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| colored operads: Colored Operads Donald Yau, 2016-02-29 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. |
| colored operads: Operads of Wiring Diagrams Donald Yau, 2018-09-19 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. |
| colored operads: Nonsymmetric Operads in Combinatorics Samuele Giraudo, 2019-01-04 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. |
| colored operads: Infinity Operads And Monoidal Categories With Group Equivariance Donald Yau, 2021-12-02 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. |
| colored operads: Algebraic Operads Jean-Louis Loday, Bruno Vallette, 2012-08-08 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: Operads in Algebra, Topology and Physics Martin Markl, Steven Shnider, James D. Stasheff, 2002 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. |
| colored operads: Homotopical Quantum Field Theory Donald Yau, 2019-11-11 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. |
| colored operads: Homotopy of Operads and Grothendieck-Teichmuller Groups Benoit Fresse, 2017-05-22 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. |
| colored operads: Involutive Category Theory Donald Yau, 2020-11-30 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. |
| colored operads: Geometry and Dynamics of Groups and Spaces Mikhail Kapranov, Sergii Kolyada, Yu. I. Manin, Pieter Moree, Leonid Potyagailo, 2008-03-05 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. |
| colored operads: Graphs and Patterns in Mathematics and Theoretical Physics Mikhail Lyubich, Leon Armenovich Takhtadzhi︠a︡n, 2005 The Stony Brook Conference, Graphs and Patterns in Mathematics and Theoretical Physics, was dedicated to Dennis Sullivan in honor of his sixtieth birthday. The event's scientific content, which was suggested by Sullivan, was largely based on mini-courses and survey lectures. The main idea was to help researchers and graduate students in mathematics and theoretical physics who encounter graphs in their research to overcome conceptual barriers. The collection begins with Sullivan's paper, Sigma models and string topology, which describes a background algebraic structure for the sigma model based on algebraic topology and transversality. Other contributions to the volume were organized into five sections: Feynman Diagrams, Algebraic Structures, Manifolds: Invariants and Mirror Symmetry, Combinatorial Aspects of Dynamics, and Physics. These sections, along with more research-oriented articles, contain the following surveys: Feynman diagrams for pedestrians and mathematicians by M. Polyak, Notes on universal algebra by A. Voronov, Unimodal maps and hierarchical models by M. Yampolsky, and Quantum geometry in action: big bang and black holes by A. Ashtekar. This comprehensive volume is suitable for graduate students and research mathematicians interested in graph theory and its applications in mathematics and physics. |
| colored operads: Higher Structures and Operadic Calculus Bruno Vallette, 2025-05-19 This book presents the notes originating from five series of lectures given at the CRM Barcelona in 21-25 June, 2021, during the “Higher homotopical structures” programme. Since their introduction 60 years ago, the notions of infinity algebras (Stasheff, Sugawara), higher categories (Boardman-Vogt), operads (May), and model categories (Quillen) have given rise to powerful new tools which made possible the resolution of open problems and prompted revolutions in many domains like algebraic topology (rational homotopy theory, faithful algebraic invariants of the homotopy type of spaces), deformation theory (formality theorems, formal moduli problems), and mathematical physics (quantization of Poisson manifolds, quantum field theories), to name but a few. This theory of higher structures using operadic calculus is currently under rapid development. The aim of this book is to provide the community with an accessible state-of-the-art, while at the same time giving interested researchers and advanced students a brief overview on the subject. |
| colored operads: Real Homotopy of Configuration Spaces Najib Idrissi, 2022-06-11 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. |
| colored operads: The Homotopy Theory of (?,1)-Categories Julia E. Bergner, 2018-03-15 An introductory treatment to the homotopy theory of homotopical categories, presenting several models and comparisons between them. |
| colored operads: Deformation Theory of Algebras and Their Diagrams Martin Markl, 2012 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. |
| colored operads: Infinity Properads and Infinity Wheeled Properads Philip Hackney, Marcy Robertson, Donald Yau, 2015-09-07 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. |
| colored operads: Conférence Moshé Flato 1999 Giuseppe Dito, Daniel Sternheimer, 2000-07-31 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. |
| colored operads: Topics in Noncommutative Geometry Guillermo Cortiñas, 2012 Luis Santalo Winter Schools are organized yearly by the Mathematics Department and the Santalo Mathematical Research Institute of the School of Exact and Natural Sciences of the University of Buenos Aires (FCEN). This volume contains the proceedings of the third Luis Santalo Winter School which was devoted to noncommutative geometry and held at FCEN July 26-August 6, 2010. Topics in this volume concern noncommutative geometry in a broad sense, encompassing various mathematical and physical theories that incorporate geometric ideas to the study of noncommutative phenomena. It explores connections with several areas including algebra, analysis, geometry, topology and mathematical physics. Bursztyn and Waldmann discuss the classification of star products of Poisson structures up to Morita equivalence. Tsygan explains the connections between Kontsevich's formality theorem, noncommutative calculus, operads and index theory. Hoefel presents a concrete elementary construction in operad theory. Meyer introduces the subject of $\mathrm{C}^*$-algebraic crossed products. Rosenberg introduces Kasparov's $KK$-theory and noncommutative tori and includes a discussion of the Baum-Connes conjecture for $K$-theory of crossed products, among other topics. Lafont, Ortiz, and Sanchez-Garcia carry out a concrete computation in connection with the Baum-Connes conjecture. Zuk presents some remarkable groups produced by finite automata. Mesland discusses spectral triples and the Kasparov product in $KK$-theory. Trinchero explores the connections between Connes' noncommutative geometry and quantum field theory. Karoubi demonstrates a construction of twisted $K$-theory by means of twisted bundles. Tabuada surveys the theory of noncommutative motives. |
| colored operads: A Handbook of Model Categories Scott Balchin, 2021-10-29 This book outlines a vast array of techniques and methods regarding model categories, without focussing on the intricacies of the proofs. Quillen model categories are a fundamental tool for the understanding of homotopy theory. While many introductions to model categories fall back on the same handful of canonical examples, the present book highlights a large, self-contained collection of other examples which appear throughout the literature. In particular, it collects a highly scattered literature into a single volume. The book is aimed at anyone who uses, or is interested in using, model categories to study homotopy theory. It is written in such a way that it can be used as a reference guide for those who are already experts in the field. However, it can also be used as an introduction to the theory for novices. |
| colored operads: Symplectic, Poisson, and Noncommutative Geometry Tohru Eguchi, Yakov Eliashberg, Yoshiaki Maeda, 2014-08-25 This volume contains seven chapters based on lectures given by invited speakers at two May 2010 workshops held at the Mathematical Sciences Research Institute. |
| colored operads: A Foundation for PROPs, Algebras, and Modules Donald Yau, Mark W. Johnson, 2015-05-28 PROPs and their variants are extremely general and powerful machines that encode operations with multiple inputs and multiple outputs. In this respect PROPs can be viewed as generalizations of operads that would allow only a single output. Variants of PROPs are important in several mathematical fields, including string topology, topological conformal field theory, homotopical algebra, deformation theory, Poisson geometry, and graph cohomology. The purpose of this monograph is to develop, in full technical detail, a unifying object called a generalized PROP. Then with an appropriate choice of pasting scheme, one recovers (colored versions of) dioperads, half-PROPs, (wheeled) operads, (wheeled) properads, and (wheeled) PROPs. Here the fundamental operation of graph substitution is studied in complete detail for the first time, including all exceptional edges and loops as examples of a new definition of wheeled graphs. A notion of generators and relations is proposed which allows one to build all of the graphs in a given pasting scheme from a small set of basic graphs using graph substitution. This provides information at the level of generalized PROPs, but also at the levels of algebras and of modules over them. Working in the general context of a symmetric monoidal category, the theory applies for both topological spaces and chain complexes in characteristic zero. This book is useful for all mathematicians and mathematical physicists who want to learn this new powerful technique. |
| colored operads: Factorization Algebras in Quantum Field Theory Kevin Costello, Owen Gwilliam, 2017 This first volume develops factorization algebras with a focus upon examples exhibiting their use in field theory, which will be useful for researchers and graduates. |
| colored operads: Higher Structures in Topology, Geometry, and Physics Ralph M. Kaufmann, Martin Markl, Alexander A. Voronov, 2024-07-03 This volume contains the proceedings of the AMS Special Session on Higher Structures in Topology, Geometry, and Physics, held virtually on March 26–27, 2022. The articles give a snapshot survey of the current topics surrounding the mathematical formulation of field theories. There is an intricate interplay between geometry, topology, and algebra which captures these theories. The hallmark are higher structures, which one can consider as the secondary algebraic or geometric background on which the theories are formulated. The higher structures considered in the volume are generalizations of operads, models for conformal field theories, string topology, open/closed field theories, BF/BV formalism, actions on Hochschild complexes and related complexes, and their geometric and topological aspects. |
| colored operads: 2016 MATRIX Annals Jan de Gier, Cheryl E. Praeger, Terence Tao, 2018-04-10 MATRIX is Australia’s international, residential mathematical research institute. It facilitates new collaborations and mathematical advances through intensive residential research programs, each lasting 1-4 weeks. This book is a scientific record of the five programs held at MATRIX in its first year, 2016: - Higher Structures in Geometry and Physics - Winter of Disconnectedness - Approximation and Optimisation - Refining C*-Algebraic Invariants for Dynamics using KK-theory - Interactions between Topological Recursion, Modularity, Quantum Invariants and Low- dimensional Topology The MATRIX Scientific Committee selected these programs based on their scientific excellence and the participation rate of high-profile international participants. Each program included ample unstructured time to encourage collaborative research; some of the longer programs also included an embedded conference or lecture series. The articles are grouped into peer-reviewed contributions and other contributions. The peer-reviewed articles present original results or reviews on selected topics related to the MATRIX program; the remaining contributions are predominantly lecture notes based on talks or activities at MATRIX. |
| colored operads: Representations of Algebras, Geometry and Physics Kiyoshi Igusa, Alex Martsinkovsky, Gordana Todorov, 2021-05-17 This volume contains selected expository lectures delivered at the 2018 Maurice Auslander Distinguished Lectures and International Conference, held April 25–30, 2018, at the Woods Hole Oceanographic Institute, Woods Hole, MA. Reflecting recent developments in modern representation theory of algebras, the selected topics include an introduction to a new class of quiver algebras on surfaces, called “geodesic ghor algebras”, a detailed presentation of Feynman categories from a representation-theoretic viewpoint, connections between representations of quivers and the structure theory of Coxeter groups, powerful new applications of approximable triangulated categories, new results on the heart of a t t-structure, and an introduction to methods of constructive category theory. |
| colored operads: Alpine Perspectives on Algebraic Topology Christian Ausoni, Kathryn Hess, Jérôme Scherer, 2009 Contains the proceedings of the Third Arolla Conference on Algebraic Topology, which took place in Arolla, Switzerland, on August 18-24, 2008. This title includes research papers on stable homotopy theory, the theory of operads, localization and algebraic K-theory, as well as survey papers on the Witten genus and localization techniques. |
| colored operads: Extended Abstracts Spring 2015 Dolors Herbera, Wolfgang Pitsch, Santiago Zarzuela, 2016-11-30 This book includes 33 expanded abstracts of selected talks given at the two workshops Homological Bonds Between Commutative Algebra and Representation Theory and Brave New Algebra: Opening Perspectives, and the conference Opening Perspectives in Algebra, Representations, and Topology, held at the Centre de Recerca Matemàtica (CRM) in Barcelona between January and June 2015. These activities were part of the one-semester intensive research program Interactions Between Representation Theory, Algebraic Topology and Commutative Algebra (IRTATCA). Most of the abstracts present preliminary versions of not-yet published results and cover a large number of topics (including commutative and non commutative algebra, algebraic topology, singularity theory, triangulated categories, representation theory) overlapping with homological methods. This comprehensive book is a valuable resource for the community of researchers interested in homological algebra in a broad sense, and those curious to learn the latest developments in the area. It appeals to established researchers as well as PhD and postdoctoral students who want to learn more about the latest advances in these highly active fields of research. |
| colored operads: Generalized Bialgebras and Triples of Operads Jean-Louis Loday, 2008 This book introduces the notion of generalized bialgebra, which includes the classical notion of bialgebra (Hopf algebra) and many others, among them the tensor algebra equipped with the deconcatenation as coproduct. The author proves that, under some mild conditions, a connected generalized bialgebra is completely determined by its primitive part. This structure theorem extends the classical Poincare-Birkhoff-Witt theorem and Cartier-Milnor-Moore theorem, valid for cocommutative bialgebras, to a large class of generalized bialgebras. Technically, the author works in the theory of operads which allows him to state his main theorem and permits him to give it a conceptual proof. A generalized bialgebra type is determined by two operads: one for the coalgebra structure $\mathcal{C}$ and one for the algebra structure $\mathcal{A}$. There is also a compatibility relation relating the two. Under some conditions, the primitive part of such a generalized bialgebra is an algebra over some sub-operad of $\mathcal{A}$, denoted $\mathcal{P}$ . The structure theorem gives conditions under which a connected generalized bialgebra is cofree (as a connected $\mathcal{C}$-coalgebra) and can be reconstructed out of its primitive part by means of an enveloping functor from $\mathcal{P}$-algebras to $\mathcal{A}$-algebras. The classical case is $(\mathcal {C, A, P})=(Com, As, Lie)$. This structure theorem unifies several results, generalizing the PBW and the CMM theorems, scattered in the literature. The author treats many explicit examples and suggests a few conjectures. |
| colored operads: Homotopy of Operads and Grothendieck-Teichmuller Groups Benoit Fresse, 2017-04-21 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. |
| colored operads: Supplemento Ai Rendiconti Del Circolo Matematico Di Palermo Circolo matematico di Palermo, 1999 |
| colored operads: Algebraic & Geometric Topology , 2007 |
| colored operads: The Homology of the Open-closed Riemann Surface Dioperad and Open-closed String Topology Eric Harrelson, 2006 |
| colored operads: Séminaire Bourbaki Société mathématique de France, 2019 This 69th volume of the Bourbaki Seminar contains the texts of the fifteen survey lectures done during the year 2016/2017. Topics addressed covered Langlands correspondence, NIP property in model theory, Navier-Stokes equation, algebraic and complex analytic geometry, algorithmic and geometric questions in knot theory, analytic number theory formal moduli problems, general relativity, sofic entropy, sphere packings, subriemannian geometry. -- Prové de l'editor. |
| colored operads: Modules Over Operads and Functors Benoit Fresse, 2009-03-26 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. |
| colored operads: Subfactors, Planar Algebras and Rotations Michael E. Burns, 2003 |
| colored operads: Higher Operads, Higher Categories Tom Leinster, 2004-07-22 Foundations of higher dimensional category theory for graduate students and researchers in mathematics and mathematical physics. |
| colored operads: New Trends in Algebras and Combinatorics K. P. Shum, 2020 |
| colored operads: An Invitation to Applied Category Theory Brendan Fong, David I. Spivak, 2019-07-18 Category theory reveals commonalities between structures of all sorts. This book shows its potential in science, engineering, and beyond. |
| colored operads: Algebraic Operads Murray R. Bremner, Vladimir Dotsenko, 2016 This book presents a systematic treatment of Gröbner bases in several contexts. The book builds up to the theory of Gröbner bases for operads due to the second author and Khoroshkin as well as various applications of the corresponding diamond lemmas in algebra. Throughout the book, both the mathematical theory and computational methods are emphasized and numerous algorithms, examples, and exercises are provided to clarify and illustrate the concrete meaning of abstract theory. |
| colored operads: Diagrammatic Morphisms and Applications David E. Radford, David N. Yetter, 2003 The technique of diagrammatic morphisms is an important ingredient in comprehending and visualizing certain types of categories with structure. It was widely used in this capacity in many areas of algebra, low-dimensional topology and physics. It was also applied to problems in classical and quantum information processing and logic. This volume contains articles based on talks at the Special Session, ``Diagrammatic Morphisms in Algebra, Category Theory, and Topology'', at the AMS Sectional Meeting in San Francisco. The articles describe recent achievements in several aspects of diagrammatic morphisms and their applications. Some of them contain detailed expositions on various diagrammatic techniques. The introductory article by D. Yetter is a thorough account of the subject in a historical perspective. |
COLORED Definition & Meaning - Merriam-Webster
The meaning of COLORED is having color. How to use colored in a sentence. Usage of Colored: Usage Guide.
Colored - Wikipedia
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COLORED Synonyms: 192 Similar and Opposite Words | Merriam ...
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COLORED definition: 1. US spelling of coloured 2. US spelling of -coloured 3. having or producing a color or colors : . Learn more.
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Definition of colored adjective in Oxford Advanced Learner's Dictionary. Meaning, pronunciation, picture, example sentences, grammar, usage notes, synonyms and more.
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COLORED Definition & Meaning - Merriam-Webster
The meaning of COLORED is having color. How to use colored in a sentence. Usage of Colored: Usage …
Colored - Wikipedia
Colored (or coloured) is a racial descriptor historically used in the United States during the Jim Crow era to refer to an African American. In many places, it may be considered a …
COLORED Synonyms: 192 Similar and Opposite Words …
Synonyms for COLORED: colorful, varied, rainbow, various, striped, multicolored, vibrant, varicolored; Antonyms of COLORED: colorless, achromatic, faded, solid, bleached, …
Colored Lines aka Winlines - Games for the Brain
Build rows of 5 or more balls of one color to score. | Restart... [Original concept by Olga Demina, also known as Winlinez and Color Linez.]
COLORED | English meaning - Cambridge Dictionary
COLORED definition: 1. US spelling of coloured 2. US spelling of -coloured 3. having or producing a color or …
