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Operads. --- Algebra, Homological. --- Knot theory.

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Topological groups. Lie groups --- Homotopy theory. --- Operads --- Loop spaces. --- Homotopie --- Opérades --- Espaces de lacets --- Operads. --- Mathematics--series --- Opérades

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Foncteurs, Théorie des. --- Espaces topologiques. --- Opérades. --- Functor theory. --- Topological spaces. --- Operads.

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In many areas of mathematics some higher operations  are arising. These have become so important that several research projects refer to such expressions. Higher operations form new types of algebras. The key to understanding and comparing them, to creating invariants of their action is operad theory. This is a point of view that is 40 years old in algebraic topology, but the new trend is its appearance in several other areas, such as algebraic geometry, mathematical physics, differential geometry, and combinatorics. The present volume is the first comprehensive and systematic approach to algebraic operads. An operad is an algebraic device that serves to study all kinds of algebras (associative, commutative, Lie, Poisson, A-infinity, etc.) from a conceptual point of view. The book presents this topic with an emphasis on Koszul duality theory. After a modern treatment of Koszul duality for associative algebras, the theory is extended to operads. Applications to homotopy algebra are given, for instance the HomotopyTransfer Theorem. Although the necessary notions of algebra are recalled, readers areexpected to be familiar with elementary homological algebra. Each chapter ends with a helpful summary and exercises. A full chapter is devoted to examples, and numerous figures are included. After an elementary chapter on classical algebra, accessible to undergraduate students, the level increases gradually through the book. However, the authors have done their best to make it suitable for graduate students: three appendices  review the basic results needed in order to understand the various  chapters. Since higher algebra is becoming essential in several research areas like deformation theory, algebraic geometry, representation theory, differential geometry, algebraic combinatorics, and mathematical physics, the book can also be used as a reference work by researchers.  

Category theory. Homological algebra --- Ordered algebraic structures --- Algebra --- Algebraic topology --- Differential topology --- algebra --- topologie (wiskunde) --- topologie --- Mathematics --- Operads --- Categories (Mathematics)

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Covering an exceptional range of topics, this text provides a unique overview of the Maurer-Cartan methods in algebra, geometry, topology, and mathematical physics. It offers a new conceptual treatment of the twisting procedure, guiding the reader through various versions with the help of plentiful motivating examples for graduate students as well as researchers. Topics covered include a novel approach to the twisting procedure for operads leading to Kontsevich graph homology and a description of the twisting procedure for (homotopy) associative algebras or (homotopy) Lie algebras using the biggest deformation gauge group ever considered. The book concludes with concise surveys of recent applications in areas including higher category theory and deformation theory.

Lie algebras. --- Twist mappings (Mathematics) --- Operads. --- Deformations of singularities. --- Àlgebres de Lie --- Teoria dels tuistors --- Singularitats (Matemàtica)