Joachim Kock (Univ. AutÃ²noma de Barcelona)
The incidence comodule bialgebra of the Baez--Dolan construction
Starting from an operad P, one can consider on one hand the free operad on P, and on the other hand the Baez--Dolan construction on P, which I will explain in detail. These two new operads have the same space of operations, but with very different notions of arity and substitution. Together the incidence bialgebras of the two operads constitute a comodule bialgebra. If P is the terminal operad, then the result is the Calaque--Ebrahimi-Fard--Manchon comodule bialgebra (except that it is with operadic trees instead of combinatorial trees). Another example is to take as P any monoid, considered as a one-coloured operad with only unary operations. In this case the resulting comodule bialgebra is the dual of the near-semiring of moulds under product and composition.