An Introduction to $L_{\mathrm{\infty }}$-Algebras and their Homotopy Theory

In this review we give a detailed introduction to the theory of (curved) $L_{\mathrm{\infty }}$-algebras and $L_{\mathrm{\infty }}$-morphisms. In particular, we recall the notion of (curved) Maurer-Cartan elements, their equivalence classes and the twisting procedure. The main focus is then the study of the homotopy theory of $L_{\mathrm{\infty }}$-algebras and $L_{\mathrm{\infty }}$-modules. In particular, one can interpret $L_{\mathrm{\infty }}$-morphisms and morphisms of $L_{\mathrm{\infty }}$-modules as Maurer-Cartan elements in certain $L_{\mathrm{\infty }}$-algebras, and we show that twisting the morphisms with equivalent Maurer-Cartan elements yields homotopic morphisms.