Algebraic (geometric) $n$-stacks

Abstract

We propose a generalization of Artin's definition of algebraic stack, which we call {\em geometric $n$-stack}. The main observation is that there is an inductive structure to the definition whereby the ingredients for the definition of geometric $n$-stack involve only $n-1$-stacks and so are already previously defined. We use this inductive structure to obtain some basic properties. We look at maps from a projective variety into certain such $n$-stacks, and obtain an interpretation of the Brill-Noether locus as the set of points of a geometric $n$-stack. At the end we explain how this provides a context for looking at de Rham theory for higher nonabelian cohomology, how one can define the Hodge filtration and so on.