In addition to the standard non-singular Cobordism theories (Oriented, Complex, Spin, etc...) one can study certain "singular" Corbordism theories corresponding to manifolds with singularities. For a given set of singularity types (I'll explain exactly what this means in the talk) one can define Cobordism groups consisting of manifolds with singularities from that given set of singularity types. These singular Cobordism theories (there is one for each choice of singularity set) can be compared to each other and the non-singular theories via an exact couple known as the Bockstein-Sullivan exact couple.
Following recent work in Cobordism Categories, for a given a set of singularity types I show how to construct a Category whose objects are closed Manifolds with singularities from the given set of singularity types and whose morphisms are the singular Cobordisms connecting them. I then identify the homotopy type of the classifying space of such a singular Cobordism Category with that of the infinite loop-space of a certain spectrum, constructed out of Thom-spectra. My result is a direct generalization of the work of S. Galatius, I. Madsen, U. Tillmann, and M. Weiss in "The Homotopy Type of a Cobordism Category" and uses many similar techniques.