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Higher operads, higher categories. (English) Zbl 1160.18001

London Mathematical Society Lecture Note Series 298. Cambridge: Cambridge University Press (ISBN 0-521-53215-9/pbk). xiv, 433 p. (2004).
The book gives an exposition of the basic framework of the higher category theory. Recent years have brought a lot of activity in this area, in large extent focused on the problem of defining the notion of a weak \(n\)-category. This work resulted in several proposed definitions; see e.g. [Theory Appl. Categ. 10, 1–70 (2002; Zbl 0987.18007)], also by T. Leinster, for an overview of ten of them. The present book concentrates on the approach originated by M. Batanin [Adv. Math. 136, No. 1, 39–103 (1998; Zbl 0912.18006)] who introduced the language of higher operads and defined weak \(n\)-categories as algebras over certain higher operads. While this strategy has been recognized as a good basis for development of the higher category theory, it relies on an extensive technical machinery. As a consequence it is easy for someone who encounters this subject for the first time to lose sight, amid abstract formalism, of the underlying intuitive ideas. Fortunately in the book under review the author goes to great lengths to help the reader avoid such problems. The book emphasizes connections of the higher category theory with geometry (one of the first sections is “Motivation for topologists”) and algebra. Newly introduced notions are illustrated by several examples, and an abundance of diagrams scattered throughout the text further enhances its clarity.
The book is divided into three main parts. The first one provides a review of some standard categorical language. While a prospective reader should have prior knowledge of the basics of category theory, it makes this text largely self-contained. The second part develops the theory of generalized operads and their algebras that, in the third part, is specialized to the instance of globular operads, yielding a definition of a weak \(n\)-category. This definition is very similar to that of Batanin, although the approach of the author results in some simplifications. In the spirit of keeping emphasis on the essential aspects of the theory throughout the main body of the book, more technical constructions and proofs are relegated to several appendices.
A more condensed version of this book appeared in [Theory Appl. Categ. 12, 73–194 (2004; Zbl 1065.18006)]. It is worth to correct here a typo appearing on the cover of the book: contrary to what it states Tom Leinster is the author and not the editor of this volume.

MSC:

18-02 Research exposition (monographs, survey articles) pertaining to category theory
55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
18D50 Operads (MSC2010)
55P48 Loop space machines and operads in algebraic topology
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
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