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I had posted a question about category theory some months ago, and I got answered that there are two ways to study Category Theory.

One is to treat Category Theory as a logic system independent from Set Theory, and another is to treat Category Theory in the context of ZFC.

I have no reason to study the first one. (because even though one has proved a theorem in some logic system independent from ZFC, logically it need not to be true in ZFC) Is there any advantage of the first one over the second one?

Also, I want a nice introductory Category theory text book (in the sense of the second version). Please recommend me a textbook :) Thank you in advance !

Martin Sleziak
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Mathemagic
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I learned lots of category theory from Chapter 0 by Paolo Aluffi. It is one of the most well-written books that I have come across. Do give it a go!

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The standard textbook is Categories for the Working Mathematician by Saunders Mac Lane. Generations of mathematicians learned category theory using this book and you won't make a mistake choosing this one. In the first chapter, both foundations you mention are treated but in the rest of the book, he adds one axiom to ZFC assuming the existence of a Grothendieck Universe .

All the important topics (and more) are treated on some 300 pages. Necessarily, this has the consequece of not everything being explained in detail and proofs not always being easy to understand (to a beginner, not to a working mathematician).

A more recent book is the Handbook of Categorical Algebra by Francis Borceux. His treatment is much more voluminous (indeed there are three volumes of the handbook). This has the advantage of proofs being much more detailed and a lot more topics being treated in depth. For example the complete third volume is devoted to toposes and sheaf theory, which, in Mac Lane is mentioned only at the very end in the appendix giving some ideas of how a logical foundation of mathematics based on category theory works.

By the way, Borceux works (most of the time) in NBG allowing categories to have proper classes of objects but being locally small. One can show that this definition is basically equivalent to the definition of "Categories with small $\operatorname{Hom}$-sets" by Mac Lane. But as beginner, you really shouldn't pay too much time on size issues, but rather get an intuitive feeling of what category theory is all about.

For this purpose, I would recommend using Mac Lane and whenever something is not clear to you, going to Borceux to get a different and more detailed view on the topic. Hope this helps.

  • This helps a lot. Thank you ;) By the way, as far as I know, Grothendieck's universe is unprovable in ZFC. Hence, does this mean that it's not possible (or not yet known) to do Category theory in ZFC itself? That is, does Category theory require a strictly stronger set theory than ZFC? So that if one proves a theorem using Category theory, will some people not accept it as true? I'm asking this since I saw a post in mathoverflow discussing whether Wiles used Grothendieck Universe to prove FLT or not. – Mathemagic Sep 03 '14 at 07:01
  • You can work purely within ZFC if you like, you just have to pay a lot of attention to size issues. It only confuses beginners. – Zhen Lin Sep 03 '14 at 07:46
  • @Mathemagic: In most real world examples you can work as follows: Proof a category-theoretic theorem using universes, i.e. show that the theorem holds for a locally small category relative to a universe $U$. By definition, universes share a lot of characteristic properties with the class of all sets in NBG. So propably you can translate your proof so that it becomes a theorem in NBG. It is well known that all theorems provable in NBG involving only sets also hold in ZFC so in algebraic/topologic/analytic applications you will be able to use your theorem in ZFC. –  Sep 03 '14 at 07:51
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Abstract and Concrete Categories is free and I use it to refresh my knowledge. I guess it wouldn't be free if it was the best, but it is readable and with a lot of examples.

A classic text book is Saunders Mc Lane 'Categories for the Working Mathematician'. But it seems to be a fast development in category theory, so I'm sure there are some good modern books.

Lehs
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    "I guess it wouldn't be free if it was the best [...]" : this is not a valid argument. Take Algebraic Topology of Hatcher which has imposed itself as one of the classic introduction for undergraduates. And even as some higher level (pun intended) with Higher Topos Theory of Lurie. Freeness in culture, especially in academics, does not mean cheap : it is the consequence of the will of certain authors to make knowledge universally accessible. (That being said, I'm not a big fan of The joys of cats.) – Pece Sep 03 '14 at 06:55
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    @Pece: You are right, but I guess there are more popular books. And being said that, I like it. – Lehs Sep 03 '14 at 07:02
  • I really like ACC, though it should be mentioned it doesn't have quite the breadth of Mac Lane. Though it also covers some concepts I haven't seen in other book-length works. – Malice Vidrine Sep 03 '14 at 08:15