Subject: categories: Re: dualities

From: Vaughan Pratt <[email-masked]>
Date: Sun, 30 Apr 2006 12:28:44 -0700

In addition to Peter's nice collection of self-dualities there are the
Kleisli and Eilenberg-Moore categories of "the" covariant power-set monad
(there are really two such monads but either will do), respectively Rel
and complete semilattices, both self-dual.  One that Mike Barr introduced
me to is the subcategory of Rel whose morphisms are the partial
injections, those binary relations such that if (x,y) and (x,z) are both
present then y = z and likewise for their converses.  My personal
favorites are finite chains with bottom (showing that \Delta, as the base
category of the presheaf category of simplicial sets, comes very close to
being self-dual; had \Delta itself been self-dual, Set^{\Delta\op} and
Set^\Delta would have been the same thing), and semilattices with a top
and all nonempty sups (my candidate for a self-dual system of event/state
structures before I replaced it with Chu spaces).

One should also mention the topological vector spaces in Barr's book on
*-autonomous categories, whose self-duality does for the
finite-dimensional vector spaces mentioned by Peter what Pontryagin
duality does for finite abelian groups.

A feature of Chu spaces I particularly like is that each of the above,
as well as finite-dimensional vector spaces and finite abelian groups,
can be described as that full subcategory of Chu(Set,K) (K = 2 except
for vector spaces and abelian groups) consisting of biextensional Chu
spaces whose rows and columns satisfy the same closure conditions.  For
example the category of finite-dimensional vector spaces over GF(2) (a
sneaky way to stick to Chu(Set,2)) embeds in Chu(Set,2) as precisely
those finite biextensional Chu spaces whose rows and columns, viewed as
bit vectors in the sense a machine-language programmer understands the
concept, are both closed under bitwise XOR.  This example is given as an
exercise at the end of Chapter 2 of
http://boole.stanford.edu/pub/coimbra.pdf, my notes for the July 1999
Coimbra School cotaught with John Baez and Cristina Pedicchio.
Proposition 2.2 in the same chapter obtains complete semilattices (of
any cardinality) as those Chu spaces whose rows and columns are closed
under bitwise OR, with the self-duality of CSLat as the immediate
Corollary 2.3.  This shows that the self-dualities CSLat and
Vct_{GF(2)}, at least for finite objects, arise identically except for
how they combine their bit-vectors, namely with respectively OR and XOR.

Vaughan

Peter Freyd wrote:
> On the subject of favorite dualities:
>
> Surely the most important are the self-dualities and the most
> important of these (so important we stop noticing it as we age) is the
> category of finite-dimensional vector spaces over a given field.
>
> Next is Pontryagin's: the category of locally compact groups.  The
> original Pontryagin duality easily generalizes: the category of
> locally compact modules over a given commutative ring is self-dual.
> (In the non-commutative case one also obtains a duality but not a
> self-duality -- unless, of course, the ring is self-dual.) A corollary
> is that the category of discrete left  R-modules is dual to the
> category of compact right  R-modules. (For 50 years I've been trying to
> turn this into an exercise in abelian categories. There's a nice
> reduction down to the proposition that  R/Z  is a cogenerator for the
> category of compact abelian groups, but that fact seems to require
> some non-trivial functional analysis.) Strange that two of the most
> important "dualities" are both Pontryagin's. The other is in algebraic
> topology theory.
>
> Then, of course there's my present favorite: the category of finitely
> presented group-valued functors from the category of finitely
> presented modules over a commutative ring.
>
>