Download or read book Coherence for Tricategories written by Robert Gordon and published by American Mathematical Soc.. This book was released on 1995 with total page 94 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work defines the concept of tricategory as the natural 3-dimensional generalization of bicategory. Trihomomorphism and triequivalence for tricategories are also defined so as to extend the concepts of homomorphism and biequivalence for bicategories.
Download or read book Coherence in Three Dimensional Category Theory written by Nick Gurski and published by Cambridge University Press. This book was released on 2013-03-21 with total page 287 pages. Available in PDF, EPUB and Kindle. Book excerpt: Dimension three is an important test-bed for hypotheses in higher category theory and occupies something of a unique position in the categorical landscape. At the heart of matters is the coherence theorem, of which this book provides a definitive treatment, as well as covering related results. Along the way the author treats such material as the Gray tensor product and gives a construction of the fundamental 3-groupoid of a space. The book serves as a comprehensive introduction, covering essential material for any student of coherence and assuming only a basic understanding of higher category theory. It is also a reference point for many key concepts in the field and therefore a vital resource for researchers wishing to apply higher categories or coherence results in fields such as algebraic topology or theoretical computer science.
Download or read book Coherence in Three Dimensional Category Theory written by Nick Gurski and published by . This book was released on 2014-05-14 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt: Serves as an introduction to higher categories as well as a reference point for many key concepts in the field.
Download or read book The Joy of Abstraction written by Eugenia Cheng and published by Cambridge University Press. This book was released on 2022-10-13 with total page 440 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematician and popular science author Eugenia Cheng is on a mission to show you that mathematics can be flexible, creative, and visual. This joyful journey through the world of abstract mathematics into category theory will demystify mathematical thought processes and help you develop your own thinking, with no formal mathematical background needed. The book brings abstract mathematical ideas down to earth using examples of social justice, current events, and everyday life – from privilege to COVID-19 to driving routes. The journey begins with the ideas and workings of abstract mathematics, after which you will gently climb toward more technical material, learning everything needed to understand category theory, and then key concepts in category theory like natural transformations, duality, and even a glimpse of ongoing research in higher-dimensional category theory. For fans of How to Bake Pi, this will help you dig deeper into mathematical concepts and build your mathematical background.
Download or read book Categories in Algebra Geometry and Mathematical Physics written by Alexei Davydov and published by American Mathematical Soc.. This book was released on 2007 with total page 482 pages. Available in PDF, EPUB and Kindle. Book excerpt: Category theory has become the universal language of modern mathematics. This book is a collection of articles applying methods of category theory to the areas of algebra, geometry, and mathematical physics. Among others, this book contains articles on higher categories and their applications and on homotopy theoretic methods. The reader can learn about the exciting new interactions of category theory with very traditional mathematical disciplines.
Download or read book 2 Dimensional Categories written by Niles Johnson and published by Oxford University Press. This book was released on 2021-01-31 with total page 476 pages. Available in PDF, EPUB and Kindle. Book excerpt: Category theory emerged in the 1940s in the work of Samuel Eilenberg and Saunders Mac Lane. It describes relationships between mathematical structures. Outside of pure mathematics, category theory is an important tool in physics, computer science, linguistics, and a quickly-growing list of other sciences. This book is about 2-dimensional categories, which add an extra dimension of richness and complexity to category theory. 2-Dimensional Categories is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories, pasting diagrams, lax functors, 2-/bilimits, the Duskin nerve, 2-nerve, internal adjunctions, monads in bicategories, 2-monads, biequivalences, the Bicategorical Yoneda Lemma, and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.
Download or read book Simplicial Methods for Higher Categories written by Simona Paoli and published by Springer. This book was released on 2019-06-03 with total page 343 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph presents a new model of mathematical structures called weak n-categories. These structures find their motivation in a wide range of fields, from algebraic topology to mathematical physics, algebraic geometry and mathematical logic. While strict n-categories are easily defined in terms associative and unital composition operations they are of limited use in applications, which often call for weakened variants of these laws. The author proposes a new approach to this weakening, whose generality arises not from a weakening of such laws but from the very geometric structure of its cells; a geometry dubbed weak globularity. The new model, called weakly globular n-fold categories, is one of the simplest known algebraic structures yielding a model of weak n-categories. The central result is the equivalence of this model to one of the existing models, due to Tamsamani and further studied by Simpson. This theory has intended applications to homotopy theory, mathematical physics and to long-standing open questions in category theory. As the theory is described in elementary terms and the book is largely self-contained, it is accessible to beginning graduate students and to mathematicians from a wide range of disciplines well beyond higher category theory. The new model makes a transparent connection between higher category theory and homotopy theory, rendering it particularly suitable for category theorists and algebraic topologists. Although the results are complex, readers are guided with an intuitive explanation before each concept is introduced, and with diagrams showing the interconnections between the main ideas and results.
Download or read book Encyclopaedia of Mathematics written by Michiel Hazewinkel and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 639 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the second supplementary volume to Kluwer's highly acclaimed eleven-volume Encyclopaedia of Mathematics. This additional volume contains nearly 500 new entries written by experts and covers developments and topics not included in the previous volumes. These entries are arranged alphabetically throughout and a detailed index is included. This supplementary volume enhances the existing eleven volumes, and together these twelve volumes represent the most authoritative, comprehensive and up-to-date Encyclopaedia of Mathematics available.
Download or read book Hopf Algebras Quantum Groups and Yang Baxter Equations written by Florin Felix Nichita and published by MDPI. This book was released on 2019-01-31 with total page 239 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a printed edition of the Special Issue "Hopf Algebras, Quantum Groups and Yang-Baxter Equations" that was published in Axioms
Download or read book Homotopy Theory of Higher Categories written by Carlos Simpson and published by Cambridge University Press. This book was released on 2011-10-20 with total page 653 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of higher categories is attracting growing interest for its many applications in topology, algebraic geometry, mathematical physics and category theory. In this highly readable book, Carlos Simpson develops a full set of homotopical algebra techniques and proposes a working theory of higher categories. Starting with a cohesive overview of the many different approaches currently used by researchers, the author proceeds with a detailed exposition of one of the most widely used techniques: the construction of a Cartesian Quillen model structure for higher categories. The fully iterative construction applies to enrichment over any Cartesian model category, and yields model categories for weakly associative n-categories and Segal n-categories. A corollary is the construction of higher functor categories which fit together to form the (n+1)-category of n-categories. The approach uses Tamsamani's definition based on Segal's ideas, iterated as in Pelissier's thesis using modern techniques due to Barwick, Bergner, Lurie and others.
Download or read book Modeling Multi Level Systems written by Octavian Iordache and published by Springer Science & Business Media. This book was released on 2011-02-07 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to modeling of multi-level complex systems, a challenging domain for engineers, researchers and entrepreneurs, confronted with the transition from learning and adaptability to evolvability and autonomy for technologies, devices and problem solving methods. Chapter 1 introduces the multi-scale and multi-level systems and highlights their presence in different domains of science and technology. Methodologies as, random systems, non-Archimedean analysis, category theory and specific techniques as model categorification and integrative closure, are presented in chapter 2. Chapters 3 and 4 describe polystochastic models, PSM, and their developments. Categorical formulation of integrative closure offers the general PSM framework which serves as a flexible guideline for a large variety of multi-level modeling problems. Focusing on chemical engineering, pharmaceutical and environmental case studies, the chapters 5 to 8 analyze mixing, turbulent dispersion and entropy production for multi-scale systems. Taking inspiration from systems sciences, chapters 9 to 11 highlight multi-level modeling potentialities in formal concept analysis, existential graphs and evolvable designs of experiments. Case studies refer to separation flow-sheets, pharmaceutical pipeline, drug design and development, reliability management systems, security and failure analysis. Perspectives and integrative points of view are discussed in chapter 12. Autonomous and viable systems, multi-agents, organic and autonomic computing, multi-level informational systems, are revealed as promising domains for future applications. Written for: engineers, researchers, entrepreneurs and students in chemical, pharmaceutical, environmental and systems sciences engineering, and for applied mathematicians.
Download or read book Applied Differential Geometry written by and published by . This book was released on with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Applied Differential Geometry written by Vladimir G. Ivancevic and published by World Scientific. This book was released on 2007 with total page 1346 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduction -- Technical preliminaries: tensors, actions and functors -- Applied manifold geometry -- Applied bundle geometry -- Applied jet geometry -- Geometrical path integrals and their applications
Download or read book Towards Higher Categories written by John C. Baez and published by Springer Science & Business Media. This book was released on 2009-09-24 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of this book is to give background for those who would like to delve into some higher category theory. It is not a primer on higher category theory itself. It begins with a paper by John Baez and Michael Shulman which explores informally, by analogy and direct connection, how cohomology and other tools of algebraic topology are seen through the eyes of n-category theory. The idea is to give some of the motivations behind this subject. There are then two survey articles, by Julie Bergner and Simona Paoli, about (infinity,1) categories and about the algebraic modelling of homotopy n-types. These are areas that are particularly well understood, and where a fully integrated theory exists. The main focus of the book is on the richness to be found in the theory of bicategories, which gives the essential starting point towards the understanding of higher categorical structures. An article by Stephen Lack gives a thorough, but informal, guide to this theory. A paper by Larry Breen on the theory of gerbes shows how such categorical structures appear in differential geometry. This book is dedicated to Max Kelly, the founder of the Australian school of category theory, and an historical paper by Ross Street describes its development.
Download or read book Theorem Proving in Higher Order Logics written by Elsa L. Gunter and published by Springer Science & Business Media. This book was released on 1997-08-06 with total page 358 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 10th International Conference on Theorem Proving in Higher Order Logics, TPHOLs '97, held in Murray Hill, NJ, USA, in August 1997. The volume presents 19 carefully revised full papers selected from 32 submissions during a thorough reviewing process. The papers cover work related to all aspects of theorem proving in higher order logics, particularly based on secure mechanization of those logics; the theorem proving systems addressed include Coq, HOL, Isabelle, LEGO, and PVS.
Download or read book Squared Hopf Algebras written by Volodymyr V. Lyubashenko and published by American Mathematical Soc.. This book was released on 1999 with total page 197 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is intended for graduate students and research mathematicians interested in associative rings and algebras.
Download or read book Types for Proofs and Programs written by Stefano Berardi and published by Springer Science & Business Media. This book was released on 1996-10-02 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains a refereed selection of revised full papers chosen from the contributions presented during the Third Annual Workshop held under the auspices of the ESPRIT Basic Research Action 6453 Types for Proofs and Programs. The workshop took place in Torino, Italy, in June 1995. Type theory is a formalism in which theorems and proofs, specifications and programs can be represented in a uniform way. The 19 papers included in the book deal with foundations of type theory, logical frameworks, and implementations and applications; all in all they constitute a state-of-the-art survey for the area of type theory.
