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.