  We define a simplicial category called the category of derived manifolds. It
contains the category of smooth manifolds as a full discrete subcategory, and
it is closed under taking arbitrary intersections in a manifold. A derived
manifold is a space together with a sheaf of local $C^\infty$-rings that is
obtained by patching together homotopy zero-sets of smooth functions on
Euclidean spaces.
  We show that derived manifolds come equipped with a stable normal bundle and
can be imbedded into Euclidean space. We define a cohomology theory called
derived cobordism, and use a Pontrjagin-Thom argument to show that the derived
cobordism theory is isomorphic to the classical cobordism theory. This allows
us to define fundamental classes in cobordism for all derived manifolds. In
particular, the intersection $A\cap B$ of submanifolds $A,B\subset X$ exists on
the categorical level in our theory, and a cup product formula
$$[A]\smile[B]=[A\cap B]$$ holds, even if the submanifolds are not transverse.
One can thus consider the theory of derived manifolds as a {\em
categorification} of intersection theory.
