  We show that Verdier duality for certain sheaves on the moduli spaces of
graphs associated to Koszul operads corresponds to Koszul duality of operads.
This in particular gives a conceptual explanation of the appearance of graph
cohomology of both the commutative and Lie types in computations of the
cohomology of the outer automorphism group of a free group. Another consequence
is an explicit computation of dualizing sheaves on spaces of metric graphs,
thus characterizing to which extent these spaces are different from oriented
orbifolds. We also provide a relation between the cohomology of the space of
metric ribbon graphs, known to be homotopy equivalent to the moduli space of
Riemann surfaces, and the cohomology of a certain sheaf on the space of usual
metric graphs.
