The h-cobordism category is the subcategory of the cobordism category (on manifolds possibly with boundary) on all h-cobordisms. We study the homotopy type of its geometric realization and relate it to the classical h-cobordism space on a compact smooth manifold, as considered in the parametrized h-cobordism theorem. As a consequence, we show that the homotopy groups of the h-cobordism category become isomorphic to Waldhausen's higher Whitehead groups of a given manifold, after stabilization and for suitable choice of tangential structure. This is joint work with G. Raptis.