Faa di Bruno for operads

For R any operad in groupoids, the associated incidence
bialgebra is shown to contain a formal series (the 'connected
Green function' in geek speak) satisfying the Faa di Bruno
formula (dual to composition of power series).  The formula is
exhibited as the homotopy cardinality of an equivalence of
groupoids.  When R is the terminal reduced operad Comm, it is
the classical Faa di Bruno formula.  When R is the free operad
on a finitary polynomial endofunctor P, it is the Faa di Bruno
formula for P-trees of Galvez-Kock-Tonks.  This is joint work
with Mark Weber.