The incidence comodule bialgebra of the Baez–Dolan construction

Starting from any operad P, one can consider on hand the free and other Baez–Dolan construction P. These two new operads have same space of operations, but with very different notions arity substitution. The main result this paper is that incidence bialgebras two-sided bar constructions constitute together a comodule bialgebra. objective: it concerns comodule-bialgebra structures groupoid slices, proof given in terms equivalences groupoids homotopy pullbacks. Comodule usual sense are obtained by taking cardinality. simplest instances cover several current interest analysis. If P identity monad, then Faà di Bruno bialgebra (dual to multiplication substitution power series). monoid Ω (considered as one-coloured only unary operations), resulting dual near-semiring Ω-moulds under product composition, employed Écalle's theory resurgent functions local dynamical systems. terminal operad, essentially Calaque–Ebrahimi-Fard–Manchon rooted trees, composition B-series numerical analysis (Chartier–Hairer–Vilmart). full generality category theory. As holds for actually about itself, providing algebraic perspective.