Stable moduli spaces of hermitian forms

We prove that Grothendieck-Witt spaces of Poincare categories are, in many cases, group completions of certain moduli spaces of hermitian forms. This, in particular, identifies Karoubis classical hermitian and quadratic K-groups with the genuine Grothendieck-Witt groups from our joint work with Calm`es, Dotto, Harpaz, Land, Moi, Nardin and Nikolaus, and thereby completes our solution of several conjectures in hermitian K-theory. The method of proof is abstracted from work of Galatius and Randal-Williams on cobordism categories of manifolds using the identification of the Grothendieck-Witt space of a Poincare category as the homotopy type of the associated cobordism category.