  We give a construction of an L-infinity map from any L-infinity algebra into
its truncated Chevalley-Eilenberg complex as well as its cyclic and A-infinity
analogues. This map fits with the inclusion into the full Chevalley-Eilenberg
complex (or its respective analogues) to form a homotopy fiber sequence of
L-infinity-algebras. Application to deformation theory and graph homology are
given. We employ the machinery of Maurer-Cartan functors in L-infinity and
A-infinity algebras and associated twistings which should be of independent
interest.
