Operads in Derived Deformation Theory

TL;DR

This paper explores the relationship between formal moduli problems and operad algebras in derived deformation theory, extending Lurie's and Calaque-Campos-Nuiten's work with new insights and clarifications.

Contribution

It provides a pedagogical exposition of existing results and introduces original lemmas to deepen understanding of operads in derived deformation theory.

Findings

Extended the equivalence between formal moduli problems and operad algebras.

Clarified the axioms for algebras in formal moduli problems.

Introduced new lemmas supporting the theoretical framework.

Abstract

A theorem by Pridham and Lurie provides an equivalence between formal moduli problems and Lie algebras in characteristic zero. In his work, Lurie has distilled the axioms that the algebras appearing in the formal moduli problem need to satisfy, and worked out the case of $\mathbb{E}_\infty$-algebras using an incarnation of the Koszul duality, in the setting of $\infty$-operads. The more recent work of Calaque-Campos-Nuiten extends Lurie's work to obtain an equivalence between formal moduli problem parameterized by a colored operad, and algebras over its Koszul dual operad. This manuscript is both, a pedagogical exposition, and a questioning of their work, with modest, but original, supporting lemmas.