Colored Operads

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Colored Operads

Donald Yau
Colored Operads Book PDF

Author : Donald Yau
Publisher : American Mathematical Soc.
Page : 428 pages
File Size : 47,8 Mb
Release : 2016-02-29
Category : Algebra, Homological
ISBN : 9781470427238

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Colored Operads by Donald Yau Pdf

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.

Operads of Wiring Diagrams

Donald Yau
Operads of Wiring Diagrams Book PDF

Author : Donald Yau
Publisher : Springer
Page : 308 pages
File Size : 54,5 Mb
Release : 2018-09-19
Category : Mathematics
ISBN : 9783319950013

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Operads of Wiring Diagrams by Donald Yau Pdf

Wiring diagrams form a kind of graphical language that describes operations or processes with multiple inputs and outputs, and shows how such operations are wired together to form a larger and more complex operation. This monograph presents a comprehensive study of the combinatorial structure of the various operads of wiring diagrams, their algebras, and the relationships between these operads. The book proves finite presentation theorems for operads of wiring diagrams as well as their algebras. These theorems describe the operad in terms of just a few operadic generators and a small number of generating relations. The author further explores recent trends in the application of operad theory to wiring diagrams and related structures, including finite presentations for the propagator algebra, the algebra of discrete systems, the algebra of open dynamical systems, and the relational algebra. A partial verification of David Spivak’s conjecture regarding the quotient-freeness of the relational algebra is also provided. In the final part, the author constructs operad maps between the various operads of wiring diagrams and identifies their images. Assuming only basic knowledge of algebra, combinatorics, and set theory, this book is aimed at advanced undergraduate and graduate students as well as researchers working in operad theory and its applications. Numerous illustrations, examples, and practice exercises are included, making this a self-contained volume suitable for self-study.

Nonsymmetric Operads in Combinatorics

Samuele Giraudo
Nonsymmetric Operads in Combinatorics Book PDF

Author : Samuele Giraudo
Publisher : Springer
Page : 172 pages
File Size : 45,8 Mb
Release : 2019-01-04
Category : Mathematics
ISBN : 9783030020743

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Nonsymmetric Operads in Combinatorics by Samuele Giraudo Pdf

Operads are algebraic devices offering a formalization of the concept of operations with several inputs and one output. Such operations can be naturally composed to form more complex ones. Coming historically from algebraic topology, operads intervene now as important objects in computer science and in combinatorics. A lot of operads involving combinatorial objects highlight some of their properties and allow to discover new ones. This book portrays the main elements of this theory under a combinatorial point of view and exposes the links it maintains with computer science and combinatorics. Examples of operads appearing in combinatorics are studied. The modern treatment of operads consisting in considering the space of formal power series associated with an operad is developed. Enrichments of nonsymmetric operads as colored, cyclic, and symmetric operads are reviewed.

Infinity Operads And Monoidal Categories With Group Equivariance

Donald Yau
Infinity Operads And Monoidal Categories With Group Equivariance Book PDF

Author : Donald Yau
Publisher : World Scientific
Page : 486 pages
File Size : 44,5 Mb
Release : 2021-12-02
Category : Mathematics
ISBN : 9789811250941

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Infinity Operads And Monoidal Categories With Group Equivariance by Donald Yau Pdf

This monograph provides a coherent development of operads, infinity operads, and monoidal categories, equipped with equivariant structures encoded by an action operad. A group operad is a planar operad with an action operad equivariant structure. In the first three parts of this monograph, we establish a foundation for group operads and for their higher coherent analogues called infinity group operads. Examples include planar, symmetric, braided, ribbon, and cactus operads, and their infinity analogues. For example, with the tools developed here, we observe that the coherent ribbon nerve of the universal cover of the framed little 2-disc operad is an infinity ribbon operad.In Part 4 we define general monoidal categories equipped with an action operad equivariant structure and provide a unifying treatment of coherence and strictification for them. Examples of such monoidal categories include symmetric, braided, ribbon, and coboundary monoidal categories, which naturally arise in the representation theory of quantum groups and of coboundary Hopf algebras and in the theory of crystals of finite dimensional complex reductive Lie algebras.

Operads in Algebra, Topology and Physics

Martin Markl,Steven Shnider,James D. Stasheff
Operads in Algebra Topology and Physics Book PDF

Author : Martin Markl,Steven Shnider,James D. Stasheff
Publisher : American Mathematical Soc.
Page : 362 pages
File Size : 51,8 Mb
Release : 2002
Category : Mathematics
ISBN : 9780821843628

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Operads in Algebra, Topology and Physics by Martin Markl,Steven Shnider,James D. Stasheff Pdf

Operads are mathematical devices which describe algebraic structures of many varieties and in various categories. From their beginnings in the 1960s, they have developed to encompass such areas as combinatorics, knot theory, moduli spaces, string field theory and deformation quantization.

Algebraic Operads

Jean-Louis Loday,Bruno Vallette
Algebraic Operads Book PDF

Author : Jean-Louis Loday,Bruno Vallette
Publisher : Springer Science & Business Media
Page : 649 pages
File Size : 44,6 Mb
Release : 2012-08-08
Category : Mathematics
ISBN : 9783642303623

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Algebraic Operads by Jean-Louis Loday,Bruno Vallette Pdf

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.

Homotopical Quantum Field Theory

Donald Yau
Homotopical Quantum Field Theory Book PDF

Author : Donald Yau
Publisher : World Scientific
Page : 311 pages
File Size : 43,5 Mb
Release : 2019-11-11
Category : Science
ISBN : 9789811212871

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Homotopical Quantum Field Theory by Donald Yau Pdf

This book provides a general and powerful definition of homotopy algebraic quantum field theory and homotopy prefactorization algebra using a new coend definition of the Boardman-Vogt construction for a colored operad. All of their homotopy coherent structures are explained in details, along with a comparison between the two approaches at the operad level. With chapters on basic category theory, trees, and operads, this book is self-contained and is accessible to graduate students.

Homotopy of Operads and Grothendieck-Teichmuller Groups

Benoit Fresse
Homotopy of Operads and Grothendieck Teichmuller Groups Book PDF

Author : Benoit Fresse
Publisher : American Mathematical Soc.
Page : 704 pages
File Size : 43,5 Mb
Release : 2017-05-22
Category : Grothendieck groups
ISBN : 9781470434823

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Homotopy of Operads and Grothendieck-Teichmuller Groups by Benoit Fresse Pdf

The ultimate goal of this book is to explain that the Grothendieck–Teichmüller group, as defined by Drinfeld in quantum group theory, has a topological interpretation as a group of homotopy automorphisms associated to the little 2-disc operad. To establish this result, the applications of methods of algebraic topology to operads must be developed. This volume is devoted primarily to this subject, with the main objective of developing a rational homotopy theory for operads. The book starts with a comprehensive review of the general theory of model categories and of general methods of homotopy theory. The definition of the Sullivan model for the rational homotopy of spaces is revisited, and the definition of models for the rational homotopy of operads is then explained. The applications of spectral sequence methods to compute homotopy automorphism spaces associated to operads are also explained. This approach is used to get a topological interpretation of the Grothendieck–Teichmüller group in the case of the little 2-disc operad. This volume is intended for graduate students and researchers interested in the applications of homotopy theory methods in operad theory. It is accessible to readers with a minimal background in classical algebraic topology and operad theory.

Geometry and Dynamics of Groups and Spaces

Mikhail Kapranov,Sergii Kolyada,Yu. I. Manin,Pieter Moree,Leonid Potyagailo
Geometry and Dynamics of Groups and Spaces Book PDF

Author : Mikhail Kapranov,Sergii Kolyada,Yu. I. Manin,Pieter Moree,Leonid Potyagailo
Publisher : Springer Science & Business Media
Page : 742 pages
File Size : 44,9 Mb
Release : 2008-03-05
Category : Mathematics
ISBN : 3764386088

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Geometry and Dynamics of Groups and Spaces by Mikhail Kapranov,Sergii Kolyada,Yu. I. Manin,Pieter Moree,Leonid Potyagailo Pdf

Alexander Reznikov (1960-2003) was a brilliant and highly original mathematician. This book presents 18 articles by prominent mathematicians and is dedicated to his memory. In addition it contains an influential, so far unpublished manuscript by Reznikov of book length. The book further provides an extensive survey on Kleinian groups in higher dimensions and some articles centering on Reznikov as a person.

Involutive Category Theory

Donald Yau
Involutive Category Theory Book PDF

Author : Donald Yau
Publisher : Springer Nature
Page : 250 pages
File Size : 50,8 Mb
Release : 2020-11-30
Category : Mathematics
ISBN : 9783030612030

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Involutive Category Theory by Donald Yau Pdf

This monograph introduces involutive categories and involutive operads, featuring applications to the GNS construction and algebraic quantum field theory. The author adopts an accessible approach for readers seeking an overview of involutive category theory, from the basics to cutting-edge applications. Additionally, the author’s own recent advances in the area are featured, never having appeared previously in the literature. The opening chapters offer an introduction to basic category theory, ideal for readers new to the area. Chapters three through five feature previously unpublished results on coherence and strictification of involutive categories and involutive monoidal categories, showcasing the author’s state-of-the-art research. Chapters on coherence of involutive symmetric monoidal categories, and categorical GNS construction follow. The last chapter covers involutive operads and lays important coherence foundations for applications to algebraic quantum field theory. With detailed explanations and exercises throughout, Involutive Category Theory is suitable for graduate seminars and independent study. Mathematicians and mathematical physicists who use involutive objects will also find this a valuable reference.

Real Homotopy of Configuration Spaces

Najib Idrissi
Real Homotopy of Configuration Spaces Book PDF

Author : Najib Idrissi
Publisher : Springer Nature
Page : 201 pages
File Size : 43,6 Mb
Release : 2022-06-11
Category : Mathematics
ISBN : 9783031044281

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Real Homotopy of Configuration Spaces by Najib Idrissi Pdf

This volume provides a unified and accessible account of recent developments regarding the real homotopy type of configuration spaces of manifolds. Configuration spaces consist of collections of pairwise distinct points in a given manifold, the study of which is a classical topic in algebraic topology. One of this theory’s most important questions concerns homotopy invariance: if a manifold can be continuously deformed into another one, then can the configuration spaces of the first manifold be continuously deformed into the configuration spaces of the second? This conjecture remains open for simply connected closed manifolds. Here, it is proved in characteristic zero (i.e. restricted to algebrotopological invariants with real coefficients), using ideas from the theory of operads. A generalization to manifolds with boundary is then considered. Based on the work of Campos, Ducoulombier, Lambrechts, Willwacher, and the author, the book covers a vast array of topics, including rational homotopy theory, compactifications, PA forms, propagators, Kontsevich integrals, and graph complexes, and will be of interest to a wide audience.

The Homotopy Theory of (?,1)-Categories

Julia E. Bergner
The Homotopy Theory of 1 Categories Book PDF

Author : Julia E. Bergner
Publisher : Cambridge University Press
Page : 289 pages
File Size : 46,9 Mb
Release : 2018-03-15
Category : Mathematics
ISBN : 9781107101364

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The Homotopy Theory of (?,1)-Categories by Julia E. Bergner Pdf

An introductory treatment to the homotopy theory of homotopical categories, presenting several models and comparisons between them.

Deformation Theory of Algebras and Their Diagrams

Martin Markl
Deformation Theory of Algebras and Their Diagrams Book PDF

Author : Martin Markl
Publisher : American Mathematical Soc.
Page : 129 pages
File Size : 41,6 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821889794

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Deformation Theory of Algebras and Their Diagrams by Martin Markl Pdf

This book brings together both the classical and current aspects of deformation theory. The presentation is mostly self-contained, assuming only basic knowledge of commutative algebra, homological algebra and category theory. In the interest of readability, some technically complicated proofs have been omitted when a suitable reference was available. The relation between the uniform continuity of algebraic maps and topologized tensor products is explained in detail, however, as this subject does not seem to be commonly known and the literature is scarce. The exposition begins by recalling Gerstenhaber's classical theory for associative algebras. The focus then shifts to a homotopy-invariant setup of Maurer-Cartan moduli spaces. As an application, Kontsevich's approach to deformation quantization of Poisson manifolds is reviewed. Then, after a brief introduction to operads, a strongly homotopy Lie algebra governing deformations of (diagrams of) algebras of a given type is described, followed by examples and generalizations.

Infinity Properads and Infinity Wheeled Properads

Philip Hackney,Marcy Robertson,Donald Yau
Infinity Properads and Infinity Wheeled Properads Book PDF

Author : Philip Hackney,Marcy Robertson,Donald Yau
Publisher : Springer
Page : 358 pages
File Size : 46,9 Mb
Release : 2015-09-07
Category : Mathematics
ISBN : 9783319205472

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Infinity Properads and Infinity Wheeled Properads by Philip Hackney,Marcy Robertson,Donald Yau Pdf

The topic of this book sits at the interface of the theory of higher categories (in the guise of (∞,1)-categories) and the theory of properads. Properads are devices more general than operads and enable one to encode bialgebraic, rather than just (co)algebraic, structures. The text extends both the Joyal-Lurie approach to higher categories and the Cisinski-Moerdijk-Weiss approach to higher operads, and provides a foundation for a broad study of the homotopy theory of properads. This work also serves as a complete guide to the generalised graphs which are pervasive in the study of operads and properads. A preliminary list of potential applications and extensions comprises the final chapter. Infinity Properads and Infinity Wheeled Properads is written for mathematicians in the fields of topology, algebra, category theory, and related areas. It is written roughly at the second year graduate level, and assumes a basic knowledge of category theory.

Conférence Moshé Flato 1999

Giuseppe Dito,Daniel Sternheimer
Conf rence Mosh Flato 1999 Book PDF

Author : Giuseppe Dito,Daniel Sternheimer
Publisher : Springer Science & Business Media
Page : 446 pages
File Size : 49,7 Mb
Release : 2000-07-31
Category : Mathematics
ISBN : 0792365402

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Conférence Moshé Flato 1999 by Giuseppe Dito,Daniel Sternheimer Pdf

These two volumes constitute the Proceedings of the `Conférence Moshé Flato, 1999'. Their spectrum is wide but the various areas covered are, in fact, strongly interwoven by a common denominator, the unique personality and creativity of the scientist in whose honor the Conference was held, and the far-reaching vision that underlies his scientific activity. With these two volumes, the reader will be able to take stock of the present state of the art in a number of subjects at the frontier of current research in mathematics, mathematical physics, and physics. Volume I is prefaced by reminiscences of and tributes to Flato's life and work. It also includes a section on the applications of sciences to insurance and finance, an area which was of interest to Flato before it became fashionable. The bulk of both volumes is on physical mathematics, where the reader will find these ingredients in various combinations, fundamental mathematical developments based on them, and challenging interpretations of physical phenomena. Audience: These volumes will be of interest to researchers and graduate students in a variety of domains, ranging from abstract mathematics to theoretical physics and other applications. Some parts will be accessible to proficient undergraduate students, and even to persons with a minimum of scientific knowledge but enough curiosity.