Operads in algebra, topology and physics TeX Export

@book{citeulike:466573, abstract = { FEATURED REVIEW. <P> The notion of an operad formalizes the idea of an abstract space of operations, inasmuch as the notion of a group formalizes the idea of an abstract space of transformations, irrespective of where these transformations act or are being represented. Various kinds of operations occur overwhelmingly throughout mathematics and physics, and most of them perhaps do not require special thought, such as the operations of addition and multiplication of integers. Some operations, such as Steenrod squares and Massey products, proven to be useful in algebraic topology, are somewhat more complicated, and some (e.g., Stasheff's \$A\_\infty\$-structures, WDVV-algebras and other algebraic structures in string theory given by Ward identities) are so complicated that serious tools are required to handle them. May's operad theory is one such tool, perhaps the most popular today, along with Lazard's "analyseurs", Adams and Mac Lane's PROP's and PACT's, Gerstenhaber's composition algebras, and Lawvere's algebraic theories. <P> There are not very many books dedicated to the theory of operads. The first two, \ref[J. P. May, <em> The geometry of iterated loop spaces</em>, Springer, Berlin, 1972; <A HREF="/msnmain?fn=105\&fmt=doc\&r=1\&pg1=CNO\&s1=420610\&loc=fromrevtext">MR0420610 (54 \#8623b)</A>] and [J. M. Boardman\ and R. M. Vogt, <em> Homotopy invariant algebraic structures on topological spaces</em>, Lecture Notes in Math., 347, Springer, Berlin, 1973; <A HREF="/msnmain?fn=105\&fmt=doc\&r=1\&pg1=CNO\&s1=420609\&loc=fromrevtext">MR0420609 (54 \#8623a)</A>], were aimed at the study of iterated loop spaces, for which operads had been invented as a technical tool. A few years later, in [J. F. Adams, <em> Infinite loop spaces</em>, Ann. of Math. Stud., 90, Princeton Univ. Press, Princeton, N.J., 1978; <A HREF="/msnmain?fn=105\&fmt=doc\&r=1\&pg1=CNO\&s1=505692\&loc=fromrevtext">MR0505692 (80d:55001)</A>], Adams described operads as one of the (several by that time) "machines" used to manufacture infinite loop spaces. During the following twenty-some years, operads had been effectively dormant, until the mathematics community came to realize that operads might be very useful in many fields of mathematics and theoretical physics, apart from just homotopy theory. At that point a concise monograph (or, better said, an extended paper) [I. K\v r\&iacute;\v z and\ J. P. May, Ast\&eacute;risque No. 233 (1995), iv+145pp.; <A HREF="/msnmain?fn=105\&fmt=doc\&r=1\&pg1=CNO\&s1=1361938\&loc=fromrevtext">MR1361938 (96j:18006)</A>] came in handy, as it provided an introduction to operads from a more modern point of view, primarily for the purpose of constructing the category of mixed Hodge motives. A later monograph [V. A. Smirnov, <em> Simplicial and operad methods in algebraic topology</em>, Translated from the Russian manuscript by G. L. Rybnikov, Amer. Math. Soc., Providence, RI, 2001; <A HREF="/msnmain?fn=105\&fmt=doc\&r=1\&pg1=CNO\&s1=1811110\&loc=fromrevtext">MR1811110 (2002a:55013)</A>] provided an extensive study of applications of operads throughout algebraic topology. <P> The book under review is perhaps the first book whose main goal is the theory of operads per se. However, the authors admit that operads are still mainly a tool and center the book around applications of operads to algebra, topology and physics, as suggested by the title. In light of the "renaissance" of the theory of operads sparked by a 1994 paper [V. Ginzburg and M. Kapranov, Duke Math. J. <strong>76</strong> (1994), no. 1, 203--272; <A HREF="/msnmain?fn=105\&fmt=doc\&r=1\&pg1=CNO\&s1=1301191\&loc=fromrevtext">MR1301191 (96a:18004)</A>] and a subsequent realization of the ubiquity of operads on the one hand and, on the other hand, a relatively heavy load of technicalities involved in the theory, a book such as this one has been long awaited by a wide scientific readership, including mathematicians and theoretical physicists. <P> The book consists of two parts. Part I, a 32-page-long Introduction and History, provides extensive historical background and sets operad theory in context. Earlier approaches to studying operations, including Lazard's analyseurs, PROP's, and theories, are discussed. Homotopy algebras, such as \$A\_\infty\$-, \$L\_\infty\$-, and \$C\_\infty\$-algebras, and spaces are introduced. This part also touches upon a number of sporadic topics that influenced the development of operad theory in recent years: various types of tree operads, including Connes-Kreimer's yoga of Feynman diagrams, Ginzburg-Kapranov's Koszul duality, cyclic, modular, and partial operads, relation to string theory, homotopy invariance, formality, quantization, and Deligne's conjecture. <P> Part II, counting almost 300 pages, is the core of the book. Chapter 1 contains the main definitions "in the full glory of the symmetric monoidal category setting": operads, pseudo-operads (operads with no unit), algebras over an operad, the Stasheff associahedra, free operads, and triples (monads). Chapter 2 reviews classical applications of operads to topology: the recognition principle for iterated loop spaces, approximation of iterated loop spaces, \$\Gamma\$-spaces, homology operations, including Steenrod squares, Kudo-Araki, Dyer-Lashof, and Browder operations, Massey products, Boardman-Vogt's \$W\$-construction, giving a cofibrant resolution of an operad, and homotopy invariance issues are discussed. Chapter 3 introduces more modern developments that have occurred in the algebraic theory of operads: quadratic operads, Koszul operads, modules over an operad, the cobar and Koszul complexes of an operad, cohomology of operad algebras, Markl's minimal models, and homotopy algebras. Chapter 4 deals with geometric constructions of operads, related to configuration spaces of points in manifolds. One of them is the Knudsen-Deligne-Mumford compactification of the moduli space of punctured complex algebraic curves of genus zero; another one is the real version of the Fulton-MacPherson compactification discussed for points in a Euclidean space, following a preprint \ref["Operads, homotopy algebra and iterated integrals for double loop spaces", http://arXiv.org/abs/hep-th/9403055] by E. Getzler and J. D. S. Jones, and in a Riemannian manifold, following [S. Axelrod and I. M. Singer, J. Differential Geom. <strong> 39</strong> (1994), no. 1, 173--213; <A HREF="/msnmain?fn=105\&fmt=doc\&r=1\&pg1=CNO\&s1=1258919\&loc=fromrevtext">MR1258919 (95b:58163)</A>]. Chapter 5 is about generalizations, such as cyclic and modular operads, and their relation to cyclic cohomology. Getzler-Kapranov's Feynman transform, generalizing the cobar construction to modular operads, is described. The last three sections are dedicated to graph homology, moduli spaces of arbitrary genera, and algebraic structures in string theory. Most recent developments are mentioned along with references in an Epilog. <P> Given the authors' ambitious goal of writing a book on a rapidly developing subject, it would be too idealistic to expect the book to be flawless. Yet it almost is. The only flaw I have been able to pin down is an erroneous statement that the rational cohomology of the moduli space of Riemann surfaces with labeled punctures is isomorphic to that of the moduli space of Riemann surfaces with unlabeled ones, in the section on graph homology at the very end of the book. The authors have also informed me that the definition of an operad in Section II.1.2 should have explicitly stated the following standard equivariance condition: The structure morphisms \$\gamma\_{n;m\_1,\ldots,m\_n}\$ must be equivariant with respect to the right action of the \$\Sigma\_{m\_i}\$ on \${\cal P}(m\_i)\$ and on \${\cal P}(m\_1+\cdots+m\_n)\$ via the natural inclusion \$\Sigma\_{m\_i} \hookrightarrow \Sigma\_{{m\_1}+\cdots+{m\_n}}\$ for each \$i=1,\dots,n\$. <P> 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. Overall, the book is 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. <P> REVISED (May, 2003) <p>Current version of review. <a href="/msnmain?co3=AND\&amp;co4=AND\&amp;dr=all\&amp;fmt=doc\&amp;fn=105\&amp;l=100\&amp;pg3=ICN\&amp;pg4=ICN\&amp;prev=t\&amp;r=1\&amp;s3=stasheff\&amp;s4=Markl">Go to earlier version.</a></p>}, address = {Providence, RI}, author = {Markl, Martin and Shnider, Steve and Stasheff, Jim}, citeulike-article-id = {466573}, citeulike-linkout-0 = {http://www.ams.org/mathscinet-getitem?mr=1898414}, keywords = {a\_infinity, algebra, bar\_construction, cyclic\_operads, koszul\_duality, l\_infinity, loop\_space, modular\_operads, operads, physics, string\_field\_theory, topology, tree\_operads}, mrnumber = {MR1898414}, posted-at = {2006-01-17 01:51:34}, priority = {2}, publisher = {American Mathematical Society}, series = {Mathematical Surveys and Monographs}, title = {Operads in algebra, topology and physics}, url = {http://www.ams.org/mathscinet-getitem?mr=1898414}, volume = {96}, year = {2002} }

TY - BOOK ID - citeulike:466573 L3 - citeulike-article-id:466573 N2 - FEATURED REVIEW. <P> The notion of an operad formalizes the idea of an abstract space of operations, inasmuch as the notion of a group formalizes the idea of an abstract space of transformations, irrespective of where these transformations act or are being represented. Various kinds of operations occur overwhelmingly throughout mathematics and physics, and most of them perhaps do not require special thought, such as the operations of addition and multiplication of integers. Some operations, such as Steenrod squares and Massey products, proven to be useful in algebraic topology, are somewhat more complicated, and some (e.g., Stasheff's $A_\infty$-structures, WDVV-algebras and other algebraic structures in string theory given by Ward identities) are so complicated that serious tools are required to handle them. May's operad theory is one such tool, perhaps the most popular today, along with Lazard's "analyseurs", Adams and Mac Lane's PROP's and PACT's, Gerstenhaber's composition algebras, and Lawvere's algebraic theories. <P> There are not very many books dedicated to the theory of operads. The first two, \ref[J. P. May, <em> The geometry of iterated loop spaces</em>, Springer, Berlin, 1972; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=420610&loc=fromrevtext">MR0420610 (54 \#8623b)</A>] and [J. M. Boardman\ and R. M. Vogt, <em> Homotopy invariant algebraic structures on topological spaces</em>, Lecture Notes in Math., 347, Springer, Berlin, 1973; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=420609&loc=fromrevtext">MR0420609 (54 \#8623a)</A>], were aimed at the study of iterated loop spaces, for which operads had been invented as a technical tool. A few years later, in [J. F. Adams, <em> Infinite loop spaces</em>, Ann. of Math. Stud., 90, Princeton Univ. Press, Princeton, N.J., 1978; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=505692&loc=fromrevtext">MR0505692 (80d:55001)</A>], Adams described operads as one of the (several by that time) "machines" used to manufacture infinite loop spaces. During the following twenty-some years, operads had been effectively dormant, until the mathematics community came to realize that operads might be very useful in many fields of mathematics and theoretical physics, apart from just homotopy theory. At that point a concise monograph (or, better said, an extended paper) [I. K\v r&iacute;\v z and\ J. P. May, Ast&eacute;risque No. 233 (1995), iv+145pp.; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=1361938&loc=fromrevtext">MR1361938 (96j:18006)</A>] came in handy, as it provided an introduction to operads from a more modern point of view, primarily for the purpose of constructing the category of mixed Hodge motives. A later monograph [V. A. Smirnov, <em> Simplicial and operad methods in algebraic topology</em>, Translated from the Russian manuscript by G. L. Rybnikov, Amer. Math. Soc., Providence, RI, 2001; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=1811110&loc=fromrevtext">MR1811110 (2002a:55013)</A>] provided an extensive study of applications of operads throughout algebraic topology. <P> The book under review is perhaps the first book whose main goal is the theory of operads per se. However, the authors admit that operads are still mainly a tool and center the book around applications of operads to algebra, topology and physics, as suggested by the title. In light of the "renaissance" of the theory of operads sparked by a 1994 paper [V. Ginzburg and M. Kapranov, Duke Math. J. <strong>76</strong> (1994), no. 1, 203--272; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=1301191&loc=fromrevtext">MR1301191 (96a:18004)</A>] and a subsequent realization of the ubiquity of operads on the one hand and, on the other hand, a relatively heavy load of technicalities involved in the theory, a book such as this one has been long awaited by a wide scientific readership, including mathematicians and theoretical physicists. <P> The book consists of two parts. Part I, a 32-page-long Introduction and History, provides extensive historical background and sets operad theory in context. Earlier approaches to studying operations, including Lazard's analyseurs, PROP's, and theories, are discussed. Homotopy algebras, such as $A_\infty$-, $L_\infty$-, and $C_\infty$-algebras, and spaces are introduced. This part also touches upon a number of sporadic topics that influenced the development of operad theory in recent years: various types of tree operads, including Connes-Kreimer's yoga of Feynman diagrams, Ginzburg-Kapranov's Koszul duality, cyclic, modular, and partial operads, relation to string theory, homotopy invariance, formality, quantization, and Deligne's conjecture. <P> Part II, counting almost 300 pages, is the core of the book. Chapter 1 contains the main definitions "in the full glory of the symmetric monoidal category setting": operads, pseudo-operads (operads with no unit), algebras over an operad, the Stasheff associahedra, free operads, and triples (monads). Chapter 2 reviews classical applications of operads to topology: the recognition principle for iterated loop spaces, approximation of iterated loop spaces, $\Gamma$-spaces, homology operations, including Steenrod squares, Kudo-Araki, Dyer-Lashof, and Browder operations, Massey products, Boardman-Vogt's $W$-construction, giving a cofibrant resolution of an operad, and homotopy invariance issues are discussed. Chapter 3 introduces more modern developments that have occurred in the algebraic theory of operads: quadratic operads, Koszul operads, modules over an operad, the cobar and Koszul complexes of an operad, cohomology of operad algebras, Markl's minimal models, and homotopy algebras. Chapter 4 deals with geometric constructions of operads, related to configuration spaces of points in manifolds. One of them is the Knudsen-Deligne-Mumford compactification of the moduli space of punctured complex algebraic curves of genus zero; another one is the real version of the Fulton-MacPherson compactification discussed for points in a Euclidean space, following a preprint \ref["Operads, homotopy algebra and iterated integrals for double loop spaces", http://arXiv.org/abs/hep-th/9403055] by E. Getzler and J. D. S. Jones, and in a Riemannian manifold, following [S. Axelrod and I. M. Singer, J. Differential Geom. <strong> 39</strong> (1994), no. 1, 173--213; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=1258919&loc=fromrevtext">MR1258919 (95b:58163)</A>]. Chapter 5 is about generalizations, such as cyclic and modular operads, and their relation to cyclic cohomology. Getzler-Kapranov's Feynman transform, generalizing the cobar construction to modular operads, is described. The last three sections are dedicated to graph homology, moduli spaces of arbitrary genera, and algebraic structures in string theory. Most recent developments are mentioned along with references in an Epilog. <P> Given the authors' ambitious goal of writing a book on a rapidly developing subject, it would be too idealistic to expect the book to be flawless. Yet it almost is. The only flaw I have been able to pin down is an erroneous statement that the rational cohomology of the moduli space of Riemann surfaces with labeled punctures is isomorphic to that of the moduli space of Riemann surfaces with unlabeled ones, in the section on graph homology at the very end of the book. The authors have also informed me that the definition of an operad in Section II.1.2 should have explicitly stated the following standard equivariance condition: The structure morphisms $\gamma_{n;m_1,\ldots,m_n}$ must be equivariant with respect to the right action of the $\Sigma_{m_i}$ on ${\cal P}(m_i)$ and on ${\cal P}(m_1+\cdots+m_n)$ via the natural inclusion $\Sigma_{m_i} \hookrightarrow \Sigma_{{m_1}+\cdots+{m_n}}$ for each $i=1,\dots,n$. <P> 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. Overall, the book is 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. <P> REVISED (May, 2003) <p>Current version of review. <a href="/msnmain?co3=AND&amp;co4=AND&amp;dr=all&amp;fmt=doc&amp;fn=105&amp;l=100&amp;pg3=ICN&amp;pg4=ICN&amp;prev=t&amp;r=1&amp;s3=stasheff&amp;s4=Markl">Go to earlier version.</a></p> SE - Mathematical Surveys and Monographs AD - Providence, RI TI - Operads in algebra, topology and physics PB - American Mathematical Society VL - 96 KW - a_infinity KW - algebra KW - bar_construction KW - cyclic_operads KW - koszul_duality KW - l_infinity KW - loop_space KW - modular_operads KW - operads KW - physics KW - string_field_theory KW - topology KW - tree_operads AU - Markl, Martin AU - Shnider, Steve AU - Stasheff, Jim PY - 2002/// UR - http://www.ams.org/mathscinet-getitem?mr=1898414 ER -