In every cohesive (∞,1)-topos there is an intrinsic notion of ∞-Chern-Weil theory that gives rise to a notion of connection on principal ∞-bundles. We describe here details of the realization of this general abstract structure in the cohesive -topos Smooth∞Grpd of smooth ∞-groupoids. For an ∞-Lie group, a connection on a smooth -principal ∞-bundle is a structure that supports the Chern-Weil homomorphism in Smooth∞Grpd: it interpolates between the nonabelian cohomology class of the bundle and...