DUALITY IN WALDHAUSEN CATEGORIES

We develop a theory of Spanier-Whitehead duality in categories with co fibrations and weak equivalences (Waldhausen categories, for short). T his includes L-theory, the involution on K-theory introduced by [Vo] i n a special case, and a map Xi relating L-theory to the Tate spectrum of Z/2 acting on K-theory. The map Xi is a distillation of the long ex act Rothenberg sequences [Sha], [Ra1], [Ra2], including analogs involv ing higher K-groups. It goes back to [WW2] in special cases. Among the examples covered here, but not in [WW2], are categories of retractive spaces where the notion of weak equivalence involves control.