3:00 - 4:30 M Barr

A new variation on Chu

ABSTRACT: This is joint work with John Kennison and Bob Raphael.

I begin by reviewing the Chu construction, in particular the separated,
extensional subcategory chu in the context of the category of *R*-modules
with with the commutative ring *R* itself as the dualizing object. Assume
that *R* has the property that the complete ring of fractions of *R* is an
injective *R*-module. If (*A*,*X*) in **chu**, say
that a map φ: *A* → *R* is continuous if there are
finitely many φ_{1}, ... ,φ_{n} in *X* such that ker
φ contains the intersection of the ker φ_{i}. When *R* is injective, it
is easy to show that every such continuous map actually belongs to *X*. We
say that (*A*,*X*) is high if every continuous map belongs to *X* and wide if
(*X*,*A*) is high. Let *X*^{+} be *X*
augmented by adding all the continuous maps
to it and *H*(*A*,*X*) = (*A*,*X*^{+}).
Then *H* is left adjoint to the inclusion of
the high objects into **chu**. Similarly, there is a right adjoint *W* to
the inclusion of wide objects. Although these adjoints don't commute,
it is nonetheless the case that when (*A*,*X*) is wide, so
is *H*(*A*,*X*) and
dually for *W*. Also the tensor product of two high objects is high and
the result is that the full subcategory of high wide objects has a
natural *-autonomous structure.