  We construct a symmetric monoidal category $LIE^{MC}$ whose objects are
shifted L-infinity algebras equipped with a complete descending filtration.
Morphisms of this category are "enhanced" infinity morphisms between shifted
L-infinity algebras. We prove that any category enriched over $LIE^{MC}$ can be
integrated to a simplicial category whose mapping spaces are Kan complexes. The
advantage gained by using enhanced morphisms is that we can see much more of
the simplicial world from the L-infinity algebra point of view. We use this
construction in a subsequent paper to produce a simplicial model of a
$(\infty,1)$-category whose objects are homotopy algebras of a fixed type.
