Descent Theory for Higher Stacks and the Cohomology of Infinity-Topoi in Univalent Foundations

This monograph establishes a rigorous synthetic framework for descent theory within the context of Univalent Foundations, specifically targeting the cohomology of higher stacks in cohesive -topoi. We construct a univalent formulation of the -categorical Giraud axioms, enabling a direct internal treatment of hypercovers and simplicial presheaves without recourse to external model-theoretic machinery. By leveraging the univalence axiom and higher inductive types, we characterize the cohomology of an object X with coefficients in an -stack A as the connected components of the derived mapping space Map(X, Bn A). Furthermore, we prove a generalized descent theorem for principal -bundles, demonstrating that the local-to-global gluing property for higher stacks is equivalent to the contractibility of the Čech nerve of the covering in the underlying -topos. This yields a non-abelian formulation of differential cohomology theories suitable for quantum gauge fields.