Operads and equivariance

TL;DR

This paper introduces the concept of action operads, algebraic structures that encode parametrized group actions on operads, and explores their algebraic properties and categorical interpretations.

Contribution

It defines action operads and investigates their algebraic structure and their actions on operads, extending the theory of operads with group actions.

Findings

Characterization of the algebra of action operads Λ

Development of the theory of Λ-operads and their properties

Categorical interpretation of Λ-operads in the 2-category of small categories

Abstract

Operads were originally defined by May to have right actions of the symmetric groups, but later formulations have also used no groups actions at all or group actions by such families as the braid groups. We call such families action operads, as they are the algebraic objects that encode parametrized group actions on operads. In Part I of this paper, we study the basic algebra of action operads $\Lambda$ and the $\Lambda$-operads they act upon. In Part II, we study $\Lambda$-operads in the 2-category of small categories.