Faà di Bruno for operads and internal algebras

For any coloured operad R, we prove a Fa\`a di Bruno formula for the `connected Green function' in the incidence bialgebra of R. This generalises on one hand the classical Fa\`a 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 Faa di Bruno formula for P-trees of G\'alvez--Kock--Tonks (P a finitary polynomial endofunctor), which corresponds to the case where R is the free operad on P. Following G\'alvez--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 fact we establish the formula more generally in a relative situation, for algebras for one polynomial monad internal to another. This covers in particular nonsymmetric operads (for which the terminal reduced case yields the noncommutative Fa\`a di Bruno formula of Brouder--Frabetti--Krattenthaler).

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