Involutive Category Theory

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Involutive Category Theory

Donald Yau
Involutive Category Theory Book PDF

Author : Donald Yau
Publisher : Springer Nature
Page : 250 pages
File Size : 43,8 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 Chapters three through five feature previously unpublished results on coherence and strictification of involutive categories and involutive monoidal categories, showcasing the author’s state-of-the-art research.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.

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

Marco Grandis
Category Theory And Applications A Textbook For Beginners Second Edition Book PDF

Author : Marco Grandis
Publisher : World Scientific
Page : 390 pages
File Size : 52,8 Mb
Release : 2021-03-05
Category : Mathematics
ISBN : 9789811236105

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

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.

Category Theory 1991: Proceedings of the 1991 Summer Category Theory Meeting, Montreal, Canada

Robert Andrew George Seely
Category Theory 1991 Proceedings of the 1991 Summer Category Theory Meeting Montreal Canada Book PDF

Author : Robert Andrew George Seely
Publisher : American Mathematical Soc.
Page : 462 pages
File Size : 47,9 Mb
Release : 1992
Category : Mathematics
ISBN : 0821860186

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Category Theory 1991: Proceedings of the 1991 Summer Category Theory Meeting, Montreal, Canada by Robert Andrew George Seely Pdf

Representing this diversity of the field, this book contains the proceedings of an international conference on category theory. The subjects covered here range from topology and geometry to logic and theoretical computer science, from homotopy to braids and conformal field theory. Although generally aimed at experts in the various fields represented, the book will also provide an excellent opportunity for nonexperts to get a feel for the diversity of current applications of category theory.

Elements of ?-Category Theory

Emily Riehl,Dominic Verity
Elements of Category Theory Book PDF

Author : Emily Riehl,Dominic Verity
Publisher : Cambridge University Press
Page : 781 pages
File Size : 44,8 Mb
Release : 2022-02-10
Category : Mathematics
ISBN : 9781108837989

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Elements of ?-Category Theory by Emily Riehl,Dominic Verity Pdf

This book develops the theory of infinite-dimensional categories by studying the universe, or ∞-cosmos, in which they live.

Tensor Categories

Pavel Etingof,Shlomo Gelaki,Dmitri Nikshych,Victor Ostrik
Tensor Categories Book PDF

Author : Pavel Etingof,Shlomo Gelaki,Dmitri Nikshych,Victor Ostrik
Publisher : American Mathematical Soc.
Page : 344 pages
File Size : 47,9 Mb
Release : 2016-08-05
Category : Algebraic topology
ISBN : 9781470434410

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Tensor Categories by Pavel Etingof,Shlomo Gelaki,Dmitri Nikshych,Victor Ostrik Pdf

Is there a vector space whose dimension is the golden ratio? Of course not—the golden ratio is not an integer! But this can happen for generalizations of vector spaces—objects of a tensor category. The theory of tensor categories is a relatively new field of mathematics that generalizes the theory of group representations. It has deep connections with many other fields, including representation theory, Hopf algebras, operator algebras, low-dimensional topology (in particular, knot theory), homotopy theory, quantum mechanics and field theory, quantum computation, theory of motives, etc. This book gives a systematic introduction to this theory and a review of its applications. While giving a detailed overview of general tensor categories, it focuses especially on the theory of finite tensor categories and fusion categories (in particular, braided and modular ones), and discusses the main results about them with proofs. In particular, it shows how the main properties of finite-dimensional Hopf algebras may be derived from the theory of tensor categories. Many important results are presented as a sequence of exercises, which makes the book valuable for students and suitable for graduate courses. Many applications, connections to other areas, additional results, and references are discussed at the end of each chapter.

Category Theory in Context

Emily Riehl
Category Theory in Context Book PDF

Author : Emily Riehl
Publisher : Courier Dover Publications
Page : 272 pages
File Size : 45,7 Mb
Release : 2017-03-09
Category : Mathematics
ISBN : 9780486820804

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Category Theory in Context by Emily Riehl Pdf

Introduction to concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads — revisits a broad range of mathematical examples from the categorical perspective. 2016 edition.

The Theory of Quantaloids

K I Rosenthal
The Theory of Quantaloids Book PDF

Author : K I Rosenthal
Publisher : CRC Press
Page : 176 pages
File Size : 42,8 Mb
Release : 2014-07-22
Category : Mathematics
ISBN : 9781498710404

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The Theory of Quantaloids by K I Rosenthal Pdf

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.

Manifolds And Local Structures: A General Theory

Marco Grandis
Manifolds And Local Structures A General Theory Book PDF

Author : Marco Grandis
Publisher : World Scientific
Page : 374 pages
File Size : 46,8 Mb
Release : 2021-02-10
Category : Mathematics
ISBN : 9789811234019

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Manifolds And Local Structures: A General Theory by Marco Grandis Pdf

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

Grandis Marco
Higher Dimensional Categories From Double To Multiple Categories Book PDF

Author : Grandis Marco
Publisher : World Scientific
Page : 536 pages
File Size : 55,5 Mb
Release : 2019-09-09
Category : Mathematics
ISBN : 9789811205125

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Higher Dimensional Categories: From Double To Multiple Categories by Grandis Marco Pdf

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.

Grothendieck Construction of Bipermutative-Indexed Categories

Donald Yau
Grothendieck Construction of Bipermutative Indexed Categories Book PDF

Author : Donald Yau
Publisher : CRC Press
Page : 361 pages
File Size : 49,8 Mb
Release : 2023-12-06
Category : Mathematics
ISBN : 9781003807469

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Grothendieck Construction of Bipermutative-Indexed Categories by Donald Yau Pdf

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

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

Author : Donald Yau
Publisher : World Scientific
Page : 486 pages
File Size : 47,7 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.

Involutions on Manifolds

Santiago Lopez de Medrano
Involutions on Manifolds Book PDF

Author : Santiago Lopez de Medrano
Publisher : Springer Science & Business Media
Page : 114 pages
File Size : 54,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642650123

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Involutions on Manifolds by Santiago Lopez de Medrano Pdf

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.

Topological Algebras with Involution

M. Fragoulopoulou
Topological Algebras with Involution Book PDF

Author : M. Fragoulopoulou
Publisher : Elsevier
Page : 514 pages
File Size : 55,6 Mb
Release : 2005-07-26
Category : Mathematics
ISBN : 9780080461229

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Topological Algebras with Involution by M. Fragoulopoulou Pdf

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.

Categories for Quantum Theory

Chris Heunen,Jamie Vicary
Categories for Quantum Theory Book PDF

Author : Chris Heunen,Jamie Vicary
Publisher : Oxford University Press
Page : 320 pages
File Size : 52,7 Mb
Release : 2019-11-14
Category : Mathematics
ISBN : 9780191060069

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Categories for Quantum Theory by Chris Heunen,Jamie Vicary Pdf

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.

Reversible Computation

Michael Kirkedal Thomsen,Mathias Soeken
Reversible Computation Book PDF

Author : Michael Kirkedal Thomsen,Mathias Soeken
Publisher : Springer
Page : 247 pages
File Size : 40,9 Mb
Release : 2019-06-17
Category : Computers
ISBN : 9783030215002

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Reversible Computation by Michael Kirkedal Thomsen,Mathias Soeken Pdf

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.