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I am having a course in Algebraic Topology and learning some basic category theory. But I only have a very limited understanding of basic set theory. I have no idea what is ZFC, and stuff like that. Thus, I find it hard to understand some motivation of the category theory.

My question is: does one need to learn some rigorous basic set theory before learning about category theory? If yes, do you have any recommendations?

Najib Idrissi
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Keith
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    IMO, you should be familiar with working with sets, though you probably don't need an axiomatic approach to them. But it also depends upon what topics are being covered. For example, if you are covering Topos, it would be a good idea to have more set theory. Otherwise, you will likely be fine. – Ben Sheller Feb 15 '16 at 05:35
  • The notion of largeness might come trouble, but (according to my short category theory) it is only thing to need set theory. Studying set theory is not much helpful to understanding or motivating category theory. – Hanul Jeon Feb 15 '16 at 07:02
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    The question was written using the first person, but I disagree with the "seeking personal advice" closure. Asking if one needs set theory to learn category theory is IMO acceptable; the close reason was created for questions like "Should I take Set Theory II at my university before or after Topology III" which are way too localized. – Najib Idrissi Feb 15 '16 at 10:17
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    I think it really depends what you mean by set theory. A naive course in set theory up until the Cantor-Berntein-Schröder theorem and similar results of that sort is probably enough. Going into axiomatic set theory is overkill for a basic category theory course (which will in general only consider small categories). – Dan Rust Feb 15 '16 at 13:01
  • This depends on exactly what kind of topic in category theory you want to learn. If you want understand topos theory or the foundations of category theory, you will need to know basic ZFC stuff. However, if you're after a basic understanding of general concepts of category theory (limits, colimits, monads etc), you don't need any set theory at all. – user40276 Feb 15 '16 at 13:02
  • Category Theory is a good canadiate as a foundational system for mathematics and so is a competitor to Set Theory. It helps to know the competition, but to know your team you do not need to know the other team. Of course, there is a pesky, yet reasonable, tendency when learning Category Theory to use set-theoretic constructions as motivators ---of course, one can just as well construe them as Type Theoretic, which is yet another candidate! What I'm really saying is, just dive into Category Theory and look-up stuff as needed and ask here on the forum. Enjoy the cats :-) – Musa Al-hassy Feb 15 '16 at 16:45
  • Of somewhat related interest. –  Feb 21 '16 at 15:15

4 Answers4

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I don't think you really need to go to formal set theory in order to understand category theory. What you need of course is basic understanding of what sets and functions are, nothing more then it is needed to understand algebra and topology (which I assume you know otherwise why taking a course in algebraic topology) :D

That said, you have to understand that category theory was born in algebraic topology and homological algebra, not for the sake of set theory.

Category theory is an abstract framework where is possible to study with the right level of generality(abstraction) many phenomena occurring in different context like topology, algebra and geometry. In particular in category theory you can define in a uniform way what are products, coproducts and many other general constructions which arise in different fields in mathematics.

In order to understand category theory can be good having some background knowledge in algebra, geometry, topology (etc) in order to have a good list of example one can use when learning categorical concepts, although not strictly necessary. Take a look to Awodey's Category theory for an introduction on category theory with very low prerequisites.

Having some knowledge in set theory can help in finding some other interesting examples (and applications of category theory) but it is not really fundamental (at least if you ignore size issues, something many mathematicians do).

Giorgio Mossa
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  • I disagree with the last point, "to understand category theory is good to have some background knowledge in". Besides elementary algebra, as mentioned in the first paragraph, you can go a long way. I don't remember the book, but I once read a book on categories for computer science (NOT pierce) and it assumed nothing and used cats for data structures. It was a small, 100pgs, and self-contained intro for cats to CS. What I'm saying is, you don't need to know a lot of fancy-shamancy math --even though it is the orgins of cats--- to actually get a basic understanding of them and to use them! – Musa Al-hassy Feb 15 '16 at 16:49
  • @MusaAl-hassy indeed you're right. What I meant to say is that it can be useful to have that stuff to understand categorical concepts via example, but of course one could take examples from fields different from topology or geometry (computer science is full of objects where you can get examples). – Giorgio Mossa Feb 15 '16 at 20:30
  • @MusaAl-hassy thanks for the remark, I've edited the answer. – Giorgio Mossa Feb 15 '16 at 20:38
  • Are you sure about the historical remark? As far as I know categories were introduced in logic. – Asaf Karagila Feb 15 '16 at 20:48
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    @AsafKaragila Well I don't know about that but if look almost everywhere you'll find out that the most ancient paper on categories is the famous "General theory of natural equivalences" where the authors (MacLane and Eilenberg) provide a list of examples from the real of algebra and topology, no logic (with the exception of some remarks on foundations). – Giorgio Mossa Feb 15 '16 at 20:59
  • To that you could add the following piece of text from the hisorical notes in the first chapter of MacLane's Categories for the working mathematicians "Categories, functors, and natural transformations themselves were discovered by Eilenberg-Mac Lane in their study of limits (via natural transformations) for universal coefficient theorems in Cech cohomology." – Giorgio Mossa Feb 15 '16 at 21:00
  • Ah. I see. I never really looked into it before, all I remembered is that one of papers in a proceedings from a logic conference in the early 1960s was about category theory (either Eilenberg or MacLane, I don't recall), and I figured it was more or less the beginning of things. But a survey of MathSciNet shows that I was wrong. Thanks! – Asaf Karagila Feb 15 '16 at 21:03
  • You're welcome! ;) – Giorgio Mossa Feb 15 '16 at 21:11
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Many of the examples and explanations given in most category theory texts are done with the assumption that you are familiar with the language of set theory. So category theory is useful for understanding the presentation of Category theory.

However there is nothing about category theory itself that requires an understanding of set theory.

Q the Platypus
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If you can study algebraic topology without first learning category theory you shouldn't worry about the set theory. There is much more to learn about category theory in that course than about set theory. The ZFC axiom system is just a standard axiom system and I would be surprised if those axioms would be mentioned elsewhere than in the introduction to overview the classes of objects in category theory.

Lehs
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If I were you, I will "learn" set theory as much as I can. Take a course in set theory at your university and do well in that class, then take the second course before learning category. The set theory knowledge you need to master should allow you to do higher level analysis, higher modern algebra, and even topology which are required for you to become a mathematician. Cathegory theory is used to study higher modern algebra and higher number theory and that is where it will take you.

DeepSea
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  • I understood the question is not so much about the usefulness of set theory but is understanding set theory a prerequisite for understanding category theory. – Q the Platypus Feb 15 '16 at 06:33
  • Yes, you should take set theory and be able to use it in other math areas. – DeepSea Feb 15 '16 at 06:48