Conceptual Mathematics: A First Introduction to Categories

Conceptual Mathematics: A First Introduction to Categories

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By (author) F. William Lawvere, By (author) Stephen H. Schanuel

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  • Publisher: CAMBRIDGE UNIVERSITY PRESS
  • Format: Paperback | 403 pages
  • Dimensions: 173mm x 249mm x 25mm | 748g
  • Publication date: 1 September 2009
  • Publication City/Country: Cambridge
  • ISBN 10: 052171916X
  • ISBN 13: 9780521719162
  • Edition: 2, Revised
  • Edition statement: 2nd Revised edition
  • Illustrations note: 575 b/w illus. 12 tables 213 exercises
  • Sales rank: 134,690

Product description

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.

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Author information

F. William Lawvere is a Professor Emeritus of Mathematics at the State University of New York. He has previously held positions at Reed College, the University of Chicago and the City University of New York, as well as visiting Professorships at other institutions worldwide. At the 1970 International Congress of Mathematicians in Nice, Prof. Lawvere delivered an invited lecture in which he introduced an algebraic version of topos theory which united several previously 'unrelated' areas in geometry and in set theory; over a dozen books, several dozen international meetings, and hundreds of research papers have since appeared, continuing to develop the consequences of that unification. Stephen H. Schanuel is a Professor of Mathematics at the State University of New York at Buffalo. He has previously held positions at Johns Hopkins University, Institute for Advanced Study and Cornell University, as well as lecturing at institutions in Denmark, Switzerland, Germany, Italy, Colombia, Canada, Ireland, and Australia. Best known for Schanuel's Lemma in homological algebra (and related work with Bass on the beginning of algebraic K-theory), and for Schanuel's Conjecture on algebraic independence and the exponential function, his research thus wanders from algebra to number theory to analysis to geometry and topology.

Review quote

'This text, written by two experts in Category Theory and tried out carefully in courses at SUNY Buffalo, provides a simple and effective first course on conceptual mathematics.' American Mathematical Monthly '... every mathematician should know the basic ideas and techniques explained in this book ...' Monatshefte fur Mathematik 'Conceptual Mathematics provides an excellent introductory account to categories for those who are starting from scratch. It treats material which will appear simple and familiar to many philosophers, but in an unfamiliar way.' Studies in History and Philosophy of Modern Physics 'Category Theory slices across the artificial boundaries dividing algebra, arithmetic, calculus, geometry, logic, topology. If you have students you wish to introduce to the subject, I suggest this delightfully elementary book . Lawvere is one of the greatest visionaries of mathematics in the last half of the twentieth century. He characteristically digs down beneath the foundations of a concept in order to simplify its understanding. Schanuel has published research in diverse areas of Algebra, Topology, and Number Theory and is known as a great teacher. I have recommended this book to motivated high school students. I certainly suggest it for undergraduates. I even suggest it for the mathematician who needs a refresher on modern concepts.' National Association of Mathematicians Newsletter 'Conceptual Mathematics is the first book to serve both as a skeleton key to mathematics for the general reader or beginning student and as an introduction to categories for computer scientists, logicians, physicists, linguists, etc ... The fundamental ideas are illuminated in an engaging way.' L'Enseignment Mathematique

Table of contents

Foreword; Note to the reader; Preview; Part I. The Category of Sets: 1. Sets, maps, composition; Part II. The Algebra of Composition: 2. Isomorphisms; Part III. Categories of Structured Sets: 3. Examples of categories; Part IV. Elementary Universal Mapping Properties: 4. Universal mapping properties; Part V. Higher Universal Mapping Properties: 5. Map objects; 6. The contravariant parts functor; 7. The components functor; Appendix 1. Geometry of figures and algebra of functions; Appendix 2. Adjoint functors; Appendix 3. The emergence of category theory within mathematics; Appendix 4. Annotated bibliography.