Categories for the Working Mathematician

Author: Saunders MacLane
Publisher: Springer Science & Business Media
ISBN: 9781461298397
Release Date: 2013-11-11
Genre: Mathematics

Category Theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by Mathe maticians working in a variety of other fields of Mathematical research. This occurs at several levels. On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an adjoint pair of functors. This appears in many substantially equivalent forms: That of universal construction, that of direct and inverse limit, and that of pairs offunctors with a natural isomorphism between corresponding sets of arrows. All these forms, with their interrelations, are examined in Chapters III to V. The slogan is "Adjoint functors arise everywhere". Alternatively, the fundamental notion of category theory is that of a monoid -a set with a binary operation of multiplication which is associative and which has a unit; a category itself can be regarded as a sort of general ized monoid. Chapters VI and VII explore this notion and its generaliza tions. Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure (given by a tensor product) lead inter alia to the study of more convenient categories of topological spaces.

Categories for the Working Mathematician

Author: Saunders Mac Lane
Publisher: Springer Science & Business Media
ISBN: 9781475747218
Release Date: 2013-04-17
Genre: Mathematics

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.

Categories for the Working Mathematician

Author: Saunders Mac Lane
Publisher: Springer Science & Business Media
ISBN: 0387984038
Release Date: 1998-09-25
Genre: Mathematics

Categories for the Working Mathematician begins with foundations, illuminating concepts such as category, functor, natural transformation, and duality. It then continues by extensively illustrating these categorical concepts while presenting applications to more advanced topics. This second edition includes many revisions and additions.

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.

Category Theory

Author: Steve Awodey
Publisher: OUP Oxford
ISBN: 9780191612558
Release Date: 2010-06-17
Genre: Philosophy

Category theory is a branch of abstract algebra with incredibly diverse applications. This text and reference book is aimed not only at mathematicians, but also researchers and students of computer science, logic, linguistics, cognitive science, philosophy, and any of the other fields in which the ideas are being applied. Containing clear definitions of the essential concepts, illuminated with numerous accessible examples, and providing full proofs of all important propositions and theorems, this book aims to make the basic ideas, theorems, and methods of category theory understandable to this broad readership. Although assuming few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided - a must for computer scientists, logicians and linguists! This Second Edition contains numerous revisions to the original text, including expanding the exposition, revising and elaborating the proofs, providing additional diagrams, correcting typographical errors and, finally, adding an entirely new section on monoidal categories. Nearly a hundred new exercises have also been added, many with solutions, to make the book more useful as a course text and for self-study.

Topology and Geometry

Author: Glen E. Bredon
Publisher: Springer Science & Business Media
ISBN: 9781475768480
Release Date: 2013-03-09
Genre: Mathematics

This book offers an introductory course in algebraic topology. Starting with general topology, it discusses differentiable manifolds, cohomology, products and duality, the fundamental group, homology theory, and homotopy theory. From the reviews: "An interesting and original graduate text in topology and geometry...a good lecturer can use this text to create a fine course....A beginning graduate student can use this text to learn a great deal of mathematics."—-MATHEMATICAL REVIEWS

Conceptual Mathematics

Author: F. William Lawvere
Publisher: Cambridge University Press
ISBN: 9780521894852
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.

Noncommutative Algebra

Author: Benson Farb
Publisher: Springer Science & Business Media
ISBN: 9781461208891
Release Date: 2012-12-06
Genre: Mathematics

About This Book This book is meant to be used by beginning graduate students. It covers basic material needed by any student of algebra, and is essential to those specializing in ring theory, homological algebra, representation theory and K-theory, among others. It will also be of interest to students of algebraic topology, functional analysis, differential geometry and number theory. Our approach is more homological than ring-theoretic, as this leads the to many important areas of mathematics. This ap student more quickly proach is also, we believe, cleaner and easier to understand. However, the more classical, ring-theoretic approach, as well as modern extensions, are also presented via several exercises and sections in Chapter Five. We have tried not to leave any gaps on the paths to proving the main theorem- at most we ask the reader to fill in details for some of the sideline results; indeed this can be a fruitful way of solidifying one's understanding.

Algebra

Author: William A. Adkins
Publisher: Springer Science & Business Media
ISBN: 0387978399
Release Date: 1992
Genre: Mathematics

This book is designed as a text for a first-year graduate algebra course. The choice of topics is guided by the underlying theme of modules as a basic unifying concept in mathematics. The book emphasizes proofs with a maximum of insight and a minimum of computation in order to promote understanding. However, extensive material on computation (for example, computation of canonical forms) is provided.

Sheaf Theory

Author: Glen E. Bredon
Publisher: Springer Science & Business Media
ISBN: 0387949054
Release Date: 1997-01-24
Genre: Mathematics

This book is primarily concerned with the study of cohomology theories of general topological spaces with "general coefficient systems." The parts of sheaf theory covered here are those areas important to algebraic topology. There are several innovations in this book. The concept of the "tautness" of a subspace is introduced and exploited throughout the book. The fact that sheaf theoretic cohomology satisfies the homotopy property is proved for general topological spaces. Relative cohomology is introduced into sheaf theory. The reader should have a thorough background in elementary homological algebra and in algebraic topology. A list of exercises at the end of each chapter will help the student to learn the material, and solutions of many of the exercises are given in an appendix. The new edition of this classic in the field has been substantially rewritten with the addition of over 80 examples and of further explanatory material. Among the items added are new sections on Cech cohomology, the Oliver transfer, intersection theory, generalized manifolds, locally homogeneous spaces, homological fibrations and

Algebraic K Theory and Its Applications

Author: Jonathan Rosenberg
Publisher: Springer Science & Business Media
ISBN: 9781461243144
Release Date: 2012-12-06
Genre: Mathematics

Algebraic K-Theory is crucial in many areas of modern mathematics, especially algebraic topology, number theory, algebraic geometry, and operator theory. This text is designed to help graduate students in other areas learn the basics of K-Theory and get a feel for its many applications. Topics include algebraic topology, homological algebra, algebraic number theory, and an introduction to cyclic homology and its interrelationship with K-Theory.

Sets for Mathematics

Author: F. William Lawvere
Publisher: Cambridge University Press
ISBN: 0521010608
Release Date: 2003-01-27
Genre: Mathematics

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

Category Theory and Applications

Author: Marco Grandis
Publisher: World Scientific
ISBN: 9789813231085
Release Date: 2018-01-16
Genre:

Category Theory now permeates most of Mathematics, large parts of theoretical Computer Science and parts of theoretical Physics. Its unifying power brings together different branches, and leads to a deeper understanding of their roots. This book is addressed to students and researchers of these fields and can be used as a text for a first course in Category Theory. It covers its basic tools, like universal properties, limits, adjoint functors and monads. These are presented in a concrete way, starting from examples and exercises taken from elementary Algebra, Lattice Theory and Topology, then developing the theory together with new exercises and applications. Applications of Category Theory form a vast and differentiated domain. This book wants to present the basic applications and a choice of more advanced ones, based on the interests of the author. References are given for applications in many other fields. Contents: Introduction Categories, Functors and Natural Transformations Limits and Colimits Adjunctions and Monads Applications in Algebra Applications in Topology and Algebraic Topology Applications in Homological Algebra Hints at Higher Dimensional Category Theory References Indices Readership: Graduate students and researchers of mathematics, computer science, physics. Keywords: Category TheoryReview: Key Features: The main notions of Category Theory are presented in a concrete way, starting from examples taken from the elementary part of well-known disciplines: Algebra, Lattice Theory and Topology The theory is developed presenting other examples and some 300 exercises; the latter are endowed with a solution, or a partial solution, or adequate hints Three chapters and some extra sections are devoted to applications

An Introduction to Knot Theory

Author: W.B.Raymond Lickorish
Publisher: Springer Science & Business Media
ISBN: 9781461206910
Release Date: 2012-12-06
Genre: Mathematics

A selection of topics which graduate students have found to be a successful introduction to the field, employing three distinct techniques: geometric topology manoeuvres, combinatorics, and algebraic topology. Each topic is developed until significant results are achieved and each chapter ends with exercises and brief accounts of the latest research. What may reasonably be referred to as knot theory has expanded enormously over the last decade and, while the author describes important discoveries throughout the twentieth century, the latest discoveries such as quantum invariants of 3-manifolds as well as generalisations and applications of the Jones polynomial are also included, presented in an easily intelligible style. Readers are assumed to have knowledge of the basic ideas of the fundamental group and simple homology theory, although explanations throughout the text are numerous and well-done. Written by an internationally known expert in the field, this will appeal to graduate students, mathematicians and physicists with a mathematical background wishing to gain new insights in this area.