Adjusted Connections I: Differential Cocycles for Principal Groupoid Bundles with Connection

We develop a new perspective on principal bundles with connection as morphisms from the tangent bundle of the underlying manifold to a classifying dg-Lie groupoid. This groupoid can be identified with a lift of the inner homomorphisms groupoid arising in {S}evera's differentiation procedure of Lie quasi-groupoids. Our new perspective readily extends to principal groupoid bundles, but requires an adjustment, an additional datum familiar from higher gauge theory. The resulting adjusted connections naturally provide a global formulation of the kinematical data of curved Yang-Mills-Higgs theories as described by Kotov-Strobl (arXiv:1510.07654) and Fischer (arXiv:2104.02175).