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:
- Joyal's Combinatorial Species
- Grothendieck's Galois Theory
- Programming (unification as computing a coequalizer, Tatsuya Hagino's 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 from 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.
Best Answer
For a while, my answer to this question was algebraic K-theory; what little I know of it, I learned from Quillen's paper, and it was a relief to finally see an example of category theory being used in an essential way to do something that was not just linguistic. Quillen defines the higher K-groups of an exact category by forming a quite different category in some combinatorial manner that seems to strip away any vestige of a connection to something non-categorical, and then taking its geometric realization and homotopy groups. The whole process: ring to module category to Q-construction to geometric realization, was the first argument I'd seen that category theory could do more than just rephrase perfectly good theorems confusingly.
(Now my answer would be "perverse sheaves", though.)