Moduli problems for operadic algebras

Abstract: A classical principle in deformation theory asserts that any formal deformation problem over a field of characteristic zero is classified by a differential graded Lie algebra. This principle has been described more precisely by Lurie and Pridham, who establish an equivalence between dg-Lie algebras and formal moduli problems indexed by Artin commutative dg-algebras. I will discuss an extension of this result to more general pairs of Koszul dual operads over a field of characteristic zero. For example, there is an equivalence of infinity-categories between pre-Lie algebras and formal moduli problems indexed by permutative algebras. Under this equivalence, permutative deformations of a trivial algebra are classified by the usual pre-Lie structure on its deformation complex. In the case of the coloured operad for nonunital operads, a relative version of Koszul duality yields an equivalence between nonunital operads and certain kinds of operadic formal moduli problems. This is joint work with D. Calaque and R. Campos.

algebraic topologycategory theoryquantum algebra

Audience: learners

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