abstract: A derived manifold is a space with a homotopical sheaf of C\infty rings, which locally looks like the zero set of smooth functions on a manifold.
I will give an axiomatization of derived manifolds, which is designed to allow one to understand their behavior without being immersed in the technical details. From these axioms, I will show that the cobordism theory of derived manifolds is equivalent to that of smooth manifolds, and that in particular, every derived manifold has a fundamental cobordism class. In particular, a cup product formula of the form
for derived submanifolds A and B, holds with full generality.