Faa di Bruno for operads and internal algebras

Mark Weber – 14 December 2016

(joint work with Joachim Kock -- report on arXiv1609.03276)

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