Operads in algebra, topology and physics. (English) Zbl 1017.18001

Markl, Martin; Shnider, Steve; Stasheff, Jim

Mathematical Surveys and Monographs. 96. Providence, RI: American Mathematical Society (AMS). x, 349 p. (2002).

The aim of the book is to introduce operads and to present the great variety of applications for which operads turn out to be the preferred tool. Operads describe the mathematical structures of various fields in mathematics and physics. In particular they turn out to be useful for the description of hierarchies of higher homotopies in suitable categories, and especially in algebraic topology for the study of iterated loop spaces. The discovery of new relationships of operads with graph cohomology, representation theory, algebraic geometry, derived categories, Morse theory, combinatorics, knot theory, moduli spaces, and cyclic cohomology implied further development within the theory of operads and emphasized the importance of operads. Also in the area of theoretical physics operads have been successfully applied in string theory and deformation quantization. The generalization of quadratic duality and Koszulness set up in an operadic context turned out to be useful for the study of homotopic questions in an algebraic setting.
The book provides an introduction to operads and brings together the essential results of today’s literature on the topic. In addition several constructions and results are described in a more general context as found in literature. Various gaps or omissions in the available proofs are filled for the first time here.
The book is split into two parts. In Part I an extensive review of the history of operads is presented in order to give the reader a feeling of the scope of application of operads. Part II of the book starts with definitions and basic results on operads in the general context of symmetric monoidal categories. In Chapter 2 of Part II classical results of topology are reviewed; key words are: iterated loop spaces, recognition, approximation and \(\Gamma\)-spaces, homology and homotopy invariances. Algebraic constructions like the bar and cobar construction, free operads, Koszul duality, and cohomology of operad algebras are presented in Chapter 3. In Chapter 4 geometric topics will be discussed. In particular compactification of moduli spaces and configuration spaces of points in manifolds will be considered.
The last chapter is dedicated to generalizations (or specifications) of operads such as cyclic and modular operads. These constructions are motivated by applications to deformation quantization, string field theory, Gromov-Witten invariants, etc.
The book has in view researchers and students who want to get a taste of operads and their applications.

Reviewer: Bernhard Drabant (Walldorf)

MSC:

18-02	Research exposition (monographs, survey articles) pertaining to category theory
55-02	Research exposition (monographs, survey articles) pertaining to algebraic topology
55P48	Loop space machines and operads in algebraic topology
18D50	Operads (MSC2010)
81-02	Research exposition (monographs, survey articles) pertaining to quantum theory
57N10	Topology of general \(3\)-manifolds (MSC2010)
57M25	Knots and links in the \(3\)-sphere (MSC2010)
57T05	Hopf algebras (aspects of homology and homotopy of topological groups)

Keywords:

operads; iterated loop spaces; string theory; deformation quantization; quadratic duality; Koszulness; symmetric monoidal categories; compactification of moduli spaces; configuration spaces; Gromov-Witten invariants