Algebraic aspects of higher nonabelian Hodge theory

We look more closely at the higher nonabelian de Rham cohomology of a smooth projective variety or family of varieties that had been defined in some previous papers. We formalize using $n$-stacks the notion of shape underlying this nonabelian cohomology. A generalization of the de Rham construction to any appropriate formal category or family of formal categories, yields various algebraic aspects of Hodge theory for the de Rham shape, such as the Hodge filtration, the Gauss-Manin connection, Griffiths transversality, an extension of the Gauss-Manin connection with regular singularities across singular points, etc. Along the way, we develop a little bit more technology for $n$-categories, such as a canonical fibrant replacement for the $n+1$-category $nCAT$; and we pose the following general type of question: what are the properties of the nonabelian cohomology $n$-stack $Hom(X,T)$ as a function of the properties of the coefficient $n$-stack $T$ and the domain $n$-stack $X$?