  Complete filtered $A_\infty$-algebras model certain deformation problems in
the noncommutative setting. The formal deformation theory of a group
representation is a classical example. With such applications in mind, we
provide the $A_\infty$ analogs of several key theorems from the Maurer-Cartan
theory for $L_\infty$-algebras. In contrast with the $L_\infty$ case, our
results hold over a field of arbitrary characteristic. We first leverage some
abstract homotopical algebra to give a concise proof of the
$A_\infty$-Goldman-Millson theorem: The nerve functor, which assigns a
simplicial set $\mathcal{N}_{\bullet}(A)$ to an $A_\infty$-algebra $A$, sends
filtered quasi-isomorphisms to homotopy equivalences. We then characterize the
homotopy groups of $\mathcal{N}_\bullet(A)$ in terms of the cohomology algebra
$H(A)$, and its group of quasi-invertible elements. Finally, we return to the
characteristic zero case and show that the nerve of $A$ is homotopy equivalent
to the simplicial Maurer-Cartan set of its commutator $L_\infty$-algebra. This
answers a question posed by N. de Kleijn and F. Wierstra in arXiv:1809.07743.
