Basic category theory

Basic Category Theory

Volumn , Issue , 2014, Pages 1-183

  Leinster, Tom ^a  

^a UNIVERSITY OF EDINBURGH   (United Kingdom)

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[No Author keywords available]

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EID: 84952942390     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1017/CBO9781107360068     Document Type: Book

References (22)

1. This book is intentionally short. Even some topics that are included in most introductions to category theory are omitted here. I will indicate some of the topics that lie beyond the scope of this book, and suggest where you might read about them. Since there is far more written on category theory than anyone could read in a lifetime, these recommendations are necessarily subjective.
2. Saunders Mac Lane, Categories for the Working Mathematician.Springer, 1971; second edition with two new chapters, 1998.
3. It is so well-written that more than forty years on, it is still the most popular introduction to the subject. It addresses a more mature readership than this text, and covers many topics omitted here, including monads (one formalization of the idea of algebraic theory), monoidal categories (categories equipped with a tensor product), 2-categories (mentioned at the end of our Chapter 1), abelian categories (categories of modules), ends (an elegant generalization of the notion of limit), and Kan extensions (which provide the tongue-in-cheek title of the book’s final section: ‘All concepts are Kan extensions’).
4. Steve Awodey, Category Theory. Oxford University Press, 2010.
5. Awodey’s book covers less than Mac Lane’s, but is particularly strong on connections between category theory and other parts of logic. It has a full chapter on cartesian closed categories, and also covers the theory of monads.
6. Those who prefer lectures to books might try this library of 75 ten-minute introductory category theory videos:
7. Eugenia Cheng and Simon Willerton, The Catsters. Available at www.youtube.com/user/TheCatsters, 2007-2010.
8. Other than the topics treated here, they cover monads, enriched categories, internal groups (and other internal algebraic structures), string diagrams (which we touched on in Remark 2.2.9), and several more sophisticated topics.
9. For inspiration as much as instruction, here are two further recommendations.
10. Saunders Mac Lane, Mathematics: Form and Function. Springer, 1986.
11. F. William Lawvere and Stephen H. Schanuel, Conceptual Mathematics: A First Introduction to Categories. Cambridge University Press, 1997.
12. Mathematics: Form and Function is a tour through much of pure and applied mathematics, written from a categorical perspective. Its declared purpose is to present the author’s philosophy of mathematics, but it can also be enjoyed for its many excellent vignettes of exposition. (Beware of the numerous small errors.) Conceptual Mathematics is a thought-provoking text and an intriguing experiment: category theory for high-school students, complete with classroom dialogues.
13. For categorical topics beyond the scope of this book, two good general references are
14. Francis Borceux, Handbook of Categorical Algebra, Volumes 1-3. Cambridge University Press, 1994.
15. Various authors, The nLab. Available at http://ncatlab.org, 2008-present.
16. Borceux’s encyclopaedic work often takes a different point of view from the present text, but covers many, many more topics. Apart from those just mentioned in connection with other books, some of the more important ones are fibrations, bimodules (also called profunctors or distributors), Lawvere theories, Cauchy completeness, Morita equivalence, absolute colimits, and flatness.
17. The nLab is an ever-growing online resource for mathematics, focusing on category theory and operating on similar principles to Wikipedia. Individual entries can be idiosyncratic, but it has become a very useful reference for advanced categorical topics.
18. Vigorous research in category theory continues to be done. The sources listed above provide ample onward references for anyone wishing to explore.
19. Timothy Gowers, Mathematics: A Very Short Introduction. Oxford University Press, 2002.
20. G. M. Kelly, Basic Concepts of Enriched Category Theory. Cambridge University Press, 1982. Also Reprints in Theory and Applications of Categories 10 (2005), 1-136, available at www.tac.mta.ca/tac/reprints.
21. F. William Lawvere and Robert Rosebrugh, Sets for Mathematics. Cambridge University Press, 2003.
22. Tom Leinster, Rethinking set theory. American Mathematical Monthly, to appear (2014). Also available at http://arxiv.org/abs/1212. 6543.