Modules Over Operads and Functors

Modules Over Operads and Functors PDF Author: Benoit Fresse
Publisher: Springer Science & Business Media
ISBN: 3540890556
Category : Mathematics
Languages : en
Pages : 304

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Book Description
The notion of an operad supplies both a conceptual and effective device to handle a variety of algebraic structures in various situations. Operads were introduced 40 years ago in algebraic topology in order to model the structure of iterated loop spaces. Since then, operads have been used fruitfully in many fields of mathematics and physics. This monograph begins with a review of the basis of operad theory. The main purpose is to study structures of modules over operads as a new device to model functors between categories of algebras as effectively as operads model categories of algebras.

Modules Over Operads and Functors

Modules Over Operads and Functors PDF Author: Benoit Fresse
Publisher: Springer Science & Business Media
ISBN: 3540890556
Category : Mathematics
Languages : en
Pages : 304

Get Book Here

Book Description
The notion of an operad supplies both a conceptual and effective device to handle a variety of algebraic structures in various situations. Operads were introduced 40 years ago in algebraic topology in order to model the structure of iterated loop spaces. Since then, operads have been used fruitfully in many fields of mathematics and physics. This monograph begins with a review of the basis of operad theory. The main purpose is to study structures of modules over operads as a new device to model functors between categories of algebras as effectively as operads model categories of algebras.

Modules over Operads and Functors

Modules over Operads and Functors PDF Author: Benoit Fresse
Publisher: Springer
ISBN: 3540890564
Category : Mathematics
Languages : en
Pages : 304

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Book Description
This monograph presents a review of the basis of operad theory. It also studies structures of modules over operads as a new device to model functors between categories of algebras as effectively as operads model categories of algebras.

Homotopy of Operads and Grothendieck-Teichmuller Groups

Homotopy of Operads and Grothendieck-Teichmuller Groups PDF Author: Benoit Fresse
Publisher: American Mathematical Soc.
ISBN: 1470434814
Category : Mathematics
Languages : en
Pages : 581

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Book Description
The Grothendieck–Teichmüller group was defined by Drinfeld in quantum group theory with insights coming from the Grothendieck program in Galois theory. The ultimate goal of this book is to explain that this group has a topological interpretation as a group of homotopy automorphisms associated to the operad of little 2-discs, which is an object used to model commutative homotopy structures in topology. This volume gives a comprehensive survey on the algebraic aspects of this subject. The book explains the definition of an operad in a general context, reviews the definition of the little discs operads, and explains the definition of the Grothendieck–Teichmüller group from the viewpoint of the theory of operads. In the course of this study, the relationship between the little discs operads and the definition of universal operations associated to braided monoidal category structures is explained. Also provided is a comprehensive and self-contained survey of the applications of Hopf algebras to the definition of a rationalization process, the Malcev completion, for groups and groupoids. Most definitions are carefully reviewed in the book; it requires minimal prerequisites to be accessible to a broad readership of graduate students and researchers interested in the applications of operads.

Algebraic Operads

Algebraic Operads PDF Author: Jean-Louis Loday
Publisher: Springer Science & Business Media
ISBN: 3642303625
Category : Mathematics
Languages : en
Pages : 649

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Book Description
In many areas of mathematics some “higher operations” are arising. These havebecome so important that several research projects refer to such expressions. Higher operationsform 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 Homotopy Transfer Theorem. Although the necessary notions of algebra are recalled, readers are expected 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 a low-level chapter on Algebra, accessible to (advanced) 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.

Colored Operads

Colored Operads PDF Author: Donald Yau
Publisher: American Mathematical Soc.
ISBN: 1470427230
Category : Mathematics
Languages : en
Pages : 458

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Book Description
The subject of this book is the theory of operads and colored operads, sometimes called symmetric multicategories. A (colored) operad is an abstract object which encodes operations with multiple inputs and one output and relations between such operations. The theory originated in the early 1970s in homotopy theory and quickly became very important in algebraic topology, algebra, algebraic geometry, and even theoretical physics (string theory). Topics covered include basic graph theory, basic category theory, colored operads, and algebras over colored operads. Free colored operads are discussed in complete detail and in full generality. The intended audience of this book includes students and researchers in mathematics and other sciences where operads and colored operads are used. The prerequisite for this book is minimal. Every major concept is thoroughly motivated. There are many graphical illustrations and about 150 exercises. This book can be used in a graduate course and for independent study.

(Co)end Calculus

(Co)end Calculus PDF Author: Fosco Loregian
Publisher: Cambridge University Press
ISBN: 1108746128
Category : Mathematics
Languages : en
Pages : 331

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Book Description
This easy-to-cite handbook gives the first systematic treatment of the (co)end calculus in category theory and its applications.

Higher Operads, Higher Categories

Higher Operads, Higher Categories PDF Author: Tom Leinster
Publisher: Cambridge University Press
ISBN: 0521532159
Category : Mathematics
Languages : en
Pages : 451

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Book Description
Foundations of higher dimensional category theory for graduate students and researchers in mathematics and mathematical physics.

Feynman Categories

Feynman Categories PDF Author: Ralph M. Kaufmann
Publisher:
ISBN: 9782856298527
Category : Categories (Mathematics)
Languages : en
Pages : 161

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Book Description
In this book, the authors 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.

Equivariant Orthogonal Spectra and S-modules

Equivariant Orthogonal Spectra and S-modules PDF Author: M. A. Mandell
Publisher: American Mathematical Soc.
ISBN: 9780821864777
Category : Mathematics
Languages : en
Pages : 132

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Book Description
The last few years have seen a revolution in our understanding of the foundations of stable homotopy theory. Many symmetric monoidal model categories of spectra whose homotopy categories are equivalent to the stable homotopy category are now known, whereas no such categories were known before 1993. The most well-known examples are the category of $S$-modules and the category of symmetric spectra. We focus on the category of orthogonal spectra, which enjoys some of the best features of $S$-modules and symmetric spectra and which is particularly well-suited to equivariant generalization. We first complete the nonequivariant theory by comparing orthogonal spectra to $S$-modules. We then develop the equivariant theory. For a compact Lie group $G$, we construct a symmetric monoidal model category of orthogonal $G$-spectra whose homotopy category is equivalent to the classical stable homotopy category of $G$-spectra. We also complete the theory of $S_G$-modules and compare the categories of orthogonal $G$-spectra and $S_G$-modules. A key feature is the analysis of change of universe, change of group, fixed point, and orbit functors in these two highly structured categories for the study of equivariant stable homotopy theory.

Polyharmonic Boundary Value Problems

Polyharmonic Boundary Value Problems PDF Author: Filippo Gazzola
Publisher: Springer
ISBN: 3642122450
Category : Mathematics
Languages : en
Pages : 444

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Book Description
This accessible monograph covers higher order linear and nonlinear elliptic boundary value problems in bounded domains, mainly with the biharmonic or poly-harmonic operator as leading principal part. It provides rapid access to recent results and references.