Faà di Bruno for operads and internal algebras

For any coloured operad R, we prove a Faà di Bruno formula for the ‘connected Green function’ in the incidence bialgebra of R. This generalises on one hand the classical Faà di Bruno formula (dual to composition of power series), corresponding to the case where R is the terminal reduced operad, and on the other hand the Faà di Bruno formula for R -trees of Gálvez–Kock–Tonks (P a finitary polynomial endofunctor), which corresponds to the case where R is the free operad on P. Following Gálvez–Kock–Tonks, we work at the objective level of groupoid slices, hence all proofs are ‘bijective’: the formula is established as the homotopy cardinality of an explicit equivalence of groupoids, in turn derived from a certain two-sided bar construction. In fact we establish the formula more generally in a relative situation, for algebras of one polynomial monad internal to another. This covers in particular nonsymmetric operads (for which the terminal reduced case yields the noncommutative Faà di Bruno formula of Brouder–Frabetti–Krattenthaler).