in Chu(Set,2) as those
biextensional Chu spaces (biextensional = no repeated rows or columns),
of any cardinality, such that the set of rows is closed under arbitrary
AND (think of them as bit vectors) and likewise for the set of columns.
  No other conditions.  Using OR instead of AND for both rows and
columns gives sup semilattices.  But that was in my previous post, where
I also mentioned that XOR in place of OR, at least for finite Chu
spaces, embeds FinVct_{GF(2)}.  In either case the symmetry of the
conditions makes the self-duality immediate.  All that is in the 1999
Coimbra notes I cited.

Vaughan