Real K-theory for Waldhausen infinity categories with genuine duality

Abstract

We develop a new framework to study real $K$-theory in the context of $\infty$-categories. For this, we introduce Waldhausen $\infty$-categories with genuine duality, which will be the input for such $K$-theory. These are Waldhausen $\infty$-categories in the sense of Barwick equipped with a compatible duality and a refinement of their (lax) hermitian objects generalizing the concept of Poincar\'e $\infty$-categories of Lurie. They may also be thought of as a version of complete Segal spaces enriched in genuine $C_2$-spaces whose underlying $\infty$-category carries a compatible Waldhausen structure, since we show that their respective $\infty$-categories are equivalent. We define the real $K$-theory genuine $C_2$-spaces by means of an enriched version of the $S_\bullet$-construction, defined for Waldhausen $\infty$-categories with genuine duality. Moreover, we prove an Additivity Theorem for this $S_\bullet$-construction which leads to an Additivity Theorem for real $K$-theory. Furthermore, such real $K$-theory satisfy a universal property -- analogous to that proved by Barwick for algebraic $K$-theory of Waldhausen $\infty$-categories --: We prove that every theory can be universally turned into an additive theory and identify our real K-theory with the universal additive theory associated to the functor that associates to a Waldhausen $\infty$-category with genuine duality its maximal subspace. Finally, we promote the real $K$-theory genuine $C_2$-spaces to genuine $C_2$-spectra.