A cobordism model for Waldhausen K-theory

We study a categorical construction called the cobordism category, which associates to each Waldhausen category a simplicial category of cospans. We prove that this construction is homotopy equivalent to Waldhausen's S center dot-construction and therefore it defines a model for Waldhausen K-theory. As an example, we discuss this model for A-theory and show that the cobordism category of homotopy finite spaces has the homotopy type of Waldhausen's A(*). We also review the canonical map from the cobordism category of manifolds to A-theory from this viewpoint.