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 [https://mathoverflow.net/questions/292147/tensor-products-of-infty-algebras-over-operads][1], 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 $L\otimes N^*(\Delta^1)$?


  [1]: https://mathoverflow.net/questions/292147/tensor-products-of-infty-algebras-over-operads