# Set theory for category theory beginners

I am wondering how much set theory is needed to read the basics of category theory, and what (preferrably short) book would be recommended.

Usually I would just use naive set theory without worrying whether something is really a set, so it bugs me quite a lot that we have small/locally small/large category where we need to specify certain things to be a set. I have never worked with axiomatic set theory before, and I don't think I am interested in going into the foundation deeply: I just want to know enough about/be comfortable with classes etc to read category theory or homological algebra.

I wiki-ed a bit and there seems to be different axioms for set theory. As classes are involved, I guess I should be looking at NBG or MK axioms.

So can anyone briefly tell me how much knowledge in set theory would suffice, or whether there are short notes/books that would serve this purpose. Thanks!

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In contrast to what some of the other answers seem to be saying, I believe that size issues play a very important role in category theory. Consider, for instance, the notion of complete category, i.e. a category having all small limits. Most "naturally-ocurring" categories, such as sets, groups, categories, etc. are complete (and cocomplete), and the ability to construct small limits and colimits is extremely important. However, these are all large categories, and a classic proof due to Freyd shows that in fact any small complete category must be a preorder (i.e. any two parallel arrows are equal). Thus, one of the most basic notions of category theory (completeness) becomes trivial if you aren't careful with size distinctions.

I also feel that more mathematicians should pay attention to set-theoretic issues, especially in category theory, and I wrote an unfortunately lengthy note myself on the subject, akin to Murfet's and Easwaran's pages linked to in Greg's answer.

However, for purposes of learning category theory, I don't think one should pay too much attention to any of this stuff. I think all you need to know, beyond naive set theory, is that some collections are "too big to be sets" (like the collection of all sets) but we can still manipulate them more or less as if they were sets, and we call them "classes." NBG and MK formalize this nicely with the "Limitation of Size" axiom: a class is a set if and only if it is not bijective with (i.e. "is not as big as") the class of all sets.

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If we use universes, doesn't Freyd's result give us a different conclusion (namely that any category that is U-complete is still a set)? (I guess by U-complete, I mean that it contains all U-small limits.) –  Harry Gindi Nov 23 '09 at 6:01
I'm looking at your paper now, and I like it very much; the discussion in section 2 is very clarifying. While I would still advise beginners not to learn set theory, I would advise those of us who advise beginners to read your paper. –  David Speyer Nov 23 '09 at 20:11
fpqc, I'm not sure what you mean. Freyd's result as I've stated it (a category C with products indexed by Arr(C) is a preorder) doesn't directly mention size at all. Assuming a universe U, it implies that any U-small category admitting U-small limits is a preorder, which is just the translation into universe-language of the usual conclusion that any small complete category is a preorder. –  Mike Shulman Nov 24 '09 at 4:33
I remember this paper now,I found it when browsing one day. It's a terrific paper by a mathematician who's clearly given this very simple but far-reaching question a lot of thought. A shame more mathematicains aren't asking these questions-too many are content simply to shoot arrows into categories and do handwaving at "universes",which many of them I find don't understand. –  Andrew L Mar 25 '10 at 20:27
@Mike Shulman, Question: In Mac Lane, there is a section which looks like a summary of his "One universe as a foundation for category theory," where he assumes ZFC + a single universe. So far, I feel that this section is sufficient, at least for reading Mac Lane (including the AFT). Am I just deluding myself? For the current task of reading Mac Lane, if I would like to pay careful attention to size issues, do I need anything beyond this section of Mac Lane? One of the reasons I ask this, is because I don't see anybody even mention ML as a potential source. Thank you very much in advance –  user2734 May 5 '10 at 17:17

Personally I found the language of sets and classes confusing, just as you describe. I've never been sure precisely what operations on classes are allowed. For instance some textbooks mention the category of all functors from Set to Set as an example of a category which isn't locally small; but it seems to me that it's not a category at all, because its collection of objects is too large to even be a class.

I'm much happier with the formalism of Grothendieck universes and the universe axiom: Every set is contained in some Grothendieck universe. Typically we choose a universe U and agree that Set denotes the category of elements of U (or sets whose cardinality is an element of U). Then there's no problem in forming the functor category [Set, Set], and using ordinary set theory we can see that its Hom sets are indeed too big to be elements of U.

For related discussion see my question here.

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This is the best answer in every case. Distinguishing between classes and sets is nonsense, since as noted above, we're forced to introduce an ever-increasing system of classes if we want to look at functors between large categories, in which case, we're just effectively looking at universes anyway, so why not go the whole mile. –  Harry Gindi Nov 23 '09 at 18:37
I agree,as long as Grothendieck universes can be defined and developed in a manner that makes classical axiomatic set theory a special case of the generalization. I know many mathematicians that have tried to do this with mixed success. I think it's certainly a task worth attempting. –  Andrew L Mar 26 '10 at 1:53

Some of you may want simply to learn set theory, rather than learn set theory in order to do category theory. Therefore, I list here a few of the most canonical texts used by set theorists---these book are all fantastic. None of them, however, is concerned with category theory at all.

Set Theory, by Thomas Jech. (3rd Millenium edition). This book is a standard graduate introduction to set theory, and covers all the elementary theory and more, including infinite combinatorics, forcing, independence, descriptive set theory, large cardinals and so on. It is used almost universally in any serious graduate introduction to set theory. Excellent text.

The Higher Infinite, by Akihiro Kanamori. This encyclopedic account of large cardinals is simply fantastic. It contains everything you wanted to know about essentially all the most well-known large cardinals. These cardinals form a very rich structure with a highly developed theory, including surprising connections even with the structure of sets of reals and much much more. Surely any talk of "universes" in category theory would be deeply informed by knowledge of the large cardinal hierarchy, and the far more nuanced and developed structure theory it provides for these concepts.

Set Theory: An Introduction to Independence Proofs, by Kenneth Kunen. This shorter book is an excellent companion to Jech's book, in that they have different approaches to many common problems. I always recommend my graduate students to play these books off against one another.

Of course, there are many others.

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A bit more basic but excellent with wonderful discussions are Herbert Enderton's ELEMENTS OF SET THEORY and Kuratowski and Mostowski's SET THEORY. Both are a bit expensive,but both are well worth it for the budding set theorist-especially Kuratowski,which is written by one of the great masters. –  Andrew L Mar 25 '10 at 20:17
I'd also like to add one other book I'm finding extremely helpful to prepare for my cumulative final in elementary set theory: INTERMEDIATE SET THEORY by F.R.Drake and D.Singh. It will act as a perfect bridge between introductory texts and Jech/Kunen.It focuses on size issues and the basic definitions and theorums of forcing in more detail then just about any other source.It also discusses many of the philosophical issues surrounding set theoretical foundations of mathematics. A great read and sadly,very expensive now. –  Andrew L May 14 '10 at 3:36

Dan Murfet has some notes on foundations for category theory which can be found here. They contain an introduction to Grothendieck universes as well as some references for learning about NBG class theory.

If you are particularly interested in some more possible foundations and their pros and cons you might want to have a look at this blog post by Kenny Easwaran.

I think that if you only want to learn introductory category theory, especially for the purpose of doing homological algebra, one can often safely just pretend that size issues are not really issues. It is true that there are many technicalities involving size that can trip one up. However, I tend to see these as "just" technicalities especially from the point of view of how I think about category theory. Maybe this is a controversial point of view?

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I guess the problem for me is I never know precisely what I can do with sets. I heard about Russell's paradox before, so I have a vague idea that a "very large collection" may not be a set, but other than that I know nothing. Perhaps that's why I feel very uncomfortable with these (possibly) mere technicalties. Thanks for the links anyway! –  Ho Chung Siu Nov 22 '09 at 3:36
I should clarify by saying that there are plenty of things which are not true without size restrictions and that there are some genuine issues that arise. A lot of effort sometimes gets devoted to working around size problems. It is more of a philosophical thing that one should think like size is not a big deal and then worry about it if/when one wants to make precise statements about certain things. –  Greg Stevenson Nov 22 '09 at 3:42
The point is that there are rarely statements you can even formulate at all in category theory that you can't fix by doing a bunch of messing around with universes or proper classes. Basically, if you understand category theory well enough, the statements you formulate will be independent of the technical nonsense. –  Harry Gindi Nov 22 '09 at 18:27
@fpqc"....independent of the technical nonsense."You know,this is why I've never been confortable with category theory-it's because too many of us are perfectly happy shooting arrows into things they just throw into a category without looking inside the boxes. It's reminiceint of Skinnerian behaviorism,Chomskian structuralism or some of the more extreme forms of string theory:They have dogmatic answers to basic questions that seem designed to avoid the nuts and bolts of what they're doing."String theory is obviously true-why do you people question it just because we can't TEST it?"Is it me? –  Andrew L Mar 25 '10 at 20:36

In my personal opinion, you do not need any set theory to learn the basics of category theory. Well, you need to know the meaning of the symbols $\in$ and $\subset$, because any mathematical paper will probably use them, but that is all.

Every once in a while, your introduction will probably use the word "class", or remark that we are working within a fixed universe. For the purpose of learning the basics, I claim you can ignore these statements. Just think of "class" as a large set, and "working within a universe" as "we are allowed to do reasonable set-theoretic constructions".

Certainly, it's worth eventually going back and learning what those terms mean. But I just looked at my bookshelf. The most intensely category theoretic books are FGA (together with "Fundamental Algebraic Geometry: Grothendieck's FGA Explained"), "Methods of homological algebra", "Introduction to Homological Algebra", Hatcher's "Algebraic Topology" and "A guide to Quantum Groups." I claim that you can understand any of these with only the most naive set theory.

I'd be curious to know what areas of category theory can't be handled in this naive manner.

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Transfinite induction and regular cardinals (can) come up in the theory of triangulated categories. This is actually also an area where one really does need to be concerned about local smallness of categories. –  Greg Stevenson Nov 23 '09 at 21:12
Let me be a little more specific. Let's say I just saw Freyd-Mitchell in Weibel's book, and he says the theorem needs the abelian category to be small. Can I just ignore that? And in any case, I feel so strange about it: what can go wrong when the abelian category is large? Questions like this come up from time to time, and I'm just not sure if they acn be ignored. –  Ho Chung Siu Nov 24 '09 at 1:01

See Sets for Mathematics by Lawevere and Rosebrugh. Its a great book, but it is misnamed. It should have been called something like "An introduction to category theory via set theory." The point of the book really is category theory, not set theory.

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For the record, it's by Lawvere and Rosebrugh, not Lawvere alone. Also, I think the name is spot-on. It's an approach to set theory that's very closely tied to how most mathematicians use sets in their daily practice (unlike the traditional approach). –  Tom Leinster Nov 23 '09 at 21:39
Thanks for pointing out my ommision regard Rosebrugh. I added his name. Perhaps the book shows how algebraists think about sets, but it's far from how I (as an analyst) think about them. I came to the book already knowing set theory and wanting to learn a little category theory. I thought that's how most people would approach the book, but I suppose some folks might learn in the opposite order. –  John D. Cook Nov 23 '09 at 22:03
You're right; I overstated my case. When you look at the book, it doesn't immediately give the impression of "how mathematicians think about sets". What I wanted to say is that their axiomatization of set theory (using sets and functions) consists of a list of statements that correspond to features of sets as used by ordinary mathematicians every day. The same is not true of ZFC (which axiomatizes sets and membership). For instance, ZFC's Axiom of Foundation implies that there is some real number none of whose elements are real numbers. To most people, that statement is nonsensical. –  Tom Leinster Nov 24 '09 at 13:34
Lawvere and Rosebrugh IS excellent and of course,Lawvere has long been one of the most active and vocal advocates of the creation of a unified foundation for both set theory and category theory. The book is written very much with this ultimate goal in mind.My problem with the author's approach is they practically eliminate membership and equality then becomes synonomous with isomorphism. I have a HUGE problem with this-without some stronger concept of equality,all kinds of wacko things start to happen in mathematics.See the thread dealing with equality vrs.isomorphism. –  Andrew L May 14 '10 at 3:40

I recommend "Set Theory", by Pinter. It is a very concise book, and if I am not mistaken, it uses basically von Neumann's approach to classes and sets (you asked for NBG class theory). You will only need to read the relevant (short) chapter there to feel very comfortable with sets and classes. And if you will want to learn about cardinality issues - you'll find it there in concise form.

This is not the book I would reccommend to someone who knows no math and would like to learn set theory, but for you I recommend it.

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Lawvere and Schanuel's Conceptual Mathematics is a good introduction to category theory that assumes an absolute minimum of set theory. Peter Cameron's Sets, Logic and Categories, on the other hand, is a good introduction to sets and logic (albeit far too short), but his chapter on category theory is woefully short and would be unhelpful to anyone who doesn't already understand categories a little bit.

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Don't learn classical set theory if you want to do category theory. There are a few "set theories" that are better-suited for category theory, for a list, check out the page "Set Theory" on nLab.

And also, you don't really need to learn set theory for category theory. While there can be size issues etc, arrow-theoretic language is completely different from set-theoretic language. The fact is, set theory is much easier and less abstract than category theory, and you don't want to settle into your nice set theoretic world just to have it all blown away once you start doing category theory.

For example, in set theory, all injective maps ($f:X\rightarrow Y$) (monomorphisms in the category of sets) admit a left inverse ($g:Y\rightarrow X$ such that $g \circ f = id_X$). This is not true in a general category. That is, the morphisms with this property in a general category aren't only monomorphisms, they're split monomorphisms, so set theoretic intuition absolutely fails here. There are countless other examples, but I'm sure you see my point.

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I certainly disagree with the above answer, but I do think that there is room for different opinions in these matters. –  Orr Shalit Nov 22 '09 at 14:13
@fpqc: I suggest growing slightly thicker skin -- it's acceptable for people to downvote without explaining why, although it's certainly both more polite and more helpful if they leave a comment too. –  Scott Morrison Nov 22 '09 at 21:37
Please bring this over to meta: tea.mathoverflow.net/discussion/56/… –  Scott Morrison Nov 23 '09 at 7:36
I think discussing the category of sets (which is what your split epi discussion is about) is not that relevant to the OP's question, which is about how much set theory to learn to handle hom-sets and hom-classes. I'm also not quite sure what your answer is claiming about set theory –  Yemon Choi Nov 23 '09 at 8:41
I would imagine that any set theoretic intuition one has is going to fail in any non-concrete category. –  Sean Tilson Jun 18 '10 at 13:46

Several generations of mathematicians, working and otherwise, learned their portion of axiomatic set theory and foundations from the appendix to John L. Kelley's General Topology. It was still in print the last time I checked, and I remember liking it a lot.

Note on provenance. "The system of axioms adopted is a variant of systems of Skolem and of A.P. Morse and owes much to the Hilbert–Bernays–von Neumann system as formulated by Gödel."

If you like that sort of thing …

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This is the axiom system usually called MK, for "Morse-Kelley" (this being the Kelley, and presumably the Morse being the one he refers to). –  Mike Shulman Nov 23 '09 at 15:17
Hey! That makes a good motto: MK, Non Plus Ultra! –  Jon Awbrey Nov 23 '09 at 15:50