Atiyah classes of Lie algebroid homotopy modules

For a Lie algebroid pair A L we study cocycles constructed from the extension to L of the higher connection forms of a representation up to homotopy E of the Lie algebroid A. We show that there exists a cohomology class with values in the endomorphism bundle of E that is independent of the extension above and vanishes whenever a homotopy A-compatible extension exists. Whenever the representation up to homotopy E is the resolution of a Lie algebroid representation K of A, it is shown that there exists a quasi-isomorphism sending the new Atiyah class to the classical one, associated to extensions to L of the Lie algebroid representation K.