  In this paper we study the topology of cobordism categories of manifolds with
corners. Specifically, if {Cob}_{d,<k>} is the category whose objets are a
fixed dimension d, with corners of codimension less than or equal to k, then we
identify the homotopy type of the classifying space B{Cob}_{d,<k>} as the zero
space of a homotopy colimit of certain diagram of Thom spectra. We also
identify the homotopy type of the corresponding cobordism category when extra
tangential structure is assumed on the manifolds. These results generalize the
results of Galatius, Madsen, Tillmann and Weiss, and the proofs are an
adaptation of the their methods. As an application we describe the homotopy
type of the category of open and closed strings with a background space X, as
well as its higher dimensional analogues. This generalizes work of
Baas-Cohen-Ramirez and Hanbury.
