31 Jan 2006
2:30 - 4:00

M Barr

Title: On duality of Freyd's free abelian category

Abstract: In a recent internet posting, Freyd sketched an argument that
when R is a commutative ring, the free abelian category generated by R (as
a category) is *-autonomous.  He suggested that the same ought to be true
for a ring with an anti-involution.  I think that is not true, but since R
is isomorphic to R^\op (the very definition of an anti-involution), the
free abelian category is necessarily self-dual.  The purpose of this talk
is to describe an explicit duality for that case, which is considerably
simpler than Freyd's argument in the commutative case.  It uses some
elementary homological algebra.  Since it takes only about five minutes, I
will use the remaining eighty five minutes for a primer on homological
algebra.