  The theories of strings and $D$-branes have motivated the development of non
Abelian cohomology techniques in differential geometry, on the purpose to find
a geometric interpretation of characteristic classes. The spaces studied here,
like orbifolds are not often smooth. In classical differential geometry, non
smooth spaces appear also naturally, for example in the theory of foliations,
the space of leaves can be an orbifold with singularities. The scheme to study
these structures is identical: classical tools used in differential geometry,
like connections, curvature are adapted. The purpose of this paper is to
present the notion of differential category which unifies all these points of
view. This enables us to provide a geometric interpretation of 5-characteristic
classes, and to interpret classical problems which appear in the theory of
$G$-structures by using gerbes.
