Homotopy morphisms between convolution homotopy Lie algebras

Abstract

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.