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

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.

arXiv comment20 pages, comments welcomed

References in corpus (2)

• Maurer-Cartan moduli and theorems of Riemann-Hilbert type
• Convolution algebras and the deformation theory of infinity-morphisms