This question first arose as I wrote an answer for the question: Is there a nice application of category theory to functional/complex/harmonic analysis?; it can also be regarded as a (hopefully) more focused version of the question Most striking applications of category theory?.
It seems that category theory began as an organizational tool in topology and algebraic geometry, but by now it has grown into an area of research in its own right with applications all over the place in mathematics. I realized as I was thinking about my answer to the question above that I did not know the statements of the main results in category theory, or even if there are "main results". In a comment, Martin Brandenburg suggested several examples:
- The general adjoint functor theorem
- Freyd's representability criterion
- Beck's monadacity theorem
- Recognition theorems for locally presentable categories
- Brown's representability theorem
He also indicated that these results have numerous unsung applications to other areas of mathematics. This question is essentially an invitation for Martin and anyone else add to this list and explain some of the applications of the items on it. I would not have asked this question on mathoverflow if I believed that such a list already existed; if I am wrong then the question should probably be closed.
Here is what I have in mind for an answer to this question. It should include the statement of a theorem in pure category theory (ideally using language which is friendly to outsiders) and at least one application to another area of mathematics. The community wiki rule "one theorem per answer" makes sense here, particularly so that others can conveniently add applications to your list.
When I say "theorem in pure category theory", I don't insist that the result be incredibly nontrivial, just that it is a result which is stated and proved in the language of category theory. For example, the statement that $\pi_1$ is a functor belongs to topology, not category theory; on the other hand the Yoneda lemma counts even though it is a "lemma" instead of a "theorem".
When I say "application to another area of mathematics" I am ideally looking for statements which can be formulated without using the language of categories and functors. I want to be clear that I am interested in applications of specific results in category theory, not just results for which categorical thinking is useful (such results are everywhere).
Best Answer
The small object argument. Essentially this states that if you have a collection of maps $f_\alpha$ in a presentable category (actually, you only need the domains of the $f_\alpha$ to be compact, along with cocompleteness of the category), then any map in the category can be functorially factored as the composite of two maps:
This was first used by Grothendieck to show (in his Tohoku paper) that a Grothendieck abelian category always has enough injectives (which, as far as I know, is not directly obvious for abelian sheaves on a site, for instance). Later it became the main tool in constructing model structures on categories, because it lets you show that the factorizations needed in the definition exist.