Duality in Waldhausen Categories

We develop a theory of Spanier–Whitehead duality in categories with cofibrations and weak equivalences (Waldhausen categories, for short). This includes L–theory, the involution on K–theory introduced by [Vo] in a special case, and a map Ξ relating L–theory to the Tate spectrum of Z/2 acting on K–theory. The map Ξ is a distillation of the long exact Rothenberg sequences [Sha], [Ra1], [Ra2], including analogs involving 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.