On the de Rham Cohomology of Differential and Algebraic Stacks

We introduce the notion of cofoliation on a stack. A cofoliation is a change of the differentiable structure which amounts to giving a full representable smooth epimorphism. Cofoliations are uniquely determined by their associated Lie algebroids. Cofoliations on stacks arise from flat connections on groupoids. Connections on groupoids generalize connections on gerbes and bundles in a natural way. A flat connection on a groupoid is an integrable distribution of the morphism space compatible with the groupoid structure and complementary to both source and target fibres. A cofoliation of a stack determines the flat groupoid up to étale equivalence. We show how a cofoliation on a stack gives rise to a refinement of the Hodge to De Rham spectral sequence, where the E1-term consists entirely of vector bundle valued cohomology groups. Our theory works for differentiable, holomorphic and algebraic stacks.