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

Abstract

In this review we give a detailed introduction to the theory of (curved) $L_{\infty }$-algebras and $L_{\infty }$-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 $L_{\infty }$-algebras and $L_{\infty }$-modules. In particular, one can interpret $L_{\infty }$-morphisms and morphisms of $L_{\infty }$-modules as Maurer-Cartan elements in certain $L_{\infty }$-algebras, and we show that twisting the morphisms with equivalent Maurer-Cartan elements yields homotopic morphisms.

62 pages, comments are welcome!