The homotopy class of twisted $L_\infty$-morphisms

The global formality of Dolgushev depends on the choice of a torsion-free covariant derivative. We prove that the globalized formalities with respect to two different covariant derivatives are homotopic. More explicitly, we derive the statement by proving a more general homotopy equivalence between $L_\infty$-morphisms that are twisted with gauge equivalent Maurer–Cartan elements.