Lie Theory for Complete Curved $A_\infty$-algebras

TL;DR

This paper extends Maurer-Cartan theory to curved $A_ abla$-algebras, constructing a simplicial set functor that models their deformation theory and forms a Kan complex, applicable over fields of characteristic zero or greater.

Contribution

It introduces an $A_ abla$-analog of the Maurer-Cartan simplicial set, enabling the study of deformation problems of $ abla$-morphisms over non-symmetric operads.

Findings

The functor produces a Kan complex for curved $A_ abla$-algebras.

It effectively models deformation theory over fields of characteristic zero or higher.

Application to $ abla$-morphisms demonstrates its utility in non-symmetric operad contexts.

Abstract

In this paper we develop the $A_\infty$-analog of the Maurer-Cartan simplicial set associated to an $L_\infty$-algebra and show how we can use this to study the deformation theory of $\infty$-morphisms of algebras over non-symmetric operads. More precisely, we define a functor from the category of (curved) $A_\infty$-algebras to simplicial sets which sends an $A_\infty$-algebra to the associated simplicial set of Maurer-Cartan elements. This functor has the property that it gives a Kan complex. We also show that this functor can be used to study deformation problems over a field of characteristic greater or equal than $0$. As a specific example of such a deformation problem we study the deformation theory of $\infty$-morphisms over non-symmetric operads.