I was wondering if someone could help me understand this result, or point me towards a reference. Suppose that $M$ and $N$ are $L_{\infty}$ modules over a dgla $L$. Suppose that $\phi: M \rightarrow N$ is a morphism of modules. Then for $\pi \in MC(L)$, one defines $T_{\pi}(\phi)(x): \sum 1/k! \phi(\pi, \cdots,\pi, x)$, which is a morphism of complexes $(M, d_{M}^{\pi}) \rightarrow (N, d_{N}^{\pi})$.
This paper: https://arxiv.org/pdf/math/0010321.pdf ("Lemma "on page 19) asserts that if $\phi$ is a quasi-isomorphism of complexes, then $T_{\pi}(\phi)$ is as well, provided that $\pi$ is small enough. Unfortunately, there is no proof provided. Could someone shed some light on this? I'd greatly appreciate it.
Thanks!
