An Introduction to Category Theory

Author: Harold Simmons
Publisher: Cambridge University Press
ISBN: 9781139503327
Release Date: 2011-09-22
Genre: Mathematics

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
Publisher: Birkhäuser
ISBN: 9783319419176
Release Date: 2017-01-05
Genre: Mathematics

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.

Basic Category Theory for Computer Scientists

Author: Benjamin C. Pierce
Publisher: MIT Press
ISBN: 0262660717
Release Date: 1991
Genre: Computers

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

Category Theory for the Sciences

Author: David I. Spivak
Publisher: MIT Press
ISBN: 9780262028134
Release Date: 2014-10-10
Genre: Computers

An introduction to category theory as a rigorous, flexible, and coherent modeling language that can be used across the sciences.

Basic Category Theory

Author: Tom Leinster
Publisher: Cambridge University Press
ISBN: 9781107044241
Release Date: 2014-07-24
Genre: Mathematics

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

Introduction to Higher Order Categorical Logic

Author: J. Lambek
Publisher: Cambridge University Press
ISBN: 0521356539
Release Date: 1988-03-25
Genre: Mathematics

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.

A Taste of Category Theory for Computer Scientists

Author: Benjamin C. Pierce
Publisher:
ISBN: OCLC:21356659
Release Date: 1990
Genre: Categories (Mathematics)

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."

Higher Category Theory

Author: Ezra Getzler
Publisher: American Mathematical Soc.
ISBN: 9780821810569
Release Date: 1998
Genre: Mathematics

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.

Unglaubliche Zahlen

Author: Ian Stewart
Publisher: Rowohlt Verlag GmbH
ISBN: 9783644564312
Release Date: 2016-06-24
Genre: Mathematics

In diesem Buch nimmt der britische Mathe-Guru seine Leser mit auf eine Reise durch das Reich der Zahlen – reelle, rationale, irrationale, komplexe; ganz, ganz kleine und unendlich große, Fraktale, Logarithmen, Hochzahlen, Primzahlen, Kusszahlen und viele mehr. Jedes Kapitel konzentriert sich auf eine Zahl oder Zahlengruppe und erläutert, warum sie so interessant ist. «Jede Zahl hat ihre eigene Geschichte zu erzählen», heißt es im Vorwort. Stewart erzählt sie mit Begeisterung und versteht es geschickt, diese Geschichten miteinander zu verweben, ob es um die Zahl Pi geht oder zum Schluss auch um Geheimcodes, den Rubikwürfel und Sudoku. Darüber hinaus erfährt man viel über die Geschichte der Mathematik und die Rolle, die sie für unsere Entwicklung spielt. Schließlich waren es die Zahlen, so der Autor, «die es der Menschheit ermöglicht haben, sich aus dem Schlamm zu ziehen und nach den Sternen zu greifen».

Einf hrung in die Kategorientheorie

Author: Martin Brandenburg
Publisher: Springer-Verlag
ISBN: 9783662535219
Release Date: 2016-12-15
Genre: Mathematics

Die Kategorientheorie deckt die innere Architektur der Mathematik auf. Dabei werden die strukturellen Gemeinsamkeiten zwischen mathematischen Disziplinen und ihren spezifischen Konstruktionen herausgearbeitet. Dieses Buch gibt eine systematische Einführung in die Grundbegriffe der Kategorientheorie. Zahlreiche ausführliche Erklärungstexte sowie die große Menge an Beispielen helfen beim Einstieg in diese verhältnismäßig abstrakte Theorie. Es werden viele konkrete Anwendungen besprochen, welche die Nützlichkeit der Kategorientheorie im mathematischen Alltag belegen. Jedes Kapitel wird mit einem motivierenden Text eingeleitet und mit einer großen Aufgabensammlung abgeschlossen. An Vorwissen muss der Leser lediglich ein paar Grundbegriffe des Mathematik-Studiums mitbringen. Die vorliegende zweite vollständig durchgesehene Auflage ist um ausführliche Lösungen zu ausgewählten Aufgaben ergänzt.

Grundkurs Topologie

Author: Gerd Laures
Publisher: Springer-Verlag
ISBN: 9783662459539
Release Date: 2015-04-07
Genre: Mathematics

Die Topologie beschäftigt sich mit den qualitativen Eigenschaften geometrischer Objekte. Ihr Begriffsapparat ist so mächtig, dass kaum eine mathematische Struktur nicht mit Gewinn topologisiert wurde. Dieses Buch versteht sich als Brücke von den einführenden Vorlesungen der Analysis und Linearen Algebra zu den fortgeschrittenen Vorlesungen der Algebraischen und Geometrischen Topologie. Es eignet sich besonders für Studierende in einem Bachelor- oder Masterstudiengang der Mathematik, kann aber auch zum Selbststudium für mathematisch interessierte Naturwissenschaftler dienen. Die Autoren legen besonderen Wert auf eine moderne Sprache, welche die vorgestellten Ideen vereinheitlicht und damit erleichtert. Definitionen werden stets mit vielen Beispielen unterlegt und neue Konzepte werden mit zahlreichen Bildern illustriert. Über 170 Übungsaufgaben (mit Lösungen zu ausgewählten Aufgaben auf der Website zum Buch) helfen, die vermittelten Inhalte einzuüben und zu vertiefen. Viele Abschnitte werden ergänzt durch kurze Einblicke in weiterführende Themen, die einen Ausgangspunkt für Studienarbeiten oder Seminarthemen bieten. Neben dem üblichen Stoff zur mengentheoretischen Topologie, der Theorie der Fundamentalgruppen und der Überlagerungen werden auch Bündel, Garben und simpliziale Methoden angesprochen, welche heute zu den Grundbegriffen der Geometrie und Topologie gehören.

Conceptual Mathematics

Author: F. William Lawvere
Publisher: Cambridge University Press
ISBN: 9781139643962
Release Date: 2009-07-30
Genre: Mathematics

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.