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Homotopical properties of the simplicial Maurer-Cartan functor

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.