I would like to understand the $L_\infty$ structure on the tensor product of an $L_\infty$ algebra (over $\mathbb{R}$) $L$ with the normalized cochains on the one-simplex $N^*(\Delta^1)$. This latter object is an $E_\infty$ algebra, and so, by the answer in Tensor products of $\infty$-algebras over operads, there is an $L_\infty$-algebra structure on $L\otimes N^*(\Delta^1)$. Is there an explicit (and hopefully simple) formula for such an $L_\infty$ structure on $L\otimes N^*(\Delta^1)$?