The cobordism category and Waldhausen's K-theory

Understand

This paper examines the category C^k_{d,n} whose morphisms are d-dimensional smooth manifolds that are properly embedded in the product of a k-dimensional cube with an (d+n-k)-dimensional Euclidean space.

• There are k directions to compose k-dimensional cubes, so C^k_{d,n} is a (strict) k-tuple category.
• The geometric realization of the k-dimensional multi-nerve is the classifying space BC^k_{d,n}.
• At the end of the paper we construct an infinite loop map to Waldhausens K-theory.