  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.
