Higher-dimensional category theory is the study of a zoo of exotic structures: operads, n-categories, multicategories, monoidal categories, braided monoidal categories, and more. It is intertwined with the study of structures such as homotopy algebras (A∞ -categories, L∞-algebras, Γ-spaces, …), n-stacks, and n-vector spaces, and draws it inspiration from areas as diverse as topology, quantum algebra, mathematical physics, logic, and theoretical computer science.

No surprise, then, that the subject has developed chaotically. The rush towards formalizing certain commonly-imagined concepts has resulted in an extraordinary mass of ideas, employing diverse techniques from most of the subject areas mentioned. What is needed is a transparent, natural, and practical language in which to express these ideas. The main aim of this book is to present one. It is the language of generalized operads. It is introduced carefully, then used to give simple descriptions of a variety of higher categorical structures.

I hope that by the end, the reader will be convinced that generalized operads provide as appropriate a language for higher-dimensional category theory as vector spaces do for linear algebra, or sheaves for algebraic geometry.