6 October 2009
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: AR is continuous if there are finitely many φ1, ... ,φn in X such that ker φ contains the intersection of the ker φi. 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.