The incidence comodule bialgebra of the Baez--Dolan construction

Abstract:

Starting from any operad P, one can consider on one hand the free operad on P, and on the other hand the Baez–Dolan construction on P. These two new operads have the same space of operations, but with very different notions of arity and substitution. The main result is that the incidence bialgebras of the two-sided bar constructions of the two operads constitute together a comodule bialgebra. The motivating example is the Calaque–Ebrahimi-Fard–Manchon comodule bialgebra of trees from numerical analysis. The purpose of the talk is to explain all the concepts mentioned in the abstract, and also some background on polynomial monads, simplicial groupoids, and homotopy combinatorics.

Reference: http://arxiv.org/abs/1912.11320