An involution on the $K$–theory of bimonoidal categories with anti-involution

Abstract

We construct a combinatorially defined involution on the algebraic $K$–theory of the ring spectrum associated to a bimonoidal category with anti-involution. Particular examples of such are braided bimonoidal categories. We investigate examples such as $K\left(ku\right)$, $K\left(ko\right)$ and Waldhausen’s $A$–theory of spaces of the form $BBG$, for abelian groups $G$. We show that the involution agrees with the classical one for a bimonoidal category associated to a ring and prove that it is not trivial in the above mentioned examples.