Involutive Category Theory

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Involutive Category Theory

Author : Donald Yau
Publisher : Springer Nature
ISBN 13 : 3030612031
Total Pages : 250 pages
Book Rating : 4.0/5 (36 download)

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Book Synopsis Involutive Category Theory by : Donald Yau

Download or read book Involutive Category Theory written by Donald Yau and published by Springer Nature. This book was released on 2020-11-30 with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 Chapters three through five feature previously unpublished results on coherence and strictification of involutive categories and involutive monoidal categories, showcasing the author’s state-of-the-art research.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.

Elements of ?-Category Theory

Author : Emily Riehl
Publisher : Cambridge University Press
ISBN 13 : 1108837980
Total Pages : 781 pages
Book Rating : 4.1/5 (88 download)

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Book Synopsis Elements of ?-Category Theory by : Emily Riehl

Download or read book Elements of ?-Category Theory written by Emily Riehl and published by Cambridge University Press. This book was released on 2022-02-10 with total page 781 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book develops the theory of infinite-dimensional categories by studying the universe, or ∞-cosmos, in which they live.

Category Theory And Applications: A Textbook For Beginners (Second Edition)

Author : Marco Grandis
Publisher : World Scientific
ISBN 13 : 9811236100
Total Pages : 390 pages
Book Rating : 4.8/5 (112 download)

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Book Synopsis Category Theory And Applications: A Textbook For Beginners (Second Edition) by : Marco Grandis

Download or read book Category Theory And Applications: A Textbook For Beginners (Second Edition) written by Marco Grandis and published by World Scientific. This book was released on 2021-03-05 with total page 390 pages. Available in PDF, EPUB and Kindle. Book excerpt: Category Theory now permeates most of Mathematics, large parts of theoretical Computer Science and parts of theoretical Physics. Its unifying power brings together different branches, and leads to a better understanding of their roots.This book is addressed to students and researchers of these fields and can be used as a text for a first course in Category Theory. It covers the basic tools, like universal properties, limits, adjoint functors and monads. These are presented in a concrete way, starting from examples and exercises taken from elementary Algebra, Lattice Theory and Topology, then developing the theory together with new exercises and applications.A reader should have some elementary knowledge of these three subjects, or at least two of them, in order to be able to follow the main examples, appreciate the unifying power of the categorical approach, and discover the subterranean links brought to light and formalised by this perspective.Applications of Category Theory form a vast and differentiated domain. This book wants to present the basic applications in Algebra and Topology, with a choice of more advanced ones, based on the interests of the author. References are given for applications in many other fields.In this second edition, the book has been entirely reviewed, adding many applications and exercises. All non-obvious exercises have now a solution (or a reference, in the case of an advanced topic); solutions are now collected in the last chapter.

Manifolds And Local Structures: A General Theory

Author : Marco Grandis
Publisher : World Scientific
ISBN 13 : 9811234019
Total Pages : 374 pages
Book Rating : 4.8/5 (112 download)

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Book Synopsis Manifolds And Local Structures: A General Theory by : Marco Grandis

Download or read book Manifolds And Local Structures: A General Theory written by Marco Grandis and published by World Scientific. This book was released on 2021-02-10 with total page 374 pages. Available in PDF, EPUB and Kindle. Book excerpt: Local structures, like differentiable manifolds, fibre bundles, vector bundles and foliations, can be obtained by gluing together a family of suitable 'elementary spaces', by means of partial homeomorphisms that fix the gluing conditions and form a sort of 'intrinsic atlas', instead of the more usual system of charts living in an external framework.An 'intrinsic manifold' is defined here as such an atlas, in a suitable category of elementary spaces: open euclidean spaces, or trivial bundles, or trivial vector bundles, and so on.This uniform approach allows us to move from one basis to another: for instance, the elementary tangent bundle of an open Euclidean space is automatically extended to the tangent bundle of any differentiable manifold. The same holds for tensor calculus.Technically, the goal of this book is to treat these structures as 'symmetric enriched categories' over a suitable basis, generally an ordered category of partial mappings.This approach to gluing structures is related to Ehresmann's one, based on inductive pseudogroups and inductive categories. A second source was the theory of enriched categories and Lawvere's unusual view of interesting mathematical structures as categories enriched over a suitable basis.

Higher Dimensional Categories: From Double To Multiple Categories

Author : Marco Grandis
Publisher : World Scientific
ISBN 13 : 9811205124
Total Pages : 535 pages
Book Rating : 4.8/5 (112 download)

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Book Synopsis Higher Dimensional Categories: From Double To Multiple Categories by : Marco Grandis

Download or read book Higher Dimensional Categories: From Double To Multiple Categories written by Marco Grandis and published by World Scientific. This book was released on 2019-09-09 with total page 535 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of higher dimensional categories has mostly been developed in the globular form of 2-categories, n-categories, omega-categories and their weak versions. Here we study a different form: double categories, n-tuple categories and multiple categories, with their weak and lax versions.We want to show the advantages of this form for the theory of adjunctions and limits. Furthermore, this form is much simpler in higher dimension, starting with dimension three where weak 3-categories (also called tricategories) are already quite complicated, much more than weak or lax triple categories.This book can be used as a textbook for graduate and postgraduate studies, and as a basis for research. Notions are presented in a 'concrete' way, with examples and exercises; the latter are endowed with a solution or hints. Part I, devoted to double categories, starts at basic category theory and is kept at a relatively simple level. Part II, on multiple categories, can be used independently by a reader acquainted with 2-dimensional categories.

Categories for Quantum Theory

Author : Chris Heunen
Publisher : Oxford University Press
ISBN 13 : 0191060062
Total Pages : 320 pages
Book Rating : 4.1/5 (91 download)

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Book Synopsis Categories for Quantum Theory by : Chris Heunen

Download or read book Categories for Quantum Theory written by Chris Heunen and published by Oxford University Press. This book was released on 2019-11-14 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: Monoidal category theory serves as a powerful framework for describing logical aspects of quantum theory, giving an abstract language for parallel and sequential composition, and a conceptual way to understand many high-level quantum phenomena. This text lays the foundation for this categorical quantum mechanics, with an emphasis on the graphical calculus which makes computation intuitive. Biproducts and dual objects are introduced and used to model superposition and entanglement, with quantum teleportation studied abstractly using these structures. Monoids, Frobenius structures and Hopf algebras are described, and it is shown how they can be used to model classical information and complementary observables. The CP construction, a categorical tool to describe probabilistic quantum systems, is also investigated. The last chapter introduces higher categories, surface diagrams and 2-Hilbert spaces, and shows how the language of duality in monoidal 2-categories can be used to reason about quantum protocols, including quantum teleportation and dense coding. Prior knowledge of linear algebra, quantum information or category theory would give an ideal background for studying this text, but it is not assumed, with essential background material given in a self-contained introductory chapter. Throughout the text links with many other areas are highlighted, such as representation theory, topology, quantum algebra, knot theory, and probability theory, and nonstandard models are presented, such as sets and relations. All results are stated rigorously, and full proofs are given as far as possible, making this book an invaluable reference for modern techniques in quantum logic, with much of the material not available in any other textbook.

Grothendieck Construction of Bipermutative-Indexed Categories

Author : Donald Yau
Publisher : CRC Press
ISBN 13 : 1003807461
Total Pages : 361 pages
Book Rating : 4.0/5 (38 download)

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Book Synopsis Grothendieck Construction of Bipermutative-Indexed Categories by : Donald Yau

Download or read book Grothendieck Construction of Bipermutative-Indexed Categories written by Donald Yau and published by CRC Press. This book was released on 2023-12-06 with total page 361 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is the first and only book-length reference for this material. Contents of Chapter 2, Chapter 3, Part 2, and Part 3 is new, not having appeared in any of the research literature. The book will appeal to mathematicians interested in topology. Book shelved as a reference title.

Infinity Operads And Monoidal Categories With Group Equivariance

Author : Donald Yau
Publisher : World Scientific
ISBN 13 : 9811250944
Total Pages : 486 pages
Book Rating : 4.8/5 (112 download)

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Book Synopsis Infinity Operads And Monoidal Categories With Group Equivariance by : Donald Yau

Download or read book Infinity Operads And Monoidal Categories With Group Equivariance written by Donald Yau and published by World Scientific. This book was released on 2021-12-02 with total page 486 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Topological Algebras with Involution

Author : M. Fragoulopoulou
Publisher : Elsevier
ISBN 13 : 0080461220
Total Pages : 514 pages
Book Rating : 4.0/5 (84 download)

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Book Synopsis Topological Algebras with Involution by : M. Fragoulopoulou

Download or read book Topological Algebras with Involution written by M. Fragoulopoulou and published by Elsevier. This book was released on 2005-07-26 with total page 514 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book familiarizes both popular and fundamental notions and techniques from the theory of non-normed topological algebras with involution, demonstrating with examples and basic results the necessity of this perspective. The main body of the book is focussed on the Hilbert-space (bounded) representation theory of topological *-algebras and their topological tensor products, since in our physical world, apart from the majority of the existing unbounded operators, we often meet operators that are forced to be bounded, like in the case of symmetric *-algebras. So, one gets an account of how things behave, when the mathematical structures are far from being algebras endowed with a complete or non-complete algebra norm. In problems related with mathematical physics, such instances are, indeed, quite common.Key features:- Lucid presentation- Smooth in reading- Informative- Illustrated by examples- Familiarizes the reader with the non-normed *-world- Encourages the hesitant- Welcomes new comers.- Well written and lucid presentation.- Informative and illustrated by examples.- Familiarizes the reader with the non-normed *-world.

The Theory of Quantaloids

Author : K I Rosenthal
Publisher : CRC Press
ISBN 13 : 1498710409
Total Pages : 160 pages
Book Rating : 4.4/5 (987 download)

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Book Synopsis The Theory of Quantaloids by : K I Rosenthal

Download or read book The Theory of Quantaloids written by K I Rosenthal and published by CRC Press. This book was released on 2014-07-22 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a detailed account of the theory of quantaloids, a natural generalization of quantales. The basic theory, examples and construction are given and particular emphasis is placed on the free quantaloid construction, as well as on the perspective provided by enriched categories.

Involutions on Manifolds

Author : Santiago Lopez de Medrano
Publisher : Springer Science & Business Media
ISBN 13 : 3642650120
Total Pages : 114 pages
Book Rating : 4.6/5 (426 download)

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Book Synopsis Involutions on Manifolds by : Santiago Lopez de Medrano

Download or read book Involutions on Manifolds written by Santiago Lopez de Medrano and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains the results of work done during the years 1967-1970 on fixed-point-free involutions on manifolds, and is an enlarged version of the author's doctoral dissertation [54J written under the direction of Professor William Browder. The subject of fixed-paint-free involutions, as part of the subject of group actions on manifolds, has been an important source of problems, examples and ideas in topology for the last four decades, and receives renewed attention every time a new technical development suggests new questions and methods ([62, 8, 24, 63J). Here we consider mainly those properties of fixed-point-free involutions that can be best studied using the techniques of surgery on manifolds. This approach to the subject was initiated by Browder and Livesay. Special attention is given here to involutions of homotopy spheres, but even for this particular case, a more general theory is very useful. Two important related topics that we do not touch here are those of involutions with fixed points, and the relationship between fixed-point-free involutions and free Sl-actions. For these topics, the reader is referred to [23J, and to [33J, [61J, [82J, respectively. The two main problems we attack are those of classification of involutions, and the existence and uniqueness of invariant submanifolds with certain properties. As will be seen, these problems are closely related. If (T, l'n) is a fixed-point-free involution of a homotopy sphere l'n, the quotient l'n/Tis called a homotopy projective space.

Algebra and Coalgebra in Computer Science

Author : Reiko Heckel
Publisher : Springer
ISBN 13 : 3642402062
Total Pages : 370 pages
Book Rating : 4.6/5 (424 download)

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Book Synopsis Algebra and Coalgebra in Computer Science by : Reiko Heckel

Download or read book Algebra and Coalgebra in Computer Science written by Reiko Heckel and published by Springer. This book was released on 2013-08-27 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 5th International Conference on Algebra and Coalgebra in Computer Science, CALCO 2013, held in Warsaw, Poland, in September 2013. The 18 full papers presented together with 4 invited talks were carefully reviewed and selected from 33 submissions. The papers cover topics in the fields of abstract models and logics, specialized models and calculi, algebraic and coalgebraic semantics, system specification and verification, as well as corecursion in programming languages, and algebra and coalgebra in quantum computing. The book also includes 6 papers from the CALCO Tools Workshop, co-located with CALCO 2013 and dedicated to tools based on algebraic and/or coalgebraic principles.

Homological Algebra

Author : Marco Grandis
Publisher : World Scientific
ISBN 13 : 9814407070
Total Pages : 382 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis Homological Algebra by : Marco Grandis

Download or read book Homological Algebra written by Marco Grandis and published by World Scientific. This book was released on 2012 with total page 382 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book we want to explore aspects of coherence in homological algebra, that already appear in the classical situation of abelian groups or abelian categories. Lattices of subobjects are shown to play an important role in the study of homological systems, from simple chain complexes to all the structures that give rise to spectral sequences. A parallel role is played by semigroups of endorelations. These links rest on the fact that many such systems, but not all of them, live in distributive sublattices of the modular lattices of subobjects of the system. The property of distributivity allows one to work with induced morphisms in an automatically consistent way, as we prove in a 'Coherence Theorem for homological algebra'. (On the contrary, a 'non-distributive' homological structure like the bifiltered chain complex can easily lead to inconsistency, if one explores the interaction of its two spectral sequences farther than it is normally done.) The same property of distributivity also permits representations of homological structures by means of sets and lattices of subsets, yielding a precise foundation for the heuristic tool of Zeeman diagrams as universal models of spectral sequences. We thus establish an effective method of working with spectral sequences, called 'crossword chasing', that can often replace the usual complicated algebraic tools and be of much help to readers that want to apply spectral sequences in any field.

Issues in General Physics Research: 2013 Edition

Author :
Publisher : ScholarlyEditions
ISBN 13 : 1490106901
Total Pages : 1211 pages
Book Rating : 4.4/5 (91 download)

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Book Synopsis Issues in General Physics Research: 2013 Edition by :

Download or read book Issues in General Physics Research: 2013 Edition written by and published by ScholarlyEditions. This book was released on 2013-05-01 with total page 1211 pages. Available in PDF, EPUB and Kindle. Book excerpt: Issues in General Physics Research / 2013 Edition is a ScholarlyEditions™ book that delivers timely, authoritative, and comprehensive information about Quantum Physics. The editors have built Issues in General Physics Research: 2013 Edition on the vast information databases of ScholarlyNews.™ You can expect the information about Quantum Physics in this book to be deeper than what you can access anywhere else, as well as consistently reliable, authoritative, informed, and relevant. The content of Issues in General Physics Research: 2013 Edition has been produced by the world’s leading scientists, engineers, analysts, research institutions, and companies. All of the content is from peer-reviewed sources, and all of it is written, assembled, and edited by the editors at ScholarlyEditions™ and available exclusively from us. You now have a source you can cite with authority, confidence, and credibility. More information is available at http://www.ScholarlyEditions.com/.

Theoretical Computer Science

Author : Christian S. Calude
Publisher : Springer Science & Business Media
ISBN 13 : 3642152392
Total Pages : 397 pages
Book Rating : 4.6/5 (421 download)

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Book Synopsis Theoretical Computer Science by : Christian S. Calude

Download or read book Theoretical Computer Science written by Christian S. Calude and published by Springer Science & Business Media. This book was released on 2010-08-23 with total page 397 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 6th FIP WG 2.2 International Conference, TCS 2010, held as a part of the 21th World Computer Congress, WCC 2010, in Brisbane, Australia, in September 2010. The 23 revised full papers presented, together with 4 invited talks, were carefully reviewed and selected from 39 submissions. TCS 2010 deals with topics focused at but not limited to algorithms, complexity, models of computation, logic, semantics, specification and verification, power-awareness issues in wireless networks, data mining, knowledge discovery, multiprocessor issues as well as AI issues.

Reversible Computation

Author : Michael Kirkedal Thomsen
Publisher : Springer
ISBN 13 : 3030215008
Total Pages : 246 pages
Book Rating : 4.0/5 (32 download)

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Book Synopsis Reversible Computation by : Michael Kirkedal Thomsen

Download or read book Reversible Computation written by Michael Kirkedal Thomsen and published by Springer. This book was released on 2019-06-17 with total page 246 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 11th International Conference on Reversible Computation, RC 2019, held in Lausanne, Switzerland, in June 2019. The 12 full papers and two short papers included in this volume were carefully reviewed and selected from 22 submissions. One invited talk is also included. The papers are organized in the following topical sections: theory and foundation; programming languages; circuit synthesis; evaluation of circuit synthesis; and applications and implementations.

Enric Trillas: A Passion for Fuzzy Sets

Author : Luis Magdalena
Publisher : Springer
ISBN 13 : 3319162357
Total Pages : 340 pages
Book Rating : 4.3/5 (191 download)

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Book Synopsis Enric Trillas: A Passion for Fuzzy Sets by : Luis Magdalena

Download or read book Enric Trillas: A Passion for Fuzzy Sets written by Luis Magdalena and published by Springer. This book was released on 2015-03-30 with total page 340 pages. Available in PDF, EPUB and Kindle. Book excerpt: (Preliminary) The book is a comprehensive collection of the most recent and significant research and applications in the field of fuzzy logic. It covers fuzzy structures, systems, rules, operations as well as important applications, e.g in decision making, environmental prediction and prevention, and communication. It is dedicated to Enric Trillas as an acknowledgement for his pioneering research in the field. The book include a foreword by Lotfi A. Zadeh.