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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 Severa'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).
Fischer et al. (Mon,) studied this question.