Moduli problems for operadic algebras

Abstract

A theorem of Pridham and Lurie provides an equivalence between formal moduli problems and Lie algebras in characteristic zero. We prove a generalization of this correspondence, relating formal moduli problems parametrized by algebras over a Koszul operad to algebras over its Koszul dual operad. In particular, when the Lie algebra associated to a deformation problem is induced from a pre-Lie structure, it corresponds to a permutative formal moduli problem. As another example, we obtain a correspondence between operadic formal moduli problems and augmented operads.

Derived Symplectic Reduction and L-Equivariant Geometry

“…For example such a deformation functor assigns for each choice of Artinian local algebra A an ∞-groupoid of deformations of X along Spec(A). As the category of local Artinian algebras and the notion of "infinitesimal" object will play an important role, we will start, following [Lur11], [CG18] and [CCN20], by defining a general framework in which we can speak of deformations along such objects.…”

“…It is an old heuristic that (commutative) deformation problems (over k) are classified by Lie algebras and it is due to a kind of duality (Koszul duality context) that is in fact related to the Koszul duality of the Lie and commutative operads (see [CCN20]). This heuristic is formalized by the following theorem:…”