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Higher Operads, Higher Categories

Tom Leinster
Cambridge University Press
Publication Date: 
Number of Pages: 
London Mathematical Society Lecture Notes Series 298
[Reviewed by
Ittay Weiss
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The birth-place of operads and categories is in homotopy theory and homology, respectively. Both structures now find applications in more areas of mathematics, computer science, and physics than can be recounted in a review (or in a book of moderate length). In homotopy theory and in theoretical physics, higher dimensional variants of operads and categories are becoming more and more prominent. Such higher dimensional structures come in two flavors: strict and weak. The strict variants are of moderate complexity and there is a consensus on what is the “correct” definition. Unfortunately, it is rarely the case that the strict variants are of any use. The weak structures, on the other hand, appear to be the more common structure one would meet out there, but the complexity of the weak structures is immense and numerous definitions have been suggested, and for most of those it is unclear whether or not they are equivalent.

The book under review is a guided tour through the jungle of higher dimensional operads and categories. The book is very well-written and would be of interest to the student already interested in the subject as well as to the student who had never heard of it but would like to acquire a better understanding for what it entails and why we should care. The clear introduction and the motivation for topologists (highlighting the a relationship between higher dimensional categories and the higher homotopy groups via the phenomenon of stabilization) points at one of several paths leading into the jungle.

The tour of the jungle of higher structures is given in three parts. Part I, providing background, is a stroll through the part of the jungle occupied by rather familiar creatures. First encountering categories the tour proceeds to present monoidal categories and enrichment so as to present the first view of the complexities of higher dimensions: strict n-categories. Preparing the ground for a visit to their weak variants, bicategories are also treated. Next on the tour are operads, presented as a generalization of categories. Finally, further notions of monoidal categories are presented, illustrating the interplay between coherence and biased choices.

Part II is devoted to operads, presenting a unifying formalism which allows one to observe that what may initially appear as animals belonging to different species have in fact a common abstract ancestor. Part III is where the tour becomes much more demanding. The path is not well-marked, the trees are very dense, and the animals bite. The author does a very good job of guiding the reader through this part of the jungle, presenting several definitions of weak n-categories and discussing the many subtleties inherent to their theory.

The book, by virtue of its comprehensive bibliography and its modern perspective on operads, may easily serve as a starting point for further reading and research on the subjects of higher dimensional categories and operads.

Ittay Weiss is Lecturer of Mathematics at the School of Computing, Information and Mathematical Sciences of the University of the South Pacific in Suva, Fiji.

Part I. Background:
1. Classical categorical structures
2. Classical operads and multicategories
3. Notions of monoidal category

Part II. Operads. 4. Generalized operads and multicategories: basics
5. Example: fc-multicategories
6. Generalized operads and multicategories: further theory
7. Opetopes

Part III. n-categories:
8. Globular operads
9. A definition of weak n-category
10. Other definitions of weak n-category

A. Symmetric structures
B. Coherence for monoidal categories
C. Special Cartesian monads
D. Free multicategories
E. Definitions of trees
F. Free strict n-categories
G. Initial operad-with-contraction