Feynman categories / Ralph M. Kaufmann & Benjamin C. Ward

Auteur: Kaufmann, Ralph M. (1969-) - AuteurCo-auteur: Ward, Benjamin C. - AuteurType de document: MonographieCollection: Astérisque ; 387Langue: anglaisPays: FranceÉditeur: Paris : Société Mathématique de France, cop. 2017Description: 1 vol. (vii-161 p.) ; 24 cm ISBN: 9782856298527 ; br. Note: In this book we give a new foundational, categorical formulation for operations and relations and objects parameterizing them. This generalizes and unifies the theory of operads and all their cousins including but not limited to PROPs, modular operads, twisted (modular) operads, properads, hyperoperads, their colored versions, as well as algebras over operads and an abundance of other related structures, such as crossed simplicial groups, the augmented simplicial category or FI-modules. The usefulness of this approach is that it allows us to handle all the classical as well as more esoteric structures under a common framework and we can treat all the situations simultaneously. Many of the known constructions simply become Kan extensions. In this common framework, we also derive universal operations, such as those underlying Deligne's conjecture, construct Hopf algebras as well as perform resolutions, (co)bar transforms and Feynman transforms which are related to master equations. For these applications, we construct the relevant model category structures. This produces many new examples (SMF)Bibliographie: Bibliogr. p. [155]-161. Index. Sujets MSC: 18D10 Category theory; homological algebra -- Categories with structure -- Monoidal categories, symmetric monoidal categories, braided categories
55U35 Algebraic topology -- Applied homological algebra and category theory -- Abstract and axiomatic homotopy theory
55P48 Algebraic topology -- Homotopy theory -- Loop space machines, operads
18D50 Category theory; homological algebra -- Categories with structure -- Operads
81Q05 Quantum theory -- General mathematical topics and methods in quantum theory -- Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
En-ligne: SMF - texte intégral
Location Call Number Status Date Due
Salle R 07214-01 / Séries SMF 387 (Browse Shelf) Available

In this book we give a new foundational, categorical formulation for operations and relations and objects parameterizing them. This generalizes and unifies the theory of operads and all their cousins including but not limited to PROPs, modular operads, twisted (modular) operads, properads, hyperoperads, their colored versions, as well as algebras over operads and an abundance of other related structures, such as crossed simplicial groups, the augmented simplicial category or FI-modules. The usefulness of this approach is that it allows us to handle all the classical as well as more esoteric structures under a common framework and we can treat all the situations simultaneously. Many of the known constructions simply become Kan extensions. In this common framework, we also derive universal operations, such as those underlying Deligne's conjecture, construct Hopf algebras as well as perform resolutions, (co)bar transforms and Feynman transforms which are related to master equations. For these applications, we construct the relevant model category structures. This produces many new examples (SMF)

Bibliogr. p. [155]-161. Index

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