Manchester Geometry Seminar 2007/2008

The notion of a differential graded manifold is an extension of differential calculus on manifolds and encompasses Lie algebroids as well as their higher analogues, such as Courant algebroids and Weil algebroids. For example, "generalized geometry" in the sense of Hitchin is the geometry associated to a certain class of dg manifolds. One can think of dg manifolds as tangent objects associated to differentiable (higher) stacks; to make this precise, I will construct a functor from dg manifolds to presheaves of simplicial sets on the site of smooth manifolds. Using Jardine's local projective Quillen model structure on simplicial presheaves, one obtains the homotopy type of a dg manifold. The homotopy (sheaves of) groups of a dg manifold may fail to be representable by Lie groups, but are always nice enough to have Lie algebras. This failure is what obstructs the integrability of a dg manifold, i.e. the representability of its homotopy n-type by a Lie n-groupoid. I will derive a homotopy long exact sequence of a fibration of dg manifolds and use it to construct a non-integrable Courant algebroid.