Homotopy morphisms between convolution homotopy Lie algebras

TL;DR

This paper extends the bifunctor structure on homotopy Lie algebras to include $ abla$-morphisms, enhancing the understanding of morphisms between convolution homotopy Lie algebras and their applications to rational models.

Contribution

It generalizes the bifunctor to include $ abla$-morphisms in one slot and demonstrates limitations for simultaneous $ abla$-morphisms in both slots, with applications to rational models.

Findings

Extended bifunctor to include $ abla$-morphisms in one slot.

Provided a counterexample for simultaneous $ abla$-morphisms in both slots.

Applied theory to rational models for mapping spaces.

Abstract

In previous works by the authors, 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.