  We consider the category whose objects are filtered, or complete,
$L_\infty$-algebras and whose morphisms are $\infty$-morphisms which respect
the filtrations. We then discuss the homotopical properties of the
Getzler-Hinich simplicial Maurer-Cartan functor which associates to each
filtered $L_\infty$-algebra a Kan simplicial set, or $\infty$-groupoid. In
previous work with V. Dolgushev, we showed that this functor sends weak
equivalences of filtered $L_\infty$-algebras to weak homotopy equivalences of
simplicial sets. Here we sketch a proof of the fact that this functor also
sends fibrations to Kan fibrations. To the best of our knowledge, only special
cases of this result have previously appeared in the literature. As an
application, we show how these facts concerning the simplicial Maurer--Cartan
functor provide a simple $\infty$-categorical formulation of the Homotopy
Transfer Theorem.
