**Lawrence Breen**

University of Paris-13, France

February 4, 2010

**Combinatorial differential forms'**:
A degree n differential form on a manifold (or scheme) X is generally thought of as a linear combination of n^th exterior
products of 1-forms on X. I will discuss a more intuitive definition of such n-forms, which sheds some light on a number of their properties. This will be illustrated by considering the notion of a connection form on a principal G-bundle on X and its associated curvature 2-form. If time permits, I will then examine how these notions can be extended to a categorified context. One is then led to consider a gerbe endowed with differential data consisting of
a connection and a certain 2-form known as the B-field. Such data then geometrically determines an associated curvature 3-form satisfying a higher version of the Bianchi identity.