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

TL;DR

This paper provides a comprehensive introduction to the theory of curved $L_$-algebras, focusing on their homotopy theory, Maurer-Cartan elements, and the twisting procedure, highlighting their role in understanding morphisms and modules.

Contribution

It offers a detailed review of the homotopy theory of $L_$-algebras and modules, emphasizing the interpretation of morphisms as Maurer-Cartan elements and the twisting process.

Findings

Interpretation of $L_$-morphisms as Maurer-Cartan elements

Twisting with equivalent Maurer-Cartan elements yields homotopic morphisms

Detailed framework for the homotopy theory of $L_$-algebras and modules

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.