Backround Definition Presentation over a site Models We discuss the refinement to higher differential geometry of the concept of a circle group-principal connection (on a principal U(1)-bundle). Specifically we indicate how the general abstract definition in terms of cohesion reproduces in the context of smooth cohesion to the representation of circle -connections by cocycles in smooth Deligne cohomology (dcct). In every cohesive (∞,1)-topos there is an intrinsic notion of differential cohomology with coefficients in an abelian group object that classifies -principal ∞-bundles with ∞-connection. Here we discuss the specific realization for Smooth∞Grpd the (∞,1)-topos of smooth ∞-groupoids and the circle group. In this case the intrinsic differential cohomology reproduces ordinary differential cohomology and generalizes it to base spaces that may be smooth manifolds, diffeological spaces, orbifolds and generally smooth ∞-groupoids such as deloopings of smooth ∞-groups . Differential cocycles on the latter support the ∞-Chern-Weil homomorphism that sends nonabelian ∞-connections to circle -bundles whose curvature form realizes a characteristic class in de Rham cohomology. Let Smooth∞Grpd be the cohesive (∞,1)-topos of smooth ∞-groupoid. As usual, write for the terminal global section (∞,1)-geometric morphism with its extra left adjoint, the intrinsic fundamental ∞-groupoid functor . From this induced is the path ∞-groupoid adjunction and the intrinsic de Rham cohomology adjunction For an abelian group object there for each integer is the universal curvature characteristic form, given by a cocycle-morphism The cocycles for differential cohomology in degree with coefficients in are the points in the homotopy fiber of the morphism on cohomology induced by this. Every such cocycle we may think of as an ∞-connection on the -principal ∞-bundle classified by the underlying cocycle in . We consider these constructions in the model