The category of finite dimensional vector spaces over a field K has
many interesting properties: It is a symmetric closed monoidal (hereafter
known as autonomous) category which has an object K, with the property that
the functor (-,K), internal Hom into K, induces an equivalence with its
opposite category. Similar remarks apply to the category of finite
dimensional (real or complex) banach spaces. We call such a category *-autonomous.
Almost the same thing happens with finite abelian groups, except the "dualizing
object", ~ /~ or ~/~ , is not an object of the category. In no case is the
category involved complete, nor is there an obvious way of extending both the
closed structure and the duality to any of the completions. In studying these
phenomena, I came on a fairly general construction which allows you to begin
with one of the above categories (and some similar ones) to embed it fully into
a complete and cocomplete category which admits an autonomous structure and
which, using the original dualizing object, is *-autonomous.