I am a graduate student currently working my way through an introductory course in category theory.
A question I have for this theory is why it is usefull?
Now i know this is a rich theory that is being used in many fields in mathematics, for example we have encountered categorical notions in a (introductory) course of algebraic geometry. Here we defined a contravariant functor between the categories of k-varieties (with k an algebraically closed field) and the category of k-algebras with their respective morphisms.
However we did work with the notions of category-theory, we didn't use any specific theorems.
So to specify my question further: In what way does the work done in category theory help us in proving important theorems in other fields in mathematics? Or perhaps: Can you provide examples in which deep categorical theorems are applicable and usefull for proving hard problems in other fields of mathematics?
Any criticism on my way of posing this quesion is welcomed, this is my first post here!