# an-introduction-to-category-theory

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## An Introduction To Category Theory

**Author :**Harold Simmons

**ISBN :**9781139503327

**Genre :**Mathematics

**File Size :**53. 12 MB

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Category theory provides a general conceptual framework that has proved fruitful in subjects as diverse as geometry, topology, theoretical computer science and foundational mathematics. Here is a friendly, easy-to-read textbook that explains the fundamentals at a level suitable for newcomers to the subject. Beginning postgraduate mathematicians will find this book an excellent introduction to all of the basics of category theory. It gives the basic definitions; goes through the various associated gadgetry, such as functors, natural transformations, limits and colimits; and then explains adjunctions. The material is slowly developed using many examples and illustrations to illuminate the concepts explained. Over 200 exercises, with solutions available online, help the reader to access the subject and make the book ideal for self-study. It can also be used as a recommended text for a taught introductory course.

## An Introduction To The Language Of Category Theory

**Author :**Steven Roman

**ISBN :**9783319419176

**Genre :**Mathematics

**File Size :**36. 31 MB

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This textbook provides an introduction to elementary category theory, with the aim of making what can be a confusing and sometimes overwhelming subject more accessible. In writing about this challenging subject, the author has brought to bear all of the experience he has gained in authoring over 30 books in university-level mathematics. The goal of this book is to present the five major ideas of category theory: categories, functors, natural transformations, universality, and adjoints in as friendly and relaxed a manner as possible while at the same time not sacrificing rigor. These topics are developed in a straightforward, step-by-step manner and are accompanied by numerous examples and exercises, most of which are drawn from abstract algebra. The first chapter of the book introduces the definitions of category and functor and discusses diagrams,duality, initial and terminal objects, special types of morphisms, and some special types of categories,particularly comma categories and hom-set categories. Chapter 2 is devoted to functors and naturaltransformations, concluding with Yoneda's lemma. Chapter 3 presents the concept of universality and Chapter 4 continues this discussion by exploring cones, limits, and the most common categorical constructions – products, equalizers, pullbacks and exponentials (along with their dual constructions). The chapter concludes with a theorem on the existence of limits. Finally, Chapter 5 covers adjoints and adjunctions. Graduate and advanced undergraduates students in mathematics, computer science, physics, or related fields who need to know or use category theory in their work will find An Introduction to Category Theory to be a concise and accessible resource. It will be particularly useful for those looking for a more elementary treatment of the topic before tackling more advanced texts.

## Category Theory For The Sciences

**Author :**David I. Spivak

**ISBN :**9780262320535

**Genre :**Mathematics

**File Size :**27. 62 MB

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An introduction to category theory as a rigorous, flexible, and coherent modeling language that can be used across the sciences. Category theory was invented in the 1940s to unify and synthesize different areas in mathematics, and it has proven remarkably successful in enabling powerful communication between disparate fields and subfields within mathematics. This book shows that category theory can be useful outside of mathematics as a rigorous, flexible, and coherent modeling language throughout the sciences. Information is inherently dynamic; the same ideas can be organized and reorganized in countless ways, and the ability to translate between such organizational structures is becoming increasingly important in the sciences. Category theory offers a unifying framework for information modeling that can facilitate the translation of knowledge between disciplines. Written in an engaging and straightforward style, and assuming little background in mathematics, the book is rigorous but accessible to non-mathematicians. Using databases as an entry to category theory, it begins with sets and functions, then introduces the reader to notions that are fundamental in mathematics: monoids, groups, orders, and graphs—categories in disguise. After explaining the “big three” concepts of category theory—categories, functors, and natural transformations—the book covers other topics, including limits, colimits, functor categories, sheaves, monads, and operads. The book explains category theory by examples and exercises rather than focusing on theorems and proofs. It includes more than 300 exercises, with solutions. Category Theory for the Sciences is intended to create a bridge between the vast array of mathematical concepts used by mathematicians and the models and frameworks of such scientific disciplines as computation, neuroscience, and physics.

## Basic Category Theory

**Author :**Tom Leinster

**ISBN :**9781107044241

**Genre :**Mathematics

**File Size :**45. 91 MB

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A short introduction ideal for students learning category theory for the first time.

## Basic Category Theory For Computer Scientists

**Author :**Benjamin C. Pierce

**ISBN :**0262660717

**Genre :**Computers

**File Size :**57. 89 MB

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Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Category theory is a branch of pure mathematics that is becoming an increasingly important tool in theoretical computer science, especially in programming language semantics, domain theory, and concurrency, where it is already a standard language of discourse. Assuming a minimum of mathematical preparation, Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Four case studies illustrate applications of category theory to programming language design, semantics, and the solution of recursive domain equations. A brief literature survey offers suggestions for further study in more advanced texts. Contents Tutorial * Applications * Further Reading

## Categories Types And Structures

**Author :**Andrea Asperti

**ISBN :**UOM:39015022019742

**Genre :**Computers

**File Size :**74. 65 MB

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Category theory is a mathematical subject whose importance in several areas of computer science, most notably the semantics of programming languages and the design of programmes using abstract data types, is widely acknowledged. This book introduces category theory at a level appropriate for computer scientists and provides practical examples in the context of programming language design.

## Category Theory In Context

**Author :**Emily Riehl

**ISBN :**9780486820804

**Genre :**Mathematics

**File Size :**57. 9 MB

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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.

## An Introduction To Category Theory

**Author :**Viakalathur Sankrithi Krishnan

**ISBN :**UCAL:B4497483

**Genre :**Categories (Mathematics).

**File Size :**34. 59 MB

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## Category Theory

**Author :**Horst Herrlich

**ISBN :**STANFORD:36105031398253

**Genre :**Categories (Mathematics).

**File Size :**82. 32 MB

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## Introduction To Higher Order Categorical Logic

**Author :**J. Lambek

**ISBN :**0521356539

**Genre :**Mathematics

**File Size :**37. 35 MB

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Part I indicates that typed-calculi are a formulation of higher-order logic, and cartesian closed categories are essentially the same. Part II demonstrates that another formulation of higher-order logic is closely related to topos theory.

## Categories For Types

**Author :**Roy L. Crole

**ISBN :**0521457017

**Genre :**Computers

**File Size :**46. 98 MB

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This textbook explains the basic principles of categorical type theory and the techniques used to derive categorical semantics for specific type theories. It introduces the reader to ordered set theory, lattices and domains, and this material provides plenty of examples for an introduction to category theory, which covers categories, functors, natural transformations, the Yoneda lemma, cartesian closed categories, limits, adjunctions and indexed categories. Four kinds of formal system are considered in detail, namely algebraic, functional, polymorphic functional, and higher order polymorphic functional type theory. For each of these the categorical semantics are derived and results about the type systems are proved categorically. Issues of soundness and completeness are also considered. Aimed at advanced undergraduates and beginning graduates, this book will be of interest to theoretical computer scientists, logicians and mathematicians specializing in category theory.

## Higher Category Theory

**Author :**Ezra Getzler

**ISBN :**9780821810569

**Genre :**Mathematics

**File Size :**51. 74 MB

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This volume presents the proceedings of the workshop on higher category theory and mathematical physics held at Northwestern University. Exciting new developments were presented with the aim of making them better known outside the community of experts. In particular, presentations in the style, 'Higher Categories for the Working Mathematician', were encouraged. The volume is the first to bring together developments in higher category theory with applications. This collection is a valuable introduction to this topic - one that holds great promise for future developments in mathematics.

## Category Theory And Applications

**Author :**Marco Grandis

**ISBN :**9789813231085

**Genre :**

**File Size :**64. 73 MB

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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 deeper 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 its 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. Applications of Category Theory form a vast and differentiated domain. This book wants to present the basic applications and a choice of more advanced ones, based on the interests of the author. References are given for applications in many other fields. Contents: IntroductionCategories, Functors and Natural TransformationsLimits and ColimitsAdjunctions and MonadsApplications in AlgebraApplications in Topology and Algebraic TopologyApplications in Homological AlgebraHints at Higher Dimensional Category TheoryReferencesIndices Readership: Graduate students and researchers of mathematics, computer science, physics. Keywords: Category TheoryReview: Key Features: The main notions of Category Theory are presented in a concrete way, starting from examples taken from the elementary part of well-known disciplines: Algebra, Lattice Theory and TopologyThe theory is developed presenting other examples and some 300 exercises; the latter are endowed with a solution, or a partial solution, or adequate hintsThree chapters and some extra sections are devoted to applications

## Categories For The Working Mathematician

**Author :**Saunders Mac Lane

**ISBN :**0387984038

**Genre :**Mathematics

**File Size :**86. 14 MB

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Categories for the Working Mathematician begins with foundations, illuminating concepts such as category, functor, natural transformation, and duality. It then continues by extensively illustrating these categorical concepts while presenting applications to more advanced topics. This second edition includes many revisions and additions.

## Categories For Quantum Theory

**Author :**Chris Heunen

**ISBN :**9780198739623

**Genre :**Mathematics

**File Size :**39. 12 MB

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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.

## A Taste Of Category Theory For Computer Scientists

**Author :**Benjamin C. Pierce

**ISBN :**OCLC:21356659

**Genre :**Categories (Mathematics)

**File Size :**26. 26 MB

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Abstract: "Category theory is a branch of pure mathematics that more and more frequently touches the daily work of computer scientists, especially those with an interest in programming languages or formal specifications. This survey is an 'introduction to the introductions' to category theory--a brief answer to the questions, 'What is category theory?' 'What are its basic concepts?' 'What are computer scientists using it for?' and 'Where can I learn more?' The first section introduces the most common category-theoretic terms and idioms, assuming as little specific mathematical background as possible. the second section presents four case studies from the recent research literature applying category theory to the semantics of computation. A reading list in the third section suggests pathways into the existing literature, including textbooks, standard reference works, and selected research papers."

## Categories And Sheaves

**Author :**Masaki Kashiwara

**ISBN :**9783540279495

**Genre :**Mathematics

**File Size :**46. 21 MB

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Categories and sheaves appear almost frequently in contemporary advanced mathematics. This book covers categories, homological algebra and sheaves in a systematic manner starting from scratch and continuing with full proofs to the most recent results in the literature, and sometimes beyond. The authors present the general theory of categories and functors, emphasizing inductive and projective limits, tensor categories, representable functors, ind-objects and localization.

## Conceptual Mathematics

**Author :**F. William Lawvere

**ISBN :**9780521894852

**Genre :**Mathematics

**File Size :**72. 41 MB

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In the last 60 years, the use of the notion of category has led to a remarkable unification and simplification of mathematics. Conceptual Mathematics introduces this tool for the learning, development, and use of mathematics, to beginning students and also to practising mathematical scientists. This book provides a skeleton key that makes explicit some concepts and procedures that are common to all branches of pure and applied mathematics. The treatment does not presuppose knowledge of specific fields, but rather develops, from basic definitions, such elementary categories as discrete dynamical systems and directed graphs; the fundamental ideas are then illuminated by examples in these categories. This second edition provides links with more advanced topics of possible study. In the new appendices and annotated bibliography the reader will find concise introductions to adjoint functors and geometrical structures, as well as sketches of relevant historical developments.

## Category Theory Applied To Computation And Control

**Author :**Ernest G. Manes

**ISBN :**UCAL:B4406139

**Genre :**Automates - Congrès

**File Size :**22. 12 MB

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Presents the results of a symposium which brought together scientists interested in applying modern algebraic techniques to problems in control & computer science.

## Coherence In Three Dimensional Category Theory

**Author :**Nick Gurski

**ISBN :**9781107328792

**Genre :**Mathematics

**File Size :**31. 39 MB

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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.