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•‐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.

Topological manifold bundles and the 𝐴-theory assembly map

“…Roughly speaking, this transformation is given by viewing a chain of composable cobordisms as a filtration of their composite. Moreover, in terms of the cobordism model for A-theory presented in [10,Section 4], it may be understood as the inclusion of the θ-cobordism category into a "homotopy cobordism category", which is a category of cospans of fiberwise homotopy finite spaces over B with a structure map to p. The covariant part of this transformation was first considered by Bökstedt-Madsen [2].…”

On transfer maps in the algebraic K-theory of spaces

“…(c) It can be defined using the models for stable homotopy and A-theory from Waldhausen's manifold approach [19]. (d) It can be identified with a natural map from the cobordism category of manifolds with boundary to A-theory [13][14][15].…”

On transfer maps in the algebraic $K$-theory of spaces

“…(c) It can be defined using the models for stable homotopy and A-theory from Waldhausen's manifold approach [18]. (d) It can be identified with a natural map from the cobordism category of manifolds with boundary to A-theory [13,14,15]. The relationship between the unit map and the A-theory characteristic is explained in the following proposition.…”