  The paper contains a survey of train constructions for infinite symmetric
groups and related groups. For certain pairs (a group $G$, a subgroup $K$), we
construct categories, whose morphisms are two-dimensional surfaces tiled by
polygons and colored in a certain way. A product of morphisms is a gluing of
combinatorial bordisms. For a unitary representation of $G$ we assign a functor
from the category of bordisms to the category of Hilbert spaces and bounded
operators. The construction has numerous variations, instead of surfaces there
arise also one-dimensional objects of Brauer diagram type, multi-dimensional
pseudomanifolds, bipartite graphs
