Intertwining for semi-direct product operads

TL;DR

This paper explores the relationship between semi-direct product constructions and Borel constructions in the context of topological operads and cooperads, extending existing bar construction methods.

Contribution

It demonstrates that the semi-direct product and Borel constructions are interconnected via the topological operadic bar construction and generalizes Ching's bar construction to broader operad classes.

Findings

Established the intertwining of semi-direct and Borel constructions.

Generalized Ching's bar construction to non-reduced operads.

Provided a unified framework for operad and cooperad constructions.

Abstract

This paper shows that the semi-direct product construction for $G$-operads and the levelwise Borel construction for $G$-cooperads are intertwined by the topological operadic bar construction. En route we give a generalization of the bar construction of M. Ching from reduced to certain non-reduced topological operads.