Derived deformation theory of algebraic structures

Abstract

The main purpose of this article is to develop an explicit derived deformation theory of algebraic structures at a high level of generality, encompassing in a common framework various kinds of algebras (associative, commutative, Poisson...) or bialgebras (associative and coassociative, Lie, Frobenius...), that is algebraic structures parametrized by props. A central aspect is that we define and study moduli spaces of deformations of algebraic structures up to quasi-isomorphisms (and not just isotopies or isomorphisms). To do so, we implement methods coming from derived algebraic geometry, by encapsulating these deformation theories as classifying (pre)stacks with good infinitesimal properties and derived formal geometry, by means of derived formal moduli problems and derived formal groups. In particular, we prove that the Lie algebra describing the deformation theory of an object in a given $\infty$-category of dg algebras can be obtained equivalently as the tangent complex of loops on a derived quotient of this moduli space by the homotopy automorphims of this object. Moreover, we provide explicit formulae for such derived deformation problems of algebraic structures up to quasi-isomorphisms and relate them in a precise way to other standard deformation problems of algebraic structures. This relation is given by a fiber sequence of the associated dg-Lie algebras of their deformation complexes. Our results provide simultaneously a vast generalization of standard deformation theory of algebraic structures which is suitable (and needed) to set up algebraic deformation theory both at the $\infty$-categorical level and at a higher level of generality than algebras over operads. In addition, we study a general criterion to compare formal moduli problems of different algebraic structures and apply our formalism to $E_n$-algebras and bialgebras.