© 1978

Categories for the Working Mathematician


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

Table of contents

  1. Front Matter
    Pages i-xii
  2. Saunders Mac Lane
    Pages 1-5
  3. Saunders Mac Lane
    Pages 31-53
  4. Saunders Mac Lane
    Pages 55-78
  5. Saunders Mac Lane
    Pages 79-108
  6. Saunders Mac Lane
    Pages 109-136
  7. Saunders Mac Lane
    Pages 137-159
  8. Saunders Mac Lane
    Pages 161-190
  9. Saunders Mac Lane
    Pages 191-209
  10. Saunders Mac Lane
    Pages 211-232
  11. Saunders Mac Lane
    Pages 233-250
  12. Saunders Mac Lane
    Pages 251-266
  13. Saunders Mac Lane
    Pages 267-287
  14. Back Matter
    Pages 289-317

About this book


Categories for the Working Mathematician provides 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. The book 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 characterized 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 two new chapters on topics of active interest. One is on symmetric monoidal categories and braided monoidal categories and the coherence theorems for them. The second describes 2-categories and the higher dimensional categories which have recently come into prominence. The bibliography has also been expanded to cover some of the many other recent advances concerning categories.


Adjoint functor Morphism addition algebra theorem

Authors and affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA

Bibliographic information

  • Book Title Categories for the Working Mathematician
  • Authors Saunders Mac Lane
  • Series Title Graduate Texts in Mathematics
  • DOI
  • Copyright Information Springer Science+Business Media New York 1978
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Hardcover ISBN 978-0-387-98403-2
  • Softcover ISBN 978-1-4419-3123-8
  • eBook ISBN 978-1-4757-4721-8
  • Series ISSN 0072-5285
  • Series E-ISSN 2197-5612
  • Edition Number 2
  • Number of Pages XII, 317
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics K-Theory
  • Buy this book on publisher's site


From the reviews of the second edition:

“The book under review is an introduction to the theory of categories which, as the title suggests, is addressed to the (no-nonsense) working mathematician, thus presenting the ideas and concepts of Category Theory in a broad context of mainstream examples (primarily from algebra). … the book remains an authoritative source on the foundations of the theory and an accessible first introduction to categories. … It is very well-written, with plenty of interesting discussions and stimulating exercises.” (Ittay Weiss, MAA Reviews, July, 2014)

Second Edition

S.M. Lane

Categories for the Working Mathematician

"A very useful introduction to category theory."—INTERNATIONALE MATHEMATISCHE NACHRICHTEN