The classic paper on Koszul duality for operads is . It is very difficult for a physicist to follow, but let us make a note on a remark on page 3. They interpret the duality for operads in terms of a dualising functor hom(-,Lie)
is the operad associated to Lie algebras.
This is like the behaviour of schizophrenic
objects, meaning in this case the object Lie
. A more classical example is the object U(1) underlying Pontrjagin
 V. Ginzburg and M. Kapranov, Duke Math. J. 76
, 1 (1994) 203-272