Infinity Operads and Monoidal Categories with Group Equivariance

TL;DR

This monograph develops a unified theory of operads, infinity operads, and monoidal categories with group equivariance, providing new insights and tools for their applications in quantum algebra and Lie theory.

Contribution

It introduces the concept of infinity group operads and a unifying framework for equivariant monoidal categories with diverse examples.

Findings

Established foundations for infinity group operads.

Unified treatment of coherence and strictification.

Examples include ribbon and symmetric monoidal categories.

Abstract

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.