The homotopy theory of filtered, or complete, L\infty -algebras plays an interesting
role in areas such as the deformation theory of homotopy algebras, and the
rational homotopy theory of mapping spaces. One important tool used in these
applications is the simplicial Maurer{Cartan functor, which produces from any
ltered L\infty -algebra a Kan simplicial set, or 1 -groupoid. In this talk, I will
present some recent results that extend our previous work with V. Dolgushev,
and describe how the simplicial Maurer{Cartan functor relates the homotopy
theory of ltered L1 -algebras to that of simplicial sets. I will then provide some
applications, including simple proofs of \1 -categorical analogs" of the existence
and uniqueness statements that comprise the Homotopy Transfer Theorem.