# Examples of useful Category theory results?

I'm layman to Category theory. Trying to understand it, I just read a bit briefing on it and wiki pages of Category, Functor, Morphism. However I still could not see the merits of it.

Category theory, in layman's eyes, is another layer of abstract, similar to abstract algebra's group, ring, field or set theory's induction, poset, well-ordering. Now, after the abstraction, what's the benefit?

A possible benefit that I could imagine, is that it might bypass some restriction on algebra or set theory? for example set theory talks about "set", maybe Category Theory can talk about "class"? or, algebra talks about "group", this is again based on set, so Category Theory can talk about "group-like categories"? so maybe some results in set theory or algebra can be generalized? for example algebra talks about groups, saying that left $e$ and right $e$ is the same, and left inverse and right inverse of an element is also the same, maybe this could be generalized in Category Theory?

Anyway, could you give some examples of useful Category Theory results (theorems or approaches) that could be used in other more "concrete" mathematics fields?

• Sometimes category theory can do work for you that would otherwise be difficult. For instance, if you can show that two representable functors are isomorphic (which only requires natural bijections of sets) you get an isomorphism between representing objects. In certain contexts this may save you quite a lot of work. – Olivier Bégassat May 27 '14 at 2:19
• It would help if you were more specific (there's a lot of category theory and a lot of ways to use category theory). What kind of examples would be particularly convincing to you? Number theoretic? Group theoretic? – Qiaochu Yuan May 27 '14 at 2:32
• @QiaochuYuan i'm a math layman (only touched Riemann integral), so some example in algebra or matrix will do (too complex then i probably won't understand) – athos May 27 '14 at 3:52
• Most of the big applications of category theory are in the more abstract parts of mathematics, unfortunately. I think you shouldn't worry about it too much until you see more mathematics. – Qiaochu Yuan May 27 '14 at 4:13
• Regarding your imagination, there are things called "groupoids" and "2-groups". A direction I think you've overlooked is the notion of a "group object". – Hurkyl May 27 '14 at 9:08