Action of derived automorphisms on infinity-morphisms

In this paper we investigate how to simultaneously change homotopy algebras of a certain type and a corresponding infinity morphism between them, and show that this can be done in a homotopically unique way. More precisely, for a reduced cooperad C, given Ω(C)-algebras V and W and an infinity-morphism U from V to W, for any derivation ϕof Ω(C) we produce new Ω(C)-algebras V' and W' and a new infinity-morphism U' between them, that are unique up to homotopy. Operads play the central role in answering this question, in particular a 2-colored operad Cyl(C) that governs pairs of homotopy algebras and infinity-morphisms between them.

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