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Author : Tom Leinster
Publisher : Cambridge University Press
Page : 193 pages
File Size : 51,6 Mb
Release : 2014-07-24
Category : Mathematics
ISBN : 9781107044241
A short introduction ideal for students learning category theory for the first time.
Author : F.C. Brentano
Publisher : Springer Science & Business Media
Page : 276 pages
File Size : 51,6 Mb
Release : 2012-12-06
Category : Philosophy
ISBN : 9789400981898
This book contains the definitive statement of Franz Brentano's views on meta physics. It is made up of essays which were dictated by Brentano during the last ten years of his life, between 1907 and 1917. These dictations were assembled and edited by Alfred Kastil and first published by the Felix Meiner Verlag in 1933 under the title Kategorienlehre. Kastil added copious notes to Brentano's text. These notes have been included, with some slight omissions, in the present edition; the bibliographical references have been brought up to date. Brentano's approach to philosophy is unfamiliar to many contemporay readers. I shall discuss below certain fundamental points which such readers are likely to find the most difficult. I believe that once these points are properly understood, then what Brentano has to say will be seen to be of first importance to philosophy. THE PRIMACY OF THE INTENTIONAL To understand Brentano's theory of being, one must realize that he appeals to what he calls inner perception for his paradigmatic uses of the word "is". For inner perception, according to Brentano, is the source of our knowledge of the nature of being, just as it is the source of our knowledge of the nature of truth and of the nature of good and evil. And what can be said about the being of things that are not apprehended in inner perception can be understood only by analogy with what we are able to say about ourselves as thinking subjects.
Author : Emily Riehl
Publisher : Courier Dover Publications
Page : 272 pages
File Size : 44,6 Mb
Release : 2017-03-09
Category : Mathematics
ISBN : 9780486820804
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.
Author : David I. Spivak
Publisher : MIT Press
Page : 495 pages
File Size : 55,8 Mb
Release : 2014-10-17
Category : Mathematics
ISBN : 9780262320535
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.
Author : Saunders Mac Lane
Publisher : Springer Science & Business Media
Page : 320 pages
File Size : 52,8 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9781475747218
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.
Author : Steve Awodey
Publisher : Oxford University Press
Page : 328 pages
File Size : 41,9 Mb
Release : 2010-06-17
Category : Mathematics
ISBN : 9780199587360
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.
Author : Anonim
Publisher : Academic Press
Page : 272 pages
File Size : 43,7 Mb
Release : 1965-01-01
Category : Mathematics
ISBN : 0080873294
Theory of Categories
Author : Birgit Richter
Publisher : Cambridge University Press
Page : 401 pages
File Size : 46,9 Mb
Release : 2020-04-16
Category : Mathematics
ISBN : 9781108479622
Bridge the gap between category theory and its applications in homotopy theory with this guide for graduate students and researchers.
Author : Steven Roman
Publisher : Birkhäuser
Page : 169 pages
File Size : 50,9 Mb
Release : 2017-01-05
Category : Mathematics
ISBN : 9783319419176
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.
Author : Emily Riehl,Dominic Verity
Publisher : Cambridge University Press
Page : 781 pages
File Size : 44,5 Mb
Release : 2022-02-10
Category : Mathematics
ISBN : 9781108837989
This book develops the theory of infinite-dimensional categories by studying the universe, or ∞-cosmos, in which they live.
Author : Benjamin C. Pierce
Publisher : MIT Press
Page : 117 pages
File Size : 45,5 Mb
Release : 1991-08-07
Category : Computers
ISBN : 9780262326452
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
Author : Harold Simmons
Publisher : Cambridge University Press
Page : 128 pages
File Size : 42,8 Mb
Release : 2011-09-22
Category : Mathematics
ISBN : 9781139503327
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.
Author : Brendan Fong,David I. Spivak
Publisher : Cambridge University Press
Page : 351 pages
File Size : 46,8 Mb
Release : 2019-07-18
Category : Computers
ISBN : 9781108482295
Category theory reveals commonalities between structures of all sorts. This book shows its potential in science, engineering, and beyond.
Author : Paolo Aluffi
Publisher : American Mathematical Soc.
Page : 713 pages
File Size : 46,8 Mb
Release : 2021-11-09
Category : Education
ISBN : 9781470465711
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.
Author : F. William Lawvere,Robert Rosebrugh
Publisher : Cambridge University Press
Page : 280 pages
File Size : 45,7 Mb
Release : 2003-01-27
Category : Mathematics
ISBN : 0521010608
In this book, first published in 2003, categorical algebra is used to build a foundation for the study of geometry, analysis, and algebra.