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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!

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It is a typo. The map $f$ should only be assumed to be a morphism of the underlying graded $\mathbb S$-modules.

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  • $\begingroup$ 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? $\endgroup$
    – groupoid
    Commented Nov 20, 2023 at 9:32
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    $\begingroup$ @groupoid yes indeed! $\endgroup$
    – Dan Petersen
    Commented Nov 20, 2023 at 17:21

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