Higher $K$-theory of forms III: from chain complexes to derived categories

TL;DR

This paper establishes a canonical equivalence between hermitian K-theory spectra of exact form categories and their derived Poincaré ∞-categories, extending the understanding of quadratic functors without assuming 2-invertibility.

Contribution

It introduces a new equivalence linking hermitian K-theory of exact form categories to derived categories, and models the nonabelian derived functor of quadratic functors.

Findings

Canonical equivalence between hermitian K-theory spectra and derived Poincaré ∞-categories

Model for the nonabelian derived functor of quadratic functors

No assumptions on the invertibility of 2

Abstract

We exhibit a canonical equivalence between the hermitian $K$-theory (alias Grothendieck-Witt) spectrum of an exact form category and that of its derived Poincar\'e $\infty$-category, with no assumptions on the invertibility of $2$. Along the way, we obtain a model for the nonabelian derived functor of a nondegenerate quadratic functor on an exact category.