Operads and equivariance

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.