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Sounds like you might want to petition for an exception to the prerequisites. I don't think Lang's Abstract Algebra is probably your best bet (stick to something decidedly undergraduate)undergraduate -- maybe Gallian?), nor do I think that trying to digest either all of abstract algebra or all of category theory is your best bet. I'd aim for one major result in abstract algebra which has an analogous statement in a variety of other categories, and then see what carries over to the category-theoretic framework. One idea would be to understand the classification of finite abelian groups in your abstract algebra work, and try to understand how the result and the proof techniques carry over/generalize.

p.s. The answer to the title question is definitely yes. :)

Edit: Let me add on what I think is almost certainly the best place for you to start on categories, which is Lawvere and Schanuel's "Conceptual Mathematics: A First Introduction to Categories" (double edit: which I see mathphysicist also listed). In fact, with this book in mind, it's actually the abstract algebra part of your project that now sounds the most daunting -- is that negotiable? Their discussion of Brouwer's fixed-point theorem would make an excellent topic.

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Sounds like you might want to petition for an exception to the prerequisites. I don't think Lang's Abstract Algebra is probably your best bet (stick to something decidedly undergraduate), nor do I think that trying to digest either all of abstract algebra or all of category theory is your best bet. I'd aim for one major result in abstract algebra which has an analogous statement in a variety of other categories, and then see what carries over to the category-theoretic framework. One idea would be to understand the classification of finite abelian groups in your abstract algebra work, and try to understand how the result and the proof techniques carry over/generalize.

p.s. The answer to the title question is definitely yes. :)