Universality of the Grothendieck-Witt group via the homotopy theory of dg-categories with duality

We prove that the Grothendieck-Witt group functor is weakly universal among the additive invariants of dg-categories with duality, providing a hermitian counterpart of a result of G. Tabuada. We adapt the strategy he established to prove the universality of the Grothendieck group functor. First we construct a localization ∗-Hqe of the category of dg-categories with duality endowed with a suitable class of weak equivalences. This localization is inherited from the quasi-equivalences model structure on dg-categories. We then show that the set of isometry classes of symmetric spaces in the homotopy category of a dg-category A with duality, is co-represented in ∗-Hqe by its monoidal unit. Finally we show that an additive invariant of dg-categories with duality factors through ∗-Hqe and we prove our universality theorem.