A cobordism model for Waldhausen $K$-theory

TL;DR

This paper introduces a cobordism category model for Waldhausen K-theory, demonstrating its homotopy equivalence to the classical S•-construction and applying it to A-theory and spaces.

Contribution

It establishes a new cobordism category framework for Waldhausen K-theory, providing a geometric perspective and connecting it to A-theory and manifold cobordisms.

Findings

Cobordism category is homotopy equivalent to Waldhausen's S•-construction.

The model applies to A-theory and homotopy finite spaces.

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