Compatible $L_{\mathrm{\infty }}$-algebras

A compatible $L_{\mathrm{\infty }}$-algebra is a graded vector space together with two compatible $L_{\mathrm{\infty }}$-algebra structures on it. Given a graded vector space, we construct a graded Lie algebra whose Maurer-Cartan elements are precisely compatible $L_{\mathrm{\infty }}$-algebra structures on it. We provide examples of compatible $L_{\mathrm{\infty }}$-algebras arising from Nijenhuis operators, compatible $V$-datas and compatible Courant algebroids. We define the cohomology of a compatible $L_{\mathrm{\infty }}$-algebra and as an application, we study formal deformations. Next, we classify strict' and skeletal’ compatible $L_{\mathrm{\infty }}$-algebras in terms of crossed modules and cohomology of compatible Lie algebras. Finally, we introduce compatible Lie $2$-algebras and find their relationship with compatible $L_{\mathrm{\infty }}$-algebras.