  This paper exhibits fundamental structure underlying Lie algebra homology
with coefficients in tensor products of the adjoint representation, mostly
focusing upon the case of free Lie algebras.
  The main result yields a DG category that is constructed from the PROP
associated to the Lie operad. Underlying this is a two-term complex of
bimodules over this PROP; it is a quotient of the universal Chevalley-Eilenberg
complex.
  The homology of this DG category is intimately related to outer functors over
free groups (introduced in earlier joint work with Vespa). This uses the
author's previous results relating functors on free groups to representations
of the PROP associated to the Lie operad.
  This gives a direct algebraic explanation as to why the degree one homology
should correspond to an outer functor. Hitherto, the only known argument relied
upon the relationship with the higher Hochschild homology functors that arise
from the work of Turchin and Willwacher.

Algebraic Topology