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I'm reading Aluffi, Algebra: Chapter 0, beginning with category theory and then use it for groups, rings,...

My question is:

Is it possible to learn algebra (in University) beginning with category theory, or is it too difficult in undergraduate school ?

Does someone try this way ? Any university tried that ?

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closed as primarily opinion-based by rschwieb, Zachary Selk, Claude Leibovici, 91500, Kevin Dong Dec 2 '15 at 9:20

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise.If this question can be reworded to fit the rules in the help center, please edit the question.

I think categories inherently have group-like properties, so I'd suggest you learn abstract algebra first. Though strictly speaking, neither relies on the other, so you can give it a shot. – akkkk Jan 5 '13 at 14:37
Florida State University (the university at which Dr. Aluffi works) uses this textbook for their first year graduate algebra sequence. Undergraduates have been known to take this class with minimal prior exposure to algebra. – anonymous Jan 5 '13 at 14:38
I don't think it would be good to do so, exposure to Abstract algebra is very helpful for better understanding of Cat. theory. – Ram Jan 5 '13 at 14:41
I am also trying to teach myself using Aluffi's book. I do not think category theory is difficult, but before some maturity, it seems abstract nonsense. I do not think there are many first year undergraduates who are willing to sit there and read about category theory. It is something that makes things easier, but you have to first know the hard way to appreciate this simplicity. – Hui Yu Jan 5 '13 at 14:42
@Ram, a key motivation for the development of cat theory is the unifying perspective on various algebraic structures (as well as non-algebraic ones), in which arrows (and then functors, and natural maps) play a leading role. – alancalvitti Jan 5 '13 at 14:46
up vote 19 down vote accepted

It strikes me as somewhat silly. While algebra and category theory have a good deal in common, one of the major selling points of category theory is in its ability to unify common structures found across a variety of disciplines. The reason category theory isn't typically introduced until graduate school is that a prerequisite for appreciating its power is having such a variety of disciplines already in hand. Some of the most amazing results phrased in category-theoretic language are equivalences of categories, relating two categories -- typically, one very algebraic in nature, and one decidedly not so (at least, a priori) -- you just can't get to these results if you're trying to learn abstract algebra in a category-theoretic setting. All you can do is use fancy words to describe not-all-that fancy material (okay, I'm exaggerating that point).

It seems vastly preferrable to me to learn algebra first, and only afterwards realize that many of the things you learned along the way could be rephrased in category-theoretic language.

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I agree with you, but re-learning algebra in point of view of category theory don't make you fell like you could have learned it better ? – Alan Simonin Jan 5 '13 at 15:04
Sure it feels that way, but only because you learned it once already. Learning is not something you can do once and be done with it. In a sense, you could've learned subtraction "better" if you had known about negative numbers in the first place, but that doesn't mean we should introduce negative numbers before subtraction. – Cam McLeman Jan 5 '13 at 15:07
Yes, in a pedagogical point of view, learning stuff with two points of view help understanding better. – Alan Simonin Jan 5 '13 at 15:15
@CamMcLeman, not regarding this topic, Topology was introduced in my school (With Munkres Only) even before that time I don't know what is limit point.. – Ram Jan 5 '13 at 15:15

I think that the main thing preventing teaching category theory (the basics that is) early on (and I do mean first year undergrad) is historical reasons rather than any intrinsic difficulty of the subject matter.

In fact, in many respects category theory is easier than group theory (which is first year material in some universities, but is more commonly second year). The reason being that you can draw many examples of small categories while you can't really draw small groups. You can write out their multiplication tables but that's not the same. Diagram chases can be done by anybody and provide a visual way to argue algebraically. In short, category theory offers visual ways present arguments and proofs.

There is a lot to gain from having the language of category theory at hand from the get go, though I'm not aware of universities that teach category theory as a first year course.

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Algebra by Mac Lane and Birkoff will teach you undergraduate abstract algebra while introducing you to categorical ideas, especially the role of universals. Highly recommended.

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