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

TL;DR

This paper introduces a combinatorial involution on the algebraic K-theory of bimonoidal categories with anti-involution, connecting abstract theory with classical cases and specific examples like topological K-theory.

Contribution

It constructs a new involution on K-theory for bimonoidal categories with anti-involution and demonstrates its non-triviality in key examples, linking to classical involutions.

Findings

Involution matches classical case for ring-associated bimonoidal categories

Involution is non-trivial in algebraic K-theory of complex and real topological K-theory

Involution provides new insights into K-theory of spaces like BBG for abelian G

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 algebraic K-theory of connective complex and real topological K-theory and Waldhausen's K-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.