Higher \(K\)-theory of forms III: from chain complexes to derived categories

We exhibit a natural equivalence between the Grothendieck-Witt space associated to an exact form category, and that of its derived Poincaré category, and show that these equivalences deloop to an equivalence of (non- connective) Grothendieck-Witt spectra. In contrast to existing approaches, our argument proceeds via an analysis of an explicit model for the derived functor of a quadratic functor on a certain class of well-behaved (complicial) exact categories with weak equivalences (which includes categories of complexes and pretriangulated dg-categories), showing that hermitian constructions on these descend along the localisation functor in a way that preserves the Grothendieck-Witt theory. We also demonstrate that the genuine symmetric Poincaré structure on the bounded derived ∞-category of an ordinary exact category, coincides with the derived Poincaré structure of on-the-nose symmetric forms.