Higher K-theory of forms II. From exact categories to chain complexes

TL;DR

This paper advances Hermitian K-theory by extending quadratic functors to chain complexes within exact categories, preserving Grothendieck-Witt spaces, and bridging classical and infinity-categorical frameworks.

Contribution

It introduces a method to extend quadratic functors to chain complexes without altering Grothendieck-Witt spaces, facilitating comparison between classical and infinity-categorical Hermitian K-theory.

Findings

Extension of quadratic functor preserves Grothendieck-Witt spaces.

Establishes a link between classical and infinity-categorical Hermitian K-theory.

Provides foundational results for future comparisons in Hermitian K-theory.

Abstract

We prove basic statements about the Hermitian K-theory of exact form categories with weak equivalences. Notably, we extend a quadratic functor with values in abelian groups from an exact category to its category of bounded chain complexes in a way that does not change Grothendieck-Witt spaces. This is used in joint work with Marlowe for the comparison of the classical 1-categorical version of the Hermitian K-theory of exact categories with the infinity-categorical version of Calmes-Dotto-Harpaz-Hebestreit-Land-Moi-Nardin-Nikolaus-Steimle.