A cobordism model for Waldhausen \(K\)-theory. (English) Zbl 1444.19002
Summary: 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.
MSC:
| 19D10 | Algebraic \(K\)-theory of spaces |
| 18A30 | Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) |
| 57R90 | Other types of cobordism |
References:
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