  Galatius, Madsen, Tillmann and Weiss have identified the homotopy type of the
classifying space of the cobordism category with objects (d-1)-dimensional
manifolds embedded in R^\infty. In this paper we apply the techniques of spaces
of manifolds, as developed by the author and Galatius, to identify the homotopy
type of the cobordism category with objects (d-1)-dimensional submanifolds of a
fixed background manifold M.
  There is a description in terms of a space of sections of a bundle over M
associated to its tangent bundle. This can be interpreted as a form of Poincare
duality, relating a space of submanifolds of M to a space of functions on M.
