dualities

[Peter Freyd <pjf@saul.cis.upenn.edu>]

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.