An Introduction To Category Theory

Author : Harold Simmons
Publisher : Cambridge University Press
Page : 128 pages
File Size : 49,5 Mb
Release : 2011-09-22
Category : Mathematics
ISBN : 9781139503327

Get Book

An Introduction to Category Theory by Harold Simmons Pdf

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 : Tom Leinster
Publisher : Cambridge University Press
Page : 193 pages
File Size : 49,6 Mb
Release : 2014-07-24
Category : Mathematics
ISBN : 9781107044241

Get Book

Basic Category Theory by Tom Leinster Pdf

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

Author : Steven Roman
Publisher : Birkhäuser
Page : 169 pages
File Size : 44,9 Mb
Release : 2017-01-05
Category : Mathematics
ISBN : 9783319419176

Get Book

An Introduction to the Language of Category Theory by Steven Roman Pdf

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
Publisher : Courier Dover Publications
Page : 272 pages
File Size : 41,5 Mb
Release : 2017-03-09
Category : Mathematics
ISBN : 9780486820804

Get Book

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.

Author : David I. Spivak
Publisher : MIT Press
Page : 495 pages
File Size : 46,6 Mb
Release : 2014-10-17
Category : Mathematics
ISBN : 9780262320535

Get Book

Category Theory for the Sciences by David I. Spivak Pdf

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 : F. William Lawvere,Robert Rosebrugh
Publisher : Cambridge University Press
Page : 280 pages
File Size : 52,7 Mb
Release : 2003-01-27
Category : Mathematics
ISBN : 0521010608

Get Book

Sets for Mathematics by F. William Lawvere,Robert Rosebrugh Pdf

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

Author : Benjamin C. Pierce
Publisher : MIT Press
Page : 117 pages
File Size : 49,8 Mb
Release : 1991-08-07
Category : Computers
ISBN : 9780262326452

Get Book

Basic Category Theory for Computer Scientists by Benjamin C. Pierce Pdf

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 : Saunders Mac Lane
Publisher : Springer Science & Business Media
Page : 320 pages
File Size : 51,8 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9781475747218

Get Book

Categories for the Working Mathematician by Saunders Mac Lane Pdf

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 : Andrea Asperti,Giuseppe Longo
Publisher : MIT Press (MA)
Page : 330 pages
File Size : 42,5 Mb
Release : 1991
Category : Computers
ISBN : UOM:39015022019742

Get Book

Categories, Types, and Structures by Andrea Asperti,Giuseppe Longo Pdf

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.

Author : Brendan Fong,David I. Spivak
Publisher : Cambridge University Press
Page : 351 pages
File Size : 49,8 Mb
Release : 2019-07-18
Category : Computers
ISBN : 9781108482295

Get Book

An Invitation to Applied Category Theory by Brendan Fong,David I. Spivak Pdf

Category theory reveals commonalities between structures of all sorts. This book shows its potential in science, engineering, and beyond.

Author : Gregory Maxwell Kelly
Publisher : CUP Archive
Page : 260 pages
File Size : 40,9 Mb
Release : 1982-02-18
Category : Mathematics
ISBN : 0521287022

Get Book

Basic Concepts of Enriched Category Theory by Gregory Maxwell Kelly Pdf

Author : Niles Johnson,Donald Yau
Publisher : Oxford University Press, USA
Page : 636 pages
File Size : 44,5 Mb
Release : 2021-01-31
Category : Mathematics
ISBN : 9780198871378

Get Book

2-Dimensional Categories by Niles Johnson,Donald Yau Pdf

2-Dimensional Categories is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory.

Author : Markus Land
Publisher : Springer Nature
Page : 300 pages
File Size : 43,8 Mb
Release : 2021-04-21
Category : Mathematics
ISBN : 9783030615246

Get Book

Introduction to Infinity-Categories by Markus Land Pdf

This textbook is an introduction to the theory of infinity-categories, a tool used in many aspects of modern pure mathematics. It treats the basics of the theory and supplies all the necessary details while leading the reader along a streamlined path from the basic definitions to more advanced results such as the very important adjoint functor theorems. The book is based on lectures given by the author on the topic. While the material itself is well-known to experts, the presentation of the material is, in parts, novel and accessible to non-experts. Exercises complement this textbook that can be used both in a classroom setting at the graduate level and as an introductory text for the interested reader.

Author : Emily Riehl,Dominic Verity
Publisher : Cambridge University Press
Page : 781 pages
File Size : 44,7 Mb
Release : 2022-02-10
Category : Mathematics
ISBN : 9781108837989

Get Book

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.

Author : F. William Lawvere,Stephen H. Schanuel
Publisher : Cambridge University Press
Page : 409 pages
File Size : 49,6 Mb
Release : 2009-07-30
Category : Mathematics
ISBN : 9780521894852

Get Book

Conceptual Mathematics by F. William Lawvere,Stephen H. Schanuel Pdf

This truly elementary book on categories introduces retracts, graphs, and adjoints to students and scientists.