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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"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.

1 [made Community Wiki]

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.