What are the most striking applications of category theory? I'm trying to motivate deeper study of category theory and I have only come across the following significant examples:
- Joyals Combinatorial Species
- Grothendieck's Galois Theory
- Programming (unification as computing a coequalizer, Tatsuya Haginos categorical construction of functional programming)
I am sure that these only touch on the surface so I would be most grateful to hear of more examples, thank you!
edit: To try and be more precise. "Application" in the context of this question means that it makes use of slightly deeper results form category theory in a natural way. So we are not just trying to make a list of 'maths that uses category theory' but some of the results which exemplify it best, and might not have been possible without it.