Elements Of Category Theory

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Elements of ∞-Category Theory

Author : Emily Riehl
Publisher : Cambridge University Press
ISBN 13 : 1108952194
Total Pages : 782 pages
Book Rating : 4.1/5 (89 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 782 pages. Available in PDF, EPUB and Kindle. Book excerpt: The language of ∞-categories provides an insightful new way of expressing many results in higher-dimensional mathematics but can be challenging for the uninitiated. To explain what exactly an ∞-category is requires various technical models, raising the question of how they might be compared. To overcome this, a model-independent approach is desired, so that theorems proven with any model would apply to them all. This text develops the theory of ∞-categories from first principles in a model-independent fashion using the axiomatic framework of an ∞-cosmos, the universe in which ∞-categories live as objects. An ∞-cosmos is a fertile setting for the formal category theory of ∞-categories, and in this way the foundational proofs in ∞-category theory closely resemble the classical foundations of ordinary category theory. Equipped with exercises and appendices with background material, this first introduction is meant for students and researchers who have a strong foundation in classical 1-category theory.

Category Theory in Context

Author : Emily Riehl
Publisher : Courier Dover Publications
ISBN 13 : 0486820807
Total Pages : 272 pages
Book Rating : 4.4/5 (868 download)

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Book Synopsis Category Theory in Context by : Emily Riehl

Download or read book Category Theory in Context written by Emily Riehl and published by Courier Dover Publications. This book was released on 2017-03-09 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Basic Category Theory

Author : Tom Leinster
Publisher : Cambridge University Press
ISBN 13 : 1107044243
Total Pages : 193 pages
Book Rating : 4.1/5 (7 download)

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Book Synopsis Basic Category Theory by : Tom Leinster

Download or read book Basic Category Theory written by Tom Leinster and published by Cambridge University Press. This book was released on 2014-07-24 with total page 193 pages. Available in PDF, EPUB and Kindle. Book excerpt: A short introduction ideal for students learning category theory for the first time.

Basic Concepts of Enriched Category Theory

Author : Gregory Maxwell Kelly
Publisher : CUP Archive
ISBN 13 : 9780521287029
Total Pages : 260 pages
Book Rating : 4.2/5 (87 download)

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Book Synopsis Basic Concepts of Enriched Category Theory by : Gregory Maxwell Kelly

Download or read book Basic Concepts of Enriched Category Theory written by Gregory Maxwell Kelly and published by CUP Archive. This book was released on 1982-02-18 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Basic Category Theory for Computer Scientists

Author : Benjamin C. Pierce
Publisher : MIT Press
ISBN 13 : 0262326450
Total Pages : 117 pages
Book Rating : 4.2/5 (623 download)

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Book Synopsis Basic Category Theory for Computer Scientists by : Benjamin C. Pierce

Download or read book Basic Category Theory for Computer Scientists written by Benjamin C. Pierce and published by MIT Press. This book was released on 1991-08-07 with total page 117 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Categorical Homotopy Theory

Author : Emily Riehl
Publisher : Cambridge University Press
ISBN 13 : 1139952633
Total Pages : 371 pages
Book Rating : 4.1/5 (399 download)

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Book Synopsis Categorical Homotopy Theory by : Emily Riehl

Download or read book Categorical Homotopy Theory written by Emily Riehl and published by Cambridge University Press. This book was released on 2014-05-26 with total page 371 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book develops abstract homotopy theory from the categorical perspective with a particular focus on examples. Part I discusses two competing perspectives by which one typically first encounters homotopy (co)limits: either as derived functors definable when the appropriate diagram categories admit a compatible model structure, or through particular formulae that give the right notion in certain examples. Emily Riehl unifies these seemingly rival perspectives and demonstrates that model structures on diagram categories are irrelevant. Homotopy (co)limits are explained to be a special case of weighted (co)limits, a foundational topic in enriched category theory. In Part II, Riehl further examines this topic, separating categorical arguments from homotopical ones. Part III treats the most ubiquitous axiomatic framework for homotopy theory - Quillen's model categories. Here, Riehl simplifies familiar model categorical lemmas and definitions by focusing on weak factorization systems. Part IV introduces quasi-categories and homotopy coherence.

From Categories to Homotopy Theory

Author : Birgit Richter
Publisher : Cambridge University Press
ISBN 13 : 1108847625
Total Pages : 402 pages
Book Rating : 4.1/5 (88 download)

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Book Synopsis From Categories to Homotopy Theory by : Birgit Richter

Download or read book From Categories to Homotopy Theory written by Birgit Richter and published by Cambridge University Press. This book was released on 2020-04-16 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: Category theory provides structure for the mathematical world and is seen everywhere in modern mathematics. With this book, the author bridges the gap between pure category theory and its numerous applications in homotopy theory, providing the necessary background information to make the subject accessible to graduate students or researchers with a background in algebraic topology and algebra. The reader is first introduced to category theory, starting with basic definitions and concepts before progressing to more advanced themes. Concrete examples and exercises illustrate the topics, ranging from colimits to constructions such as the Day convolution product. Part II covers important applications of category theory, giving a thorough introduction to simplicial objects including an account of quasi-categories and Segal sets. Diagram categories play a central role throughout the book, giving rise to models of iterated loop spaces, and feature prominently in functor homology and homology of small categories.

Categories for the Working Mathematician

Author : Saunders Mac Lane
Publisher : Springer Science & Business Media
ISBN 13 : 1475747217
Total Pages : 320 pages
Book Rating : 4.4/5 (757 download)

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Book Synopsis Categories for the Working Mathematician by : Saunders Mac Lane

Download or read book Categories for the Working Mathematician written by Saunders Mac Lane and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: An array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. It then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterised by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions. This second edition includes a number of revisions and additions, including new chapters on topics of active interest: symmetric monoidal categories and braided monoidal categories, and the coherence theorems for them, as well as 2-categories and the higher dimensional categories which have recently come into prominence.

Category Theory for the Sciences

Author : David I. Spivak
Publisher : MIT Press
ISBN 13 : 0262320533
Total Pages : 495 pages
Book Rating : 4.2/5 (623 download)

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Book Synopsis Category Theory for the Sciences by : David I. Spivak

Download or read book Category Theory for the Sciences written by David I. Spivak and published by MIT Press. This book was released on 2014-10-17 with total page 495 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Category Theory

Author : Steve Awodey
Publisher : Oxford University Press
ISBN 13 : 0199587361
Total Pages : 328 pages
Book Rating : 4.1/5 (995 download)

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Book Synopsis Category Theory by : Steve Awodey

Download or read book Category Theory written by Steve Awodey and published by Oxford University Press. This book was released on 2010-06-17 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: A comprehensive reference to category theory for students and researchers in mathematics, computer science, logic, cognitive science, linguistics, and philosophy. Useful for self-study and as a course text, the book includes all basic definitions and theorems (with full proofs), as well as numerous examples and exercises.

Theoretical Computer Science for the Working Category Theorist

Author : Noson S. Yanofsky
Publisher : Cambridge University Press
ISBN 13 : 1108890679
Total Pages : 148 pages
Book Rating : 4.1/5 (88 download)

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Book Synopsis Theoretical Computer Science for the Working Category Theorist by : Noson S. Yanofsky

Download or read book Theoretical Computer Science for the Working Category Theorist written by Noson S. Yanofsky and published by Cambridge University Press. This book was released on 2022-03-03 with total page 148 pages. Available in PDF, EPUB and Kindle. Book excerpt: Using basic category theory, this Element describes all the central concepts and proves the main theorems of theoretical computer science. Category theory, which works with functions, processes, and structures, is uniquely qualified to present the fundamental results of theoretical computer science. In this Element, readers will meet some of the deepest ideas and theorems of modern computers and mathematics, such as Turing machines, unsolvable problems, the P=NP question, Kurt Gödel's incompleteness theorem, intractable problems, cryptographic protocols, Alan Turing's Halting problem, and much more. The concepts come alive with many examples and exercises.

Category Theory for Computing Science

Author : Michael Barr
Publisher :
ISBN 13 :
Total Pages : 352 pages
Book Rating : 4.3/5 (91 download)

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Book Synopsis Category Theory for Computing Science by : Michael Barr

Download or read book Category Theory for Computing Science written by Michael Barr and published by . This book was released on 1995 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: A wide coverage of topics in category theory and computer science is developed in this text, including introductory treatments of cartesian closed categories, sketches and elementary categorical model theory, and triples. Over 300 exercises are included.

An Introduction to the Language of Category Theory

Author : Steven Roman
Publisher : Birkhäuser
ISBN 13 : 331941917X
Total Pages : 169 pages
Book Rating : 4.3/5 (194 download)

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Book Synopsis An Introduction to the Language of Category Theory by : Steven Roman

Download or read book An Introduction to the Language of Category Theory written by Steven Roman and published by Birkhäuser. This book was released on 2017-01-05 with total page 169 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 Seminar

Author : G.M. Kelly
Publisher : Springer
ISBN 13 : 3540372709
Total Pages : 386 pages
Book Rating : 4.5/5 (43 download)

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Book Synopsis Category Seminar by : G.M. Kelly

Download or read book Category Seminar written by G.M. Kelly and published by Springer. This book was released on 2006-11-15 with total page 386 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Categories, Types, and Structures

Author : Andrea Asperti
Publisher : MIT Press (MA)
ISBN 13 :
Total Pages : 330 pages
Book Rating : 4.3/5 (91 download)

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Book Synopsis Categories, Types, and Structures by : Andrea Asperti

Download or read book Categories, Types, and Structures written by Andrea Asperti and published by MIT Press (MA). This book was released on 1991 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Algebra: Chapter 0

Author : Paolo Aluffi
Publisher : American Mathematical Soc.
ISBN 13 : 147046571X
Total Pages : 713 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Algebra: Chapter 0 by : Paolo Aluffi

Download or read book Algebra: Chapter 0 written by Paolo Aluffi and published by American Mathematical Soc.. This book was released on 2021-11-09 with total page 713 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Category Theory for Programmers (New Edition, Hardcover)

Author : Bartosz Milewski
Publisher :
ISBN 13 : 9780464243878
Total Pages : pages
Book Rating : 4.2/5 (438 download)

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Book Synopsis Category Theory for Programmers (New Edition, Hardcover) by : Bartosz Milewski

Download or read book Category Theory for Programmers (New Edition, Hardcover) written by Bartosz Milewski and published by . This book was released on 2019-08-24 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Category Theory is one of the most abstract branches of mathematics. It is usually taught to graduate students after they have mastered several other branches of mathematics, like algebra, topology, and group theory. It might, therefore, come as a shock that the basic concepts of category theory can be explained in relatively simple terms to anybody with some experience in programming.That's because, just like programming, category theory is about structure. Mathematicians discover structure in mathematical theories, programmers discover structure in computer programs. Well-structured programs are easier to understand and maintain and are less likely to contain bugs. Category theory provides the language to talk about structure and learning it will make you a better programmer.