An introduction to L∞-algebras and their homotopy theory for the working mathematician

Abstract

In this paper, we give a detailed introduction to the theory of (curved) [Formula: see text]-algebras and [Formula: see text]-morphisms, avoiding the concept of operads and providing explicit formulas. 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 [Formula: see text]-algebras and [Formula: see text]-modules. In particular, one can interpret [Formula: see text]-morphisms and morphisms of [Formula: see text]-modules as Maurer–Cartan elements in certain [Formula: see text]-algebras, and we show that twisting the morphisms with equivalent Maurer–Cartan elements yields homotopic morphisms. We hope that these notes provide an accessible entry point to the theory of [Formula: see text]-algebras.