L_∞-Algebras and Derived Formal Moduli Problems

Abstract

Broadly speaking, deformation theory deals with families of structures that arise when varying a given object in dependence of some suitable parameter space, comprising the study of moduli spaces, which are spaces parameterizing equivalence classes of structures. With a formal moduli problem we thus mean the infinitesimal description of a moduli space, capturing the local structure around a given point. In this chapter we first address the classical theory of algebraic deformation problems, before explaining how formal moduli problems arise as deformation functors in algebraic geometry.