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Astérisque - Titles - 387 (2017)

Titles < 2017

Feynman Categories
Ralph M. Kaufmann, Benjamin C. Ward
Astérisque 387 (2017), x+161 pages
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Résumé :
Catégories de Feynman
Dans ce livre nous proposons une nouvelle fondation catégorielle concernant des opérations et leurs relations ainsi que leurs objets paramétrisants. Cela généralise et en même temps unifie les théories d'une part des opérades et tous leurs cousins, soit les PROPs, les opérades (modulaires) tordues, les properades, les hyperopérades, leurs variantes colorées, ou encore les algèbres sur une opérade fixée, etc., et d'autre part une multitude de structures similaires comme les groupes simpliciaux croisés, la catégorie simpliciale enrichie ou les modules FI. L'utilité de cette approche se montre dans la possibilité de traiter dans un cadre commun de la même manière toutes les structures classiques ainsi que des exemples plus ésotériques. Dans cette manière, beaucoup de constructions connues se présentent comme des extensions de Kan. Dans ce cadre commun, nous pouvons dériver des opérations universelles, comme les opérations qui font part de la conjecture de Deligne, construire des algèbres de Hopf, et aussi réaliser des résolutions, les transformes bar, cobar et Feynman, qui sont liées aux équations maîtresses. Afin de développer ces exemples, nous construisons les structures de modèle pour les catégories pertinentes. Ceci donne une abondance des exemples nouveaux.

Abstract:
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.

Keywords: Feynman category, model category, monoidal category, monoidal functor, graph, Kan extension, operads, PROPs, modular operads, twisted modular operads, universal operations, Gerstenhaber bracket,pre-Lie algebra, BV algebra, bi-algebra, Hopf algebra, bar transform, cobar transform, Feynman transform, master equation, opetopic, plus construction, enriched category, Quillen adjunction, W construction.

Class. math. : 18D10, 55U35, 18D99, 55P48, 18D50, 81Q05, 18C15, 18D20, 18D25, 18G55, 55U40, 81T30, 81T18, 16T10, 16T05.


ISBN : 978-2-85629-852-7
ISSN : 0303-1179
Publié avec le concours de : Centre National de la Recherche Scientifique

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