Confluence laws and Hopf-Borel type theorem for operads

TL;DR

This paper introduces confluence laws as a generalization of mixed distributive laws for operads, enabling new rigidity theorems and broadening the scope of Hopf-Borel type results through a novel categorical approach.

Contribution

It establishes a backward reasoning method from isomorphisms of species to confluence laws, leading to new rigidity theorems for operads and their associated bialgebras.

Findings

Existence of confluence laws for operads with the same underlying S-module.

Any conilpotent P coQ-bialgebra satisfying a confluence law is free and cofree over primitives.

Many new examples of operads and bialgebras are generated, recovering known cases via duality.

Abstract

In 2008, Loday shed light on the existence of Hopf-Boreltheorems for operads. Using the vocabulary of category theory, Livernet,Mesablishvili and Wisbauer extended such theorems to monads. In bothcases, the reasoning was to start from a mixed distributive law andthen to prove that it induces an isomorphism of species to finally geta rigidity theorem. Our reasoning goes here backward: we prove thatfrom an isomorphism of species one can get what we called a confluencelaw, which generalises mixed distributive laws, and that it is enough toobtain a rigidity theorem. This enables us to show that for any operadsP and Q having the same underlying S-module, there exists a confluencelaw $\alpha$ such that any conilpotent P coQ-bialgebra satisfying $\alpha$ is free andcofree over its primitive elements. Our reasoning permits us to generatemany new examples, while recovering the known ones by consideringdual relations.