# Categories for the Working Mathematician

Textbook

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

1. Front Matter
Pages I-IX
2. Saunders Mac Lane
Pages 1-5
3. Saunders Mac Lane
Pages 7-30
4. Saunders Mac Lane
Pages 31-53
5. Saunders Mac Lane
Pages 55-76
6. Saunders Mac Lane
Pages 77-103
7. Saunders Mac Lane
Pages 105-132
8. Saunders Mac Lane
Pages 133-155
9. Saunders Mac Lane
Pages 157-186
10. Saunders Mac Lane
Pages 187-205
11. Saunders Mac Lane
Pages 207-228
12. Saunders Mac Lane
Pages 229-246
13. Back Matter
Pages 247-262

### 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

1. 1.The University of ChicagoUSA

### Bibliographic information

• Book Title Categories for the Working Mathematician
• Authors Saunders MacLane
• Series Title Graduate Texts in Mathematics
• 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
• Hardcover ISBN 978-0-387-90035-3
• Softcover ISBN 978-0-387-90036-0
• eBook ISBN 978-1-4612-9839-7
• Series ISSN 0072-5285
• Edition Number 1
• Number of Pages IX, 262
• Number of Illustrations 14 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site