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Author : Krzysztof Ciesielski
Publisher : Cambridge University Press
Page : 256 pages
File Size : 55,7 Mb
Release : 1997-08-28
Category : Mathematics
ISBN : 0521594650
Presents those methods of modern set theory most applicable to other areas of pure mathematics.
Author : Saunders Mac Lane
Publisher : Springer Science & Business Media
Page : 320 pages
File Size : 42,9 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 : Anonim
Publisher : Univalent Foundations
Page : 484 pages
File Size : 40,9 Mb
Release : 2024-06-14
Category : Electronic
ISBN : 8210379456XXX
Author : Saunders MacLane
Publisher : Springer Science & Business Media
Page : 265 pages
File Size : 52,7 Mb
Release : 2013-11-11
Category : Mathematics
ISBN : 9781461298397
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.
Author : Emily Riehl
Publisher : Courier Dover Publications
Page : 272 pages
File Size : 53,9 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 : Paul J. Cohen
Publisher : Courier Corporation
Page : 196 pages
File Size : 40,7 Mb
Release : 2008-12-09
Category : Mathematics
ISBN : 9780486469218
This exploration of a notorious mathematical problem is the work of the man who discovered the solution. Written by an award-winning professor at Stanford University, it employs intuitive explanations as well as detailed mathematical proofs in a self-contained treatment. This unique text and reference is suitable for students and professionals. 1966 edition. Copyright renewed 1994.
Author : F. William Lawvere,Robert Rosebrugh
Publisher : Cambridge University Press
Page : 280 pages
File Size : 48,7 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.
Author : Tom Leinster
Publisher : Cambridge University Press
Page : 193 pages
File Size : 40,5 Mb
Release : 2014-07-24
Category : Mathematics
ISBN : 9781107044241
A short introduction ideal for students learning category theory for the first time.
Author : Abhijit Dasgupta
Publisher : Springer Science & Business Media
Page : 444 pages
File Size : 46,6 Mb
Release : 2013-12-11
Category : Mathematics
ISBN : 9781461488545
What is a number? What is infinity? What is continuity? What is order? Answers to these fundamental questions obtained by late nineteenth-century mathematicians such as Dedekind and Cantor gave birth to set theory. This textbook presents classical set theory in an intuitive but concrete manner. To allow flexibility of topic selection in courses, the book is organized into four relatively independent parts with distinct mathematical flavors. Part I begins with the Dedekind–Peano axioms and ends with the construction of the real numbers. The core Cantor–Dedekind theory of cardinals, orders, and ordinals appears in Part II. Part III focuses on the real continuum. Finally, foundational issues and formal axioms are introduced in Part IV. Each part ends with a postscript chapter discussing topics beyond the scope of the main text, ranging from philosophical remarks to glimpses into landmark results of modern set theory such as the resolution of Lusin's problems on projective sets using determinacy of infinite games and large cardinals. Separating the metamathematical issues into an optional fourth part at the end makes this textbook suitable for students interested in any field of mathematics, not just for those planning to specialize in logic or foundations. There is enough material in the text for a year-long course at the upper-undergraduate level. For shorter one-semester or one-quarter courses, a variety of arrangements of topics are possible. The book will be a useful resource for both experts working in a relevant or adjacent area and beginners wanting to learn set theory via self-study.
Author : Charles C Pinter
Publisher : Courier Corporation
Page : 259 pages
File Size : 42,7 Mb
Release : 2014-07-23
Category : Mathematics
ISBN : 9780486497082
"This accessible approach to set theory for upper-level undergraduates poses rigorous but simple arguments. Each definition is accompanied by commentary that motivates and explains new concepts. A historical introduction is followed by discussions of classes and sets, functions, natural and cardinal numbers, the arithmetic of ordinal numbers, and related topics. 1971 edition with new material by the author"--
Author : Peter J. Cameron
Publisher : Springer Science & Business Media
Page : 191 pages
File Size : 50,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781447105893
Set theory, logic and category theory lie at the foundations of mathematics, and have a dramatic effect on the mathematics that we do, through the Axiom of Choice, Gödel's Theorem, and the Skolem Paradox. But they are also rich mathematical theories in their own right, contributing techniques and results to working mathematicians such as the Compactness Theorem and module categories. The book is aimed at those who know some mathematics and want to know more about its building blocks. Set theory is first treated naively an axiomatic treatment is given after the basics of first-order logic have been introduced. The discussion is su pported by a wide range of exercises. The final chapter touches on philosophical issues. The book is supported by a World Wibe Web site containing a variety of supplementary material.
Author : Paul Richard Halmos
Publisher : Unknown
Page : 122 pages
File Size : 55,7 Mb
Release : 1960
Category : Mathematics
ISBN : UOM:39015006570702
From the Reviews: .,."He (the author) uses the language and notation of ordinary informal mathematics to state the basic set-theoretic facts which a beginning student of advanced mathematics needs to know. ...Because of the informal method of presentation, the book is eminently suited for use as a textbook or for self-study. The reader should derive from this volume a maximum of understanding of the theorems of set theory and of their basic importance in the study of mathematics." Philosophy and Phenomenological Research
Author : Shahn Majid
Publisher : Cambridge University Press
Page : 668 pages
File Size : 51,7 Mb
Release : 2000
Category : Group theory
ISBN : 0521648688
A graduate level text which systematically lays out the foundations of Quantum Groups.
Author : Peter Komjath,Vilmos Totik
Publisher : Springer Science & Business Media
Page : 516 pages
File Size : 48,8 Mb
Release : 2006-11-22
Category : Mathematics
ISBN : 9780387362199
This volume contains a variety of problems from classical set theory and represents the first comprehensive collection of such problems. Many of these problems are also related to other fields of mathematics, including algebra, combinatorics, topology and real analysis. Rather than using drill exercises, most problems are challenging and require work, wit, and inspiration. They vary in difficulty, and are organized in such a way that earlier problems help in the solution of later ones. For many of the problems, the authors also trace the history of the problems and then provide proper reference at the end of the solution.
Author : Winfried Just,Martin Weese
Publisher : American Mathematical Soc.
Page : 210 pages
File Size : 42,9 Mb
Release : 1996
Category : Mathematics
ISBN : 9780821802663
This book is an introduction to set theory for beginning graduate students who want to get a sound grounding in those aspects of set theory used extensively throughout other areas of mathematics. Topics covered include formal languages and models, the power and limitation of the Axiomatic Method, the Axiom of Choice, including the fascinating Banach-Tarski Paradox, applications of Zorn's Lemma, ordinal arithmetic, including transfinite induction, and cardinal arithmetic. The style of writing, more a dialogue with the reader than that of the Master indoctrinating the pupil, makes this also very suitable for self-study.