A cobordism model for Waldhausen $K$-theory

Abstract

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 $A(\ast )$. We also review the canonical map from the cobordism category of manifolds to $A$-theory from this viewpoint.

Final version, to appear in J. Lond. Math. Soc

References in corpus (1)

• An Additivity theorem for cobordism categories

Cited by in corpus (3)

• An Additivity theorem for cobordism categories
• Locally (co)Cartesian fibrations as realisation fibrations and the classifying space of cospans
• The classifying space of the one-dimensional bordism category and a cobordism model for TC of spaces