Infinite symmetric groups and two-dimensional simplicial bordisms

We consider an infinite symmetric group $G$. For some its
subgroups $K$ double cosets $K\setminus G/ K$ admit a natural  structure of
a semigroup (remark for experts: this semigroup acts in  $K$-fixed vectors
of representations of $G$).  Elements of $K\setminus  G/ K$ can be
interpreted as two-dimensional polygonal bordisms, the  multplication is
the product of bordisms. We also
produce a family of constructions in the spirit of 'topological field
theories' (i.e., for each bordism we write a matrix, product of  bordisms
corresponds to product of matrices).