Homotopy models for $L_{\mathrm{\infty }}[1]$-algebras in higher degrees

We propose a model of higher homotopy theory of $L_{\mathrm{\infty }}[1]$-morphisms as a natural generalization of the $A_{\mathrm{\infty }}$-homotopies defined by Fukaya-Oh-Ohta-Ono \cite{FOOO1}. Within this framework, we show that a filling condition holds for simplices whose vertices are assigned quasi-isomorphisms.