I recently finished a first graduate course in algebraic topology, and my institution's final undergrad course in smooth manifolds. Since courses ended, I have been studying de Rham cohomology, in the process returning to many of the topics I learned in my courses and strengthening my understanding as necessary.
Throughout, I have started to become interested in category theory. But so far, the category theory I have seen has been basic: how $\pi_n$ is a functor, how pullback is a (contravariant) functor, how (co)homology is a functor. This amounts to acknowledging certain similarities between the constructions we've been doing, but I've yet to see this ghost in the code become a primary concern.
In algebraic topology, what are the earliest instances in which category theory becomes essential to obtaining results?
I am particularly interested in answers involving topics in algebraic topology that I can conceivably understand by the end of this summer, from the starting point of de Rham cohomology and the very basics of category theory.
As to what I mean by essential: I have been thinking about what 'essential' should mean here. The problem is that I have not seen much c.t. in action in a.t., so there's some 'I don't know what I don't know' going on here.
I know, in general, that many people affirm that a student should begin learning a subject 'classically,' to some extent, so that there is motivation, and so that subtleties are not lost on the student. In this sense, withholding information is justifiable. So by 'essential,' perhaps a good interpretation is 'it would be completely unjustifiable to withhold category theory from the student learning/proving this.'