  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
$S_{\bullet}$-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.
