In previous works by the authors – [26, 31] – a bifunctor was associated to any operadic twisting morphism, taking a coalgebra over a cooperad and an algebra over an operad, and giving back the space of (graded) linear maps between them endowed with a homotopy Lie algebra structure. We build on this result by using a more general notion of \infty -morphism between (co)algebras over a (co)operad associated to a twisting morphism, and show that this bifunctor can be extended to take such \infty -morphisms in either one of its two slots. We also provide a counterexample proving that it cannot be coherently extended to accept \infty -morphisms in both slots simultaneously. We apply this theory to rational models for mapping spaces.

Given a coalgebra C over a cooperad and an algebra A over an operad, it is often possible to define a natural homotopy Lie algebra structure on hom(C, A), the space of linear maps between them, called the convolution algebra of C and A. In the present article, we use convolution algebras to define the deformation complex for ∞-morphisms of algebras over operads and coalgebras over cooperads. We also complete the study of the compatibility between convolution algebras and ∞-morphisms of algebras and coalgebras. We prove that the convolution algebra bifunctor can be extended to a bifunctor that accepts ∞morphisms in both slots and which is well defined up to homotopy, and we generalize and take a new point of view on some other already known results. This paper concludes a series of works by the two authors dealing with the investigation of convolution algebras. Contents 1 Introduction 352 2 Infinity-morphisms and convolution algebras 354 3 Compatibility between convolution algebras and infinity-morphisms 357 4 Proof of Theorems 2.4, 3.1, and 3.6 362 A A counterexample 370

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