Operads are mathematical devices which 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 first appeared in the   sixties though the formal definition and appropriate generality   waited for the seventies. These early occurrences were in   algebraic topology in the study of (iterated) loop spaces and   their chain algebras. In the nineties, there was a renaissance   and further development of the theory 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, not least, theoretical   physics, especially string field theory and deformation   quantization. The generalization of quadratic duality (e.g., Lie   algebras as dual to commutative algebras) together with the   property of Koszulness in an essentially operadic context   provided an additional computational tool for studying homotopy   properties outside of the topological setting.    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.    There is a chapter on topology, reviewing classical results with   the emphasis on the $W$-construction and homotopy invariance.   Another chapter describes the (co)homology of operad algebras,   minimal models, and homotopy algebras. A chapter on geometry   focuses on the configuration spaces and their compactifications.   A final chapter deals with cyclic and modular operads and   applications to graph complexes and moduli spaces of surfaces of   arbitrary genus.

Operads in Algebra, Topology and Physics PDF

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