Examples of useful Category theory results?

Question

I'm layman to Category theory. Trying to understand it, I just read a bit briefing on it and wiki pages of Category, Functor, Morphism. However I still could not see the merits of it.

Category theory, in layman's eyes, is another layer of abstract, similar to abstract algebra's group, ring, field or set theory's induction, poset, well-ordering. Now, after the abstraction, what's the benefit?

A possible benefit that I could imagine, is that it might bypass some restriction on algebra or set theory? for example set theory talks about "set", maybe Category Theory can talk about "class"? or, algebra talks about "group", this is again based on set, so Category Theory can talk about "group-like categories"? so maybe some results in set theory or algebra can be generalized? for example algebra talks about groups, saying that left $e$ and right $e$ is the same, and left inverse and right inverse of an element is also the same, maybe this could be generalized in Category Theory?

Anyway, could you give some examples of useful Category Theory results (theorems or approaches) that could be used in other more "concrete" mathematics fields?

Answer 1

Category theory is absolutely essential to modern algebraic geometry and number theory. In algebraic geometry, cohomology groups of spaces are defined (mostly) using the machinery of homological algebra and derived functors. All of this depends heavily on category theory.

The Artin-Grothendieck theory of étale cohomology is one of the greatest achievements of modern algebraic geometry. This theory played a key role in the proof of the Weil Conjectures, which are very concrete (as far as number theory goes) and very important. None of it would have been possible without category theory. It doesn't just help to formalize the theory: it plays a central and essential role in all of the constructions.

Ultimately, even the proof of Fermat's Last Theorem has depended indirectly but essentially on many ideas of category theory.

Answer 2

I also know very little category theory, but I found the notion of a universal property to be particularly useful. There are often several ways to construct important types of objects in mathematics, for instance bases of vector spaces, quotient spaces, tensors products etc. To me, it was not always clear from the construction what exactly we're trying to construct. Universal properties often elucidate this basically by extracting exactly what's so special about an object, hence motivating constructions and clarifying many proofs.

On a more abstract level, things like functors in a sense capture the notion of "structure perservation", as devices which convert commutative diagrams from one category to another.