A First Introduction to Categories

Author: F. William Lawvere,Stephen H. Schanuel

Publisher: Cambridge University Press

ISBN: 1139643967

Category: Mathematics

Page: N.A

View: 5785

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.
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A First Introduction to Categories

Author: F. William Lawvere,Stephen H. Schanuel

Publisher: Cambridge University Press

ISBN: 0521894859

Category: Mathematics

Page: 390

View: 2286

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.
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A First Introduction to Categories

Author: F. William Lawvere,Stephen Hoel Schanuel

Publisher: Cambridge University Press

ISBN: 9780521478175

Category: Mathematics

Page: 358

View: 332

This is an introduction to thinking about elementary mathematics from a categorial point of view. The goal is to explore the consequences of a new and fundamental insight about the nature of mathematics.
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Author: F. William Lawvere,Robert Rosebrugh

Publisher: Cambridge University Press

ISBN: 9780521010603

Category: Mathematics

Page: 261

View: 4448

In this book, first published in 2003, categorical algebra is used to build a foundation for the study of geometry, analysis, and algebra.
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Author: Harold Simmons

Publisher: Cambridge University Press

ISBN: 1139503324

Category: Mathematics

Page: N.A

View: 6744

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.
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Author: Steve Awodey

Publisher: OUP Oxford

ISBN: 0191612553

Category: Philosophy

Page: 328

View: 5270

Category theory is a branch of abstract algebra with incredibly diverse applications. This text and reference book is aimed not only at mathematicians, but also researchers and students of computer science, logic, linguistics, cognitive science, philosophy, and any of the other fields in which the ideas are being applied. Containing clear definitions of the essential concepts, illuminated with numerous accessible examples, and providing full proofs of all important propositions and theorems, this book aims to make the basic ideas, theorems, and methods of category theory understandable to this broad readership. Although assuming few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided - a must for computer scientists, logicians and linguists! This Second Edition contains numerous revisions to the original text, including expanding the exposition, revising and elaborating the proofs, providing additional diagrams, correcting typographical errors and, finally, adding an entirely new section on monoidal categories. Nearly a hundred new exercises have also been added, many with solutions, to make the book more useful as a course text and for self-study.
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Author: Tom Leinster

Publisher: Cambridge University Press

ISBN: 1107044243

Category: Mathematics

Page: 190

View: 3014

A short introduction ideal for students learning category theory for the first time.
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Author: Saunders MacLane

Publisher: Springer Science & Business Media

ISBN: 1461298393

Category: Mathematics

Page: 262

View: 6659

Category Theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by Mathe maticians working in a variety of other fields of Mathematical research. This occurs at several levels. On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an adjoint pair of functors. This appears in many substantially equivalent forms: That of universal construction, that of direct and inverse limit, and that of pairs offunctors with a natural isomorphism between corresponding sets of arrows. All these forms, with their interrelations, are examined in Chapters III to V. The slogan is "Adjoint functors arise everywhere". Alternatively, the fundamental notion of category theory is that of a monoid -a set with a binary operation of multiplication which is associative and which has a unit; a category itself can be regarded as a sort of general ized monoid. Chapters VI and VII explore this notion and its generaliza tions. Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure (given by a tensor product) lead inter alia to the study of more convenient categories of topological spaces.
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Author: Benjamin C. Pierce

Publisher: MIT Press

ISBN: 9780262660716

Category: Computers

Page: 100

View: 7842

Basic Category Theory for Computer Scientists provides a straightforward presentationof the basic constructions and terminology of category theory, including limits, functors, naturaltransformations, adjoints, and cartesian closed categories.
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Author: J. Lambek,P. J. Scott

Publisher: Cambridge University Press

ISBN: 9780521356534

Category: Mathematics

Page: 304

View: 1304

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.
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Author: R. F. C. Walters

Publisher: Cambridge University Press

ISBN: 9780521422260

Category: Computers

Page: 166

View: 9154

Category Theory has, in recent years, become increasingly important and popular in computer science, and many universities now introduce Category Theory as part of the curriculum for undergraduate computer science students. Here, the theory is developed in a straightforward way, and is enriched with many examples from computer science.
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Mathematics, Mathematics

Author: CTI Reviews

Publisher: Cram101 Textbook Reviews

ISBN: 1467243841

Category: Education

Page: 23

View: 4647

Facts101 is your complete guide to Conceptual Mathematics, A First Introduction to Categories. In this book, you will learn topics such as as those in your book plus much more. With key features such as key terms, people and places, Facts101 gives you all the information you need to prepare for your next exam. Our practice tests are specific to the textbook and we have designed tools to make the most of your limited study time.
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Author: Moshe Machover

Publisher: Cambridge University Press

ISBN: 9780521479981

Category: Mathematics

Page: 288

View: 3242

Rigorous coverage of logic and set theory for students of mathematics and philosophy.
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Author: David I. Spivak

Publisher: MIT Press

ISBN: 0262320533

Category: Mathematics

Page: 496

View: 657

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.
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Author: J. R. Hindley,J. P. Seldin

Publisher: CUP Archive

ISBN: 9780521268967

Category: Mathematics

Page: 360

View: 7523

Combinatory logic and lambda-conversion were originally devised in the 1920s for investigating the foundations of mathematics using the basic concept of 'operation' instead of 'set'. They have now developed into linguistic tools, useful in several branches of logic and computer science, especially in the study of programming languages. These notes form a simple introduction to the two topics, suitable for a reader who has no previous knowledge of combinatory logic, but has taken an undergraduate course in predicate calculus and recursive functions. The key ideas and basic results are presented, as well as a number of more specialised topics, and man), exercises are included to provide manipulative practice.
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A Textbook for Beginners

Author: Marco Grandis

Publisher: World Scientific

ISBN: 9813231084

Category:

Page: 304

View: 6319

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: Introduction Categories, Functors and Natural Transformations Limits and Colimits Adjunctions and Monads Applications in Algebra Applications in Topology and Algebraic Topology Applications in Homological Algebra Hints at Higher Dimensional Category Theory References Indices 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 Topology The theory is developed presenting other examples and some 300 exercises; the latter are endowed with a solution, or a partial solution, or adequate hints Three chapters and some extra sections are devoted to applications
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The Joy of Cats

Author: Jiri Adamek,Jiří Adámek (ing.),Horst Herrlich,George E. Strecker

Publisher: N.A

ISBN: 9780486469348

Category: Mathematics

Page: 517

View: 6084

This up-to-date introductory treatment employs the language of category theory to explore the theory of structures. Its unique approach stresses concrete categories, and each categorical notion features several examples that clearly illustrate specific and general cases. A systematic view of factorization structures, this volume contains seven chapters. The first five focus on basic theory, and the final two explore more recent research results in the realm of concrete categories, cartesian closed categories, and quasitopoi. Suitable for advanced undergraduate and graduate students, it requires an elementary knowledge of set theory and can be used as a reference as well as a text. Updated by the authors in 2004, it offers a unifying perspective on earlier work and summarizes recent developments.
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Chapter 0

Author: Paolo Aluffi

Publisher: American Mathematical Soc.

ISBN: 0821847813

Category: Algebra

Page: 713

View: 6274

Algebra: Chapter 0 is a self-contained introduction to the main topics of algebra, suitable for a first sequence on the subject at the beginning graduate or upper undergraduate level. The primary distinguishing feature of the book, compared to standard textbooks in algebra, is the early introduction of categories, used as a unifying theme in the presentation of the main topics. A second feature consists of an emphasis on homological algebra: basic notions on complexes are presented as soon as modules have been introduced, and an extensive last chapter on homological algebra can form the basis for a follow-up introductory course on the subject. Approximately 1,000 exercises both provide adequate practice to consolidate the understanding of the main body of the text and offer the opportunity to explore many other topics, including applications to number theory and algebraic geometry. This will allow instructors to adapt the textbook to their specific choice of topics and provide the independent reader with a richer exposure to algebra. Many exercises include substantial hints, and navigation of the topics is facilitated by an extensive index and by hundreds of cross-references.
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Author: Emily Riehl

Publisher: Courier Dover Publications

ISBN: 0486820807

Category: Mathematics

Page: 272

View: 1275

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.
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