Infinite symmetric group, pseudomanifolds, and combinatorial cobordism-like structures

We consider a category B whose morphisms are n-dimensional pseudomanifolds equipped with certain additional structures (coloring and labeling of some cells), multiplication of morphisms is similar to a concatenation of cobordisms. On the other hand, we consider the product G of (n + 1) copies of infinite symmetric group. We construct a correspondence between the sets of morphisms of B and double coset spaces of G with respect to certain subgroups. We show that unitary representations of G produce functors from the category of B to the category of Hilbert spaces and bounded linear operators.