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An introduction to $L_{\infty}$ -algebras and their homotopy theory for the working mathematician

Mathematisches Institut, Abteilung Reine Mathematik, Albert-Ludwigs-Universität Freiburg, Ernst-Zermelo-Str. 1, 79104 Freiburg i. Br., Germany

E-mail Address: kraft.a@protonmail.com

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and

Jonas Schnitzer

https://orcid.org/0000-0003-0348-4242

Mathematisches Institut, Abteilung Reine Mathematik, Albert-Ludwigs-Universität Freiburg, Ernst-Zermelo-Str. 1, 79104 Freiburg i. Br., Germany

E-mail Address: jonas.schnitzer@math.uni-freiburg.de

Corresponding author.

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https://doi.org/10.1142/S0129055X23300066Cited by:4 (Source: Crossref)

Abstract

In this paper, we give a detailed introduction to the theory of (curved) $L_{\infty}$ -algebras and $L_{\infty}$ -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 $L_{\infty}$ -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. We hope that these notes provide an accessible entry point to the theory of $L_{\infty}$ -algebras.

Keywords:

AMSC: 17B55, 16E45

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