An Introduction to the Language of Category Theory

Author: Steven Roman

Publisher: Birkhäuser

ISBN: 331941917X

Category: Mathematics

Page: 169

View: 7798

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.
Posted in Mathematics

An Introduction to Category Theory

Author: Harold Simmons

Publisher: Cambridge University Press

ISBN: 1139503324

Category: Mathematics

Page: N.A

View: 5135

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.
Posted in Mathematics

Basic Category Theory for Computer Scientists

Author: Benjamin C. Pierce,Benjamin C.. Pierce,Ierce Benjamin

Publisher: MIT Press

ISBN: 9780262660716

Category: Computers

Page: 100

View: 312

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
Posted in Computers

Introduction to Higher-Order Categorical Logic

Author: J. Lambek,P. J. Scott

Publisher: Cambridge University Press

ISBN: 9780521356534

Category: Mathematics

Page: 304

View: 6491

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.
Posted in Mathematics

Category Theory for the Sciences

Author: David I. Spivak

Publisher: MIT Press

ISBN: 0262028131

Category: Computers

Page: 486

View: 5091

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

Categories, Types and Data Structures

An Introduction to Category Theory for the Working Computer Scientist

Author: Andréa Asperti,G. Longo

Publisher: MIT Press (MA)

ISBN: 9780262011259

Category: Computers

Page: 306

View: 9103

Posted in Computers

Principia Mathematica.

Author: Alfred North Whitehead,Bertrand Russell

Publisher: N.A

ISBN: N.A

Category: Logic, Symbolic and mathematical

Page: 167

View: 4802

Posted in Logic, Symbolic and mathematical

Categories and Modules with K-Theory in View

Author: A. J. Berrick,M. E. Keating

Publisher: Cambridge University Press

ISBN: 9780521632768

Category: Mathematics

Page: 361

View: 5440

This book, first published in 2000, develops aspects of category theory fundamental to the study of algebraic K-theory. Ring and module theory illustrates category theory which provides insight into more advanced topics in module theory. Starting with categories in general, the text then examines categories of K-theory. This leads to the study of tensor products and the Morita theory. The categorical approach to localizations and completions of modules is formulated in terms of direct and inverse limits, prompting a discussion of localization of categories in general. Finally, local-global techniques which supply information about modules from their localizations and completions and underlie some interesting applications of K-theory to number theory and geometry are considered. Many useful exercises, concrete illustrations of abstract concepts placed in their historical settings and an extensive list of references are included. This book will help all who wish to work in K-theory to master its prerequisites.
Posted in Mathematics

Abstract and concrete categories

the joy of cats

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

Publisher: Wiley-Interscience

ISBN: N.A

Category: Mathematics

Page: 482

View: 406

A modern introduction to the theory of structures via the language of category theory. Unique to this book is the emphasis on concrete categories. Also noteworthy is the systematic treatment of factorization structures, which gives a new, unifying perspective to earlier work and summarizes recent developments. Each categorical notion is accompanied by many examples, usually moving from special cases to more general cases. Comprises seven chapters; the first five present the basic theory, while the last two contain more recent research results in the realm of concrete categories, cartesian closed categories and quasitopoi. The prerequisite is an elementary knowledge of set theory. Contains exercises.
Posted in Mathematics

Categories and Sheaves

Author: Masaki Kashiwara,Pierre Schapira

Publisher: Springer Science & Business Media

ISBN: 3540279490

Category: Mathematics

Page: 498

View: 3925

Categories and sheaves appear almost frequently in contemporary advanced mathematics. This book covers categories, homological algebra and sheaves in a systematic manner starting from scratch and continuing with full proofs to the most recent results in the literature, and sometimes beyond. The authors present the general theory of categories and functors, emphasizing inductive and projective limits, tensor categories, representable functors, ind-objects and localization.
Posted in Mathematics

An Introduction to Homological Algebra

Author: Charles A. Weibel

Publisher: Cambridge University Press

ISBN: 113964307X

Category: Mathematics

Page: N.A

View: 2694

The landscape of homological algebra has evolved over the last half-century into a fundamental tool for the working mathematician. This book provides a unified account of homological algebra as it exists today. The historical connection with topology, regular local rings, and semi-simple Lie algebras are also described. This book is suitable for second or third year graduate students. The first half of the book takes as its subject the canonical topics in homological algebra: derived functors, Tor and Ext, projective dimensions and spectral sequences. Homology of group and Lie algebras illustrate these topics. Intermingled are less canonical topics, such as the derived inverse limit functor lim1, local cohomology, Galois cohomology, and affine Lie algebras. The last part of the book covers less traditional topics that are a vital part of the modern homological toolkit: simplicial methods, Hochschild and cyclic homology, derived categories and total derived functors. By making these tools more accessible, the book helps to break down the technological barrier between experts and casual users of homological algebra.
Posted in Mathematics

Einführung in die Kategorientheorie

Mit ausführlichen Erklärungen und zahlreichen Beispielen

Author: Martin Brandenburg

Publisher: Springer-Verlag

ISBN: 3662535211

Category: Mathematics

Page: 343

View: 5281

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.
Posted in Mathematics

Introduction to Coalgebra

Author: N.A

Publisher: N.A

ISBN: 1107177898

Category:

Page: N.A

View: 3989

Posted in

An Invitation to Noncommutative Geometry

Author: Matilde Marcolli

Publisher: World Scientific

ISBN: 9812814337

Category: Mathematics

Page: 506

View: 7397

This is the first existing volume that collects lectures on this important and fast developing subject in mathematics. The lectures are given by leading experts in the field and the range of topics is kept as broad as possible by including both the algebraic and the differential aspects of noncommutative geometry as well as recent applications to theoretical physics and number theory. Sample Chapter(s). A Walk in the Noncommutative Garden (1,639 KB). Contents: A Walk in the Noncommutative Garden (A Connes & M Marcolli); Renormalization of Noncommutative Quantum Field Theory (H Grosse & R Wulkenhaar); Lectures on Noncommutative Geometry (M Khalkhali); Noncommutative Bundles and Instantons in Tehran (G Landi & W D van Suijlekom); Lecture Notes on Noncommutative Algebraic Geometry and Noncommutative Tori (S Mahanta); Lectures on Derived and Triangulated Categories (B Noohi); Examples of Noncommutative Manifolds: Complex Tori and Spherical Manifolds (J Plazas); D-Branes in Noncommutative Field Theory (R J Szabo). Readership: Researchers in mathematical and theoretical physics, geometry and topology, algebra and number theory.
Posted in Mathematics

Semantics of Programming Languages

Structures and Techniques

Author: Carl A. Gunter

Publisher: MIT Press

ISBN: 9780262570954

Category: Computers

Page: 441

View: 4342

Semantics of Programming Languages exposes the basic motivations and philosophy underlying the applications of semantic techniques in computer science. It introduces the mathematical theory of programming languages with an emphasis on higher-order functions and type systems. Designed as a text for upper-level and graduate-level students, the mathematically sophisticated approach will also prove useful to professionals who want an easily referenced description of fundamental results and calculi.Basic connections between computational behavior, denotational semantics, and the equational logic of functional programs are thoroughly and rigorously developed. Topics covered include models of types, operational semantics, category theory, domain theory, fixed point (denotational). semantics, full abstraction and other semantic correspondence criteria, types and evaluation, type checking and inference, parametric polymorphism, and subtyping. All topics are treated clearly and in depth, with complete proofs for the major results and numerous exercises.
Posted in Computers

Intelligent Agents

Theory and Applications

Author: Germano Resconi,L. C. Jain

Publisher: Springer Science & Business Media

ISBN: 9783540220039

Category: Computers

Page: 402

View: 8096

This monograph presents an integrated approach to agent theory. It introduces new concepts in the area of intelligent and adaptive agents in various contexts. Agents of different orders of complexity are presented including different levels of intelligence, uncertainty, and adaptation. The book introduces the abstract theory of adaptive agents as well as their applications to brain functions, robotics, or physical domains such as sound, gravity or electromagnetic fields.
Posted in Computers

On the Theory and Therapy of Mental Disorders

An Introduction to Logotherapy and Existential Analysis

Author: Viktor Frankl

Publisher: Routledge

ISBN: 1135930325

Category: Psychology

Page: 304

View: 6439

Logotherapy and Existential Analysis has been internationally recognized for decades as an empirically supported humanistic school of psychotherapy. Evidence for the growing significance of logotherapy includes institutes, societies and professorships in many countries of the world, as well as conferences and publications. On the Theory and Therapy of Neuroses: An Introduction to Logotherapy and Existential Analysis, the translation of Viktor Frankl's Theorie und Therapie der Neurosen by James M. DuBois, will allow for the first time English-only readers to experience this essential text on logotherapy.
Posted in Psychology

Toposes and Local Set Theories

An Introduction

Author: John L. Bell

Publisher: Courier Corporation

ISBN: 0486462862

Category: Mathematics

Page: 267

View: 1753

This text introduces topos theory, a development in category theory that unites important but seemingly diverse notions from algebraic geometry, set theory, and intuitionistic logic. Topics include local set theories, fundamental properties of toposes, sheaves, local-valued sets, and natural and real numbers in local set theories. 1988 edition.
Posted in Mathematics

Elementary Categories, Elementary Toposes

Author: Colin McLarty

Publisher: Clarendon Press

ISBN: 9780191589492

Category:

Page: 278

View: 7863

The book covers elementary aspects of category theory and topos theory. It has few mathematical prerequisites, and uses categorical methods throughout rather than beginning with set theoretic foundations. It works with key notions such as cartesian closedness, adjunctions, regular categories, and the internal logic of a topos. Full statements and elementary proofs are given for the central theorems, including the fundamental theorem of toposes, the sheafification theorem, and the construction of Grothendieck toposes over any topos as base. Three chapters discuss applications of toposes in detail, namely to sets, to basic differential geometry, and to recursive analysis. - ;Introduction; PART I: CATEGORIES: Rudimentary structures in a category; Products, equalizers, and their duals; Groups; Sub-objects, pullbacks, and limits; Relations; Cartesian closed categories; Product operators and others; PART II: THE CATEGORY OF CATEGORIES: Functors and categories; Natural transformations; Adjunctions; Slice categories; Mathematical foundations; PART III: TOPOSES: Basics; The internal language; A soundness proof for topos logic; From the internal language to the topos; The fundamental theorem; External semantics; Natural number objects; Categories in a topos; Topologies; PART IV: SOME TOPOSES: Sets; Synthetic differential geometry; The effective topos; Relations in regular categories; Further reading; Bibliography; Index. -
Posted in