Hermitian K-theory for stable $\infty$-categories II: Cobordism categories and additivity

We define Grothendieck–Witt spectra in the setting of Poincaré $\infty$-categories and show that they fit into an extension with a $\mathrm{K}$- and an $\mathrm{L}$-theoretic part. As consequences, we deduce localisation sequences for Verdier quotients and generalisations of Karoubi's fundamental and periodicity theorems for rings in which $2$ need not be invertible. Our setup allows for the uniform treatment of such algebraic examples alongside homotopy-theoretic generalisations. For example, the periodicity theorem holds for complex oriented $\mathrm{E}_1$-rings, and we show that the Grothendieck–Witt theory of parametrised spectra recovers Weiss' and Williams' $\mathrm{LA}$-theory. Our Grothendieck–Witt spectra are defined via a version of the hermitian $\mathrm{Q}$-construction, and a novel feature of our approach is to interpret the latter as a cobordism category. This perspective also allows us to give a hermitian version—along with a concise proof—of the theorem of Blumberg, Gepner and Tabuada, and provides a cobordism-theoretic description of the aforementioned $\mathrm{LA}$-spectra.