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. \Omega BC^1_{d,n}-> A(BO(d)), We believe that the map factors through
\Omega^\infty\Sigma^\infty(BO(d)_+) and that the composite B{Diff}(M^d)\to
A(BO(d)) is homotopic to the map considered by Dwyer, Williams and Weiss.