I heard a bit about Category Theory, especially when studying topology, although I never really studied it. I am quite curious about it, in particular I would like if someone could better explain:
-Is category theory a study field in its own interest, like, broadly speaking, Geometry, Analysis, Algebra and so on, or is it more a way of tackling problems any mathematician, independently from his area can embrace? Is it a subfield of Logic? Is it a way to found mathematics (alternative to set theory) ?
-Are there important applications to area of mathematics different from algebra and geometry, such as Analysis, Probability and more applied areas of mathematics, or even other Sciences?
-I really like, for the little I know, the unifying, broadening approach of Category Theory, is it possible to make a career in it, are there programs which specialize in this area? Is it a growing one or a vantage point whose possibility have already been explored?