  The paper gives a categorical approach to generalized manifolds such as orbit
spaces and leaf spaces of foliations. It is suggested to consider these spaces
as sets equipped with some additional structure which generalizes the notion of
atlas. The approach is compared with the known ones that use the Grothendieck
topos, the Haefliger classifying space and the Connes non-commutative geometry.
The main aim of this paper is to indicate that the suggested approach permits
to simplify essentially the modern theory of the leaf space of foliations and
the orbit space of diffeomorphism groups, and to obtain some new results on the
characteristic classes of foliations. As an application, it is shown that the
first Chern class is non-trivial for the leaf space of the Reeb foliation and
its geometrical meaning is indicated.
