Category theory results

Question

I am curious about the category theory results. It looks very different from others areas of math for me. It is full of different definitions (limit, product, monad etc.) as well as others maths but there are few results that are proved. In other words I can't find a lot of theorems, lemmas in the category theory.

I can mentioned Yoneda's lemma as an exception of the observation. It postulates a non-trivial result and has a trivial proof for it.

Another exception is the parametricity theorem by John C. Reynolds. That is one of the basis for functional programming languages.

I really hope that I am wrong in my observation and somebody can provide my amazing results of the theory.

Answer 1

Here's an extract from Emily Riehl's Category Theory in Context,

«The author is told with distressing regularity that “there are no theorems in category theory,” which typically means that the speaker does not know any theorems in category theory. This attitude is certainly forgivable, as it was held for the first dozen years of the subject’s existence by its two founders and might reasonably persist among those with only a casual acquaintance with this area of mathematics. But since Kan’s discovery of adjoint functors and Grothendieck’s contemporaneous work on abelian categories —innovations that led to a burst of research activity in the 1960s—without question category theory has not lacked for significant theorems.»

You can find a brief introduction to some of those results in the epilogue of the book.

Answer 2

Many nontrivial theorems are of the form "this can be represented as that, and/or the invariants associated to this are isomorphic to the invariants associated to that". Two examples:

1. Much like Yoneda lemma, topos theory hinges on the importance of presheaf categories $[C^o, Set]$; a theorem by Giraud says that every Grothendieck topos arises as a left exact reflective subcategory of some $[C^o, Set]$.
2. If you like abelian categories more (say, if you like to study module categories), then Gabriel-Popescu theorem says essentially the same thing in that universe.