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I'm a undergrad(third year, Manchester uni) that is thinking of doing a PhD in this area or category theory in general.

Just wondering, what branches of Maths should I focus on? As I've been told that topology particular Algebraic topology is the main area for this. A lecturer told me I should focus on getting down Algebraic topology before thinking of doing category theory as most of the examples of category theory are from algebraic topology.

However, another lecturer told me I should be doing logic and particular model theory. I can take a course in it this year, but would mean not doing commutative algebra.

But, I'm confused on what I should be studying. Like do I need logic or do I need analysis and algebra?

Also, do you need set theory to do category theory?

P.S. It just seems like every lecturer I asked gives me different answer. Like some even was telling me that I should be doing computer science modules and graph theory.

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Please do not cross-post: mathoverflow.net/questions/75788/… –  Qiaochu Yuan Sep 18 '11 at 22:39
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Certainly a firm grounding in classical category theory would be immensely helpful, and studying algebraic topology is indeed one of the best ways of accomplishing this. Category theory is the language underlying modern algebraic topology, so having a firm grasp of the subject would necessitate having a very good understanding of category theory.

If your goal is to study higher category theory such as in Jacob Lurie's book Higher Topos Theory, I'll direct you to this MO question, where Lurie has written an outline of what you should know before attempting to tackle the book. If this is indeed the sort of thing you wish to study, perhaps the best way to figure out what you need to learn would be to start browsing through it. Even if you can't understand the book, you'll start to get a feel for the sort of things you should be learning about.

You don't need set theory to do category theory, but there are certainly areas where it comes up. For example, there is a notion of small objects in a category (which are used, for example, in Quillen's Small Object Argument) which involves some basic notions of cardinals and ordinals from set theory. However, much of category theory can be done with only a basic understanding of set theory.

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