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Categories for the Working Mathematician

  • Saunders Mac Lane

Part of the Graduate Texts in Mathematics book series (GTM, volume 5)

Table of contents

  1. Front Matter
    Pages I-IX
  2. Saunders Mac Lane
    Pages 1-5
  3. Saunders Mac Lane
    Pages 31-53
  4. Saunders Mac Lane
    Pages 55-76
  5. Saunders Mac Lane
    Pages 77-103
  6. Saunders Mac Lane
    Pages 105-132
  7. Saunders Mac Lane
    Pages 133-155
  8. Saunders Mac Lane
    Pages 157-186
  9. Saunders Mac Lane
    Pages 187-205
  10. Saunders Mac Lane
    Pages 207-228
  11. Saunders Mac Lane
    Pages 229-246
  12. Back Matter
    Pages 247-262

About this book

Introduction

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.

Keywords

Adjoint functor Categories Coproduct algebra category theory colimit equalizer semigroup transformation

Authors and affiliations

  • Saunders Mac Lane
    • 1
  1. 1.The University of ChicagoUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-9839-7
  • Copyright Information Springer-Verlag New York 1971
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-90036-0
  • Online ISBN 978-1-4612-9839-7
  • Series Print ISSN 0072-5285
  • Buy this book on publisher's site
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