4

Background: Currently I know the most basic notions of category theory, as covered by the first chapter of Jacobson's Basic Algebra II or the last chapter of Hungerford's Algebra, up to Yoneda's lemma and adjoint functors. (However, I think I need to learn these again.)

Motivation: I'd like to study Category Theory to smooth the way for Homological Algebra, so that I'll be able to read texts such as Weibel's An Introduction to Homological Algebra (tried to read it, but got stuck due to lack of familiarity with abstract abelian categories). However, this motivation is not very much a restriction on what I want to learn; I just feel that my knowledge of Category Theory is insufficient. Therefore I'd like to read a book entirely devoted to Category Theory.

Questions:

  1. I have went through the first pages of Mac Lane's classic Categories for the Working Mathematician and find it very readable. Is Mac Lane's book suitable for my purpose? If not, are there other books on Category Theory that you may recommend? (I'd like to study categories in a systematic manner, so books entirely devoted to it are preferred.)

  2. If I choose Mac Lane's book, do I need to read from cover to cover? What are the logical interdependence of the chapters? And most importantly, which parts of the book do I need to read in order to prepare for Homological Algebra?

Thanks for any advice!

joriki
  • 238,052
Yuxiao Xie
  • 8,536
  • 5
    Have you considered Category Theory in Context by Emily Riehl? https://golem.ph.utexas.edu/category/2016/11/category_theory_in_context.html I'm currently reading it cover to cover. I love it. – Ben Blum-Smith Aug 21 '18 at 04:49
  • Obligatory reference: https://math.stackexchange.com/questions/94827/what-books-must-every-math-undergraduate-read – SK19 Aug 21 '18 at 16:15

2 Answers2

6

Mac Lane is a very good general introduction, and I can't think of a good argument not to use it as your main resource if you have a reasonable math background. I wouldn't suggest learning anything less than the material of chapters I-V (this is the core stuff that you really can't get by without), and anything else in the book will depend on these chapters. I would also generally recommend chapters VI, IX, and X as very good to learn. The only strong dependency among those latter three is that you will need some concepts from chapter IX for the material in X. The only other strong dependency in the book is that Ch. XI builds on the material of VII, but that's obvious.

I will note that while there are now gentler introductions to the above core topics (like the Riehl book mentioned in the comments), if Abelian and monoidal categories are important for homological algebra (I'm not too familiar with the field, but my impression is they probably come up a fair bit) then you're unlikely to find an introduction that is an easier read and covers those topics.

Malice Vidrine
  • 9,653
  • Thank you for a very informative answer!! So far I have read the first chapter and can understand most of the examples (except for those related to Algebraic Topology). After reading your advice I think I'll read first the chapters I-V and then go straight to chapter VIII on abelian categories. Is that okay? – Yuxiao Xie Aug 21 '18 at 15:14
  • I think you'd be just fine doing that. Glad this helped! – Malice Vidrine Aug 21 '18 at 16:05
2

Abstract and Concrete Categories is a book by Jiří Adámek, Horst Herrlich, and George Strecker. The book is nicknamed The Joy of Cats. It is an excellent reference and it has the great advantage of being freely available on the web.

J.-E. Pin
  • 40,163