  This monograph provides a coherent development of operads, infinity operads,
and monoidal categories, equipped with equivariant structures encoded by an
action operad. A group operad is a planar operad with an action operad
equivariant structure. In the first three parts of this monograph, we establish
a foundation for group operads and for their higher coherent analogues called
infinity group operads. Examples include planar, symmetric, braided, ribbon,
and cactus operads, and their infinity analogues. For example, with the tools
developed here, we observe that the coherent ribbon nerve of the universal
cover of the framed little 2-disc operad is an infinity ribbon operad.
  In Part 4 we define general monoidal categories equipped with an action
operad equivariant structure, and provide a unifying treatment of coherence and
strictification for them. Examples of such monoidal categories include
symmetric, braided, ribbon, and coboundary monoidal categories, which naturally
arise in the representation theory of quantum groups and of coboundary Hopf
algebras and in the theory of crystals of finite dimensional complex reductive
Lie algebras. Many illustrations and examples are included. Assuming only basic
category theory, this monograph is intended for graduate students and
researchers. In addition to being a coherent reference for the topics covered,
this book is also suitable for a graduate student seminar and a reading course.
