# On infinity-morphisms between algebras over algebraic operads

I posted this question in the "Mathematics" stack exchange, but it hasn't got much attention... I hope it will get more here.

Let $$P$$ be a Koszul operad.

In the book of Loday-Vallette "Algebraic Operads", an $$\infty$$-morphism between algebras $$A$$, $$B$$ over a $$P_\infty$$-operad is defined as a dg-morphism $$P^\textrm{¡}(A) \to P^{\textrm{¡}}(B)$$ of cofree dg-$$P^\textrm{¡}$$-coalgebras that respects some coderivations $$d_\varphi$$, $$d_\psi$$ (determined by the $$P_\infty$$-algebra structures on $$A$$ and $$B$$). In Theorem 10.2.6 they claim that $$\infty$$-morphisms are in bijection with morphisms $$f: P^\textrm{¡} \to \textrm{End}^A_B$$ of dg-$$\mathbb{S}$$-modules satisfying the equation $$\partial(f) = f*\varphi - \psi\stackrel{\bigcirc}{*}f.$$

My question goes as follows. If $$f$$ is a morphism of dg-$$\mathbb{S}$$-modules, shouldn't $$\partial(f) = 0$$? Then why do they write the equation in this form? I believe I'm missing the point... I would be very grateful if someone could point me out in the right direction!

It is a typo. The map $$f$$ should only be assumed to be a morphism of the underlying graded $$\mathbb S$$-modules.
• Oh thank you so much!!!! So does this mean that $\infty$-morphisms are in bijection with morphisms of $\mathbb{S}$-modules $f: P^\textrm{¡} \to \textrm{End}^A_B$ satisfying the equation? Nov 20 at 9:32