A new approach to formal moduli problems

TL;DR

This paper introduces a novel operadic calculus-based framework for formal moduli problems, establishing an adjunction between deformation problems and their Koszul duals, and providing new proofs and generalizations of existing theorems.

Contribution

It develops a new operadic approach to infinitesimal deformation problems, including an effective method for constructing models and a homological criterion for equivalences.

Findings

Established an adjunction between deformation problems and Koszul dual algebras

Provided a new proof of the Lurie--Pridham theorem

Offered concrete models for deformation problems

Abstract

The main goal of this paper is to introduce a framework for infinitesimal deformation problems, using new methods coming from operadic calculus. We construct an adjunction between infinitesimal deformation problems over some type of algebras and their Koszul dual algebras, in any characteristic. This adjunction is an equivalence if and only if some algebras are equivalent to their completions. We give a concrete homological criterion for it. It gives us a new proof of the celebrated Lurie--Pridham theorem, as well as of many other generalizations of it. Our methods are effective, meaning they directly produce point-set models for the algebras that encode infinitesimal deformation problems.