**Mathematical Surveys and Monographs**

Volume: 96;
2002;
349 pp;
Softcover

MSC: Primary 18; 55;

Print ISBN: 978-0-8218-4362-8

Product Code: SURV/96.S

List Price: $98.00

AMS Member Price: $78.40

MAA Member Price: $88.20

**Electronic ISBN: 978-1-4704-1323-1
Product Code: SURV/96.S.E**

List Price: $98.00

AMS Member Price: $78.40

MAA Member Price: $88.20

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# Operads in Algebra, Topology and Physics

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*Martin Markl; Steve Shnider; Jim Stasheff*

Operads are powerful tools, and this is the book in which to read about them.

—Bulletin of the London Mathematical
Society

Operads are mathematical devices that describe algebraic structures
of many varieties and in various categories. Operads are particularly
important in categories with a good notion of “homotopy”,
where they play a key role in organizing hierarchies of higher
homotopies. Significant examples from algebraic topology first
appeared in the sixties, although the formal definition and
appropriate generality were not forged until the seventies. In the
nineties, a renaissance and further development of the theory were
inspired by the discovery of new relationships with graph cohomology,
representation theory, algebraic geometry, derived categories, Morse
theory, symplectic and contact geometry, combinatorics, knot theory,
moduli spaces, cyclic cohomology, and, last but not least, theoretical
physics, especially string field theory and deformation
quantization.

The book contains a detailed and comprehensive historical
introduction describing the development of operad theory from the
initial period when it was a rather specialized tool in homotopy
theory to the present when operads have a wide range of applications
in algebra, topology, and mathematical physics. Many results and
applications currently scattered in the literature are brought
together here along with new results and insights. The basic
definitions and constructions are carefully explained and include many
details not found in any of the standard literature.

#### Readership

Graduate students, research mathematicians, and mathematical physicists interested in homotopy theory, gauge theory, and string theory.

#### Reviews & Endorsements

Operads are powerful tools, and this is the book in which to read about them.

-- Bulletin of the London Mathematical Society

The first book whose main goal is the theory of operads per se … a book such as this one has been long awaited by a wide scientific readership, including mathematicians and theoretical physicists … Written in a way to stimulate thought and abundant in references, spanning from 1898 through 2001, the book under review is guaranteed to contribute to the constant quest of mathematics for novel ideas and effective applications … a great piece of mathematical literature and will be helpful to anyone who needs to use operads, from graduate students to mature mathematicians and physicists.

-- Mathematical Reviews, Featured Review

#### Table of Contents

# Table of Contents

## Operads in Algebra, Topology and Physics

- Contents vii8 free
- Preface ix10 free
- Part I 112 free
- Chapter 1. Introduction and History 314
- A prehistory 314
- 1.1. Lazard's formal group laws 314
- 1.2. PROPs and PACTs 415
- 1.3. Non-Σ operads and operads 516
- 1.4. Theories 718
- 1.5. Tree operads 819
- 1.6. .A[sub(∞)]-spaces and loop spaces 920
- 1.7. E[sub(∞)]-spaces and iterated loop spaces 1223
- 1.8. A[sub(∞)]-algebras 1324
- 1.9. Partiality and A[sub(∞)]-categories 1425
- 1.10. L[sub(∞)]-algebras 1728
- 1.11. C[sub(∞)]-algebras 1930
- 1.12. n-ary algebras 1930
- 1.13. Operadic bar construction and Koszul duality 2031
- 1.14. Cyclic operads 2132
- 1.15. Moduli spaces and modular operads 2233
- 1.16. Operadic interpretation of closed string field theory 2334
- 1.17. From topological operads to dg operads 2637
- 1.18. Homotopy invariance in algebra and topology 2738
- 1.19. Formality, quantization and Deligne's conjecture 2940
- 1.20. Insertion operads 3142

- Part II 3546
- Chapter 1. Operads in a Symmetric Monoidal Category 3748
- 1.1. Symmetric monoidal categories 3748
- 1.2. Operads 4051
- 1.3. Pseudo-operads 4556
- 1.4. Operad algebras 4657
- 1.5. The pseudo-operad of labeled rooted trees 5061
- 1.6. The Stasheff associahedra 5667
- 1.7. Operads defined in terms of arbitrary finite sets 6071
- 1.8. Operads as monoids 6778
- 1.9. Free operads and free pseudo-operads 7182
- 1.10. Collections, K-collections and K-operads 8495
- 1.11. The GK-construction 8697
- 1.12. Triples 8899

- Chapter 2. Topology – Review of Classical Results 93104
- 2.1. Iterated loop spaces 93104
- 2.2. Recognition 94105
- 2.3. The bar construction: theme and variations 96107
- 2.4. Approximation 97108
- 2.5. τ-spaces 101112
- 2.6. Homology operations 102113
- 2.7. The linear isometries operad and infinite loop spaces 106117
- 2.8. W-construction 109120
- 2.9. Algebraic structures up to strong homotopy 112123

- Chapter 3. Algebra 121132
- 3.1. The cobar complex of an operad 121132
- 3.2. Quadratic operads 137148
- 3.3. Koszul operads 145156
- 3.4. A complex relating the two conditions for a Koszul operad 149160
- 3.5. Trees with levels 154165
- 3.6. The spectral sequences relating N(P) and C(P) 158169
- 3.7. Coalgebras and coderivations 165176
- 3.8. The homology and cohomology of operad algebras 173184
- 3.9. The pre-Lie structure on Coder(F[sup(c)][sub(p)](x)) 182193
- 3.10. Application: minimal models and homotopy algebras 186197

- Chapter 4. Geometry 203214
- Chapter 5. Generalization of Operads 247258

- Epilog 327338
- Bibliography 329340
- Glossary of notations 339350
- Index 345356