Recently, I was reminded in Melvyn Nathason's first year graduate algebra course of a debate I've been having both within myself and externally for some time. For better or worse,the course most students first use and learn extensive category theory and arrow chasing is in an advanced algebra course,either an honors undergraduate abstract algebra course or a first-year graduate algebra course.
(Ok,that's not entirely true,you can first learn about it also in topology.But it's really in algebra where it has the biggest impact.Topology can be done entirely without it wherareas algebra without it beyond the basics becomes rather cumbersome.Also,homological methods become pretty much impossible.)
I've never really been comfortable with category theory. It's always seemed to me that giving up elements and dealing with objects that are knowable only up to isomorphism was a huge leap of faith that modern mathematics should be beyond.But I've tried to be a good mathematican and learn it for my own good. The fact I'm deeply interested in algebra makes this more of a priority.
My question is whether or not category theory really should be introduced from jump in a serious algebra course.Professor Nathanson remarked in lecture that he recently saw his old friend Hyman Bass,and they discussed the teaching of algebra with and without category theory.Both had learned algebra in thier student days from van der Waerden (which incidently,is the main reference for the course and still his favorite algebra book despite being hopelessly outdated).Melvyn gave a categorical construction of the Fundamental Isomorphism Theorum of Abelian Groups after Bass gave a classical statement of the result.Bass said,"It's the same result expressed in 2 different languages.It really doesn't matter if we use the high-tech approach or not." Would algebracists of later generations agree with Professor Bass?
A number of my fellow graduate students think set theory should be abandoned altogether and thrown in the same bin with Newtonian infinitesimals (nonstandard constructions not withstanding) and think all students should learn category theory before learning anything else.Personally,I think category theory would be utterly mysterious to students without a considerable stock of examples to draw from. Categories and universal properties are vast generalizations of huge numbers of not only concrete examples,but certain theorums as well. As such,I believe it's much better learned after gaining a considerable fascility with mathematics-after at the very least,undergraduate courses in topology and algebra.
Paolo Aluffi's wonderful book Algebra:Chapter 0,is usually used by the opposition as a counterexample,as it uses category theory heavily from the beginning.However,I point out that Aluffi himself clearly states this is intended as a course for advanced students and he strongly advises some background in algebra first. I like the book immensely,but I agree.
What does the board think of this question? Categories early or categories late in student training?
