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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?

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  • $\begingroup$ 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. $\endgroup$ – Olivier Bégassat May 27 '14 at 2:19
  • $\begingroup$ 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? $\endgroup$ – Qiaochu Yuan May 27 '14 at 2:32
  • $\begingroup$ @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) $\endgroup$ – athos May 27 '14 at 3:52
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    $\begingroup$ 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. $\endgroup$ – Qiaochu Yuan May 27 '14 at 4:13
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    $\begingroup$ 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". $\endgroup$ – Hurkyl May 27 '14 at 9:08
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Category theory is absolutely essential to modern algebraic geometry and number theory. In algebraic geometry, cohomology groups of spaces are defined (mostly) using the machinery of homological algebra and derived functors. All of this depends heavily on category theory.

The Artin-Grothendieck theory of étale cohomology is one of the greatest achievements of modern algebraic geometry. This theory played a key role in the proof of the Weil Conjectures, which are very concrete (as far as number theory goes) and very important. None of it would have been possible without category theory. It doesn't just help to formalize the theory: it plays a central and essential role in all of the constructions.

Ultimately, even the proof of Fermat's Last Theorem has depended indirectly but essentially on many ideas of category theory.

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  • $\begingroup$ thanks for the explanation. I think it's beyond my reach today so i'll keep it for a future review, avoiding messing up my brain for now.. Alas, a little learning is a dangerous thing! $\endgroup$ – athos May 27 '14 at 6:32
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I also know very little category theory, but I found the notion of a universal property to be particularly useful. There are often several ways to construct important types of objects in mathematics, for instance bases of vector spaces, quotient spaces, tensors products etc. To me, it was not always clear from the construction what exactly we're trying to construct. Universal properties often elucidate this basically by extracting exactly what's so special about an object, hence motivating constructions and clarifying many proofs.

On a more abstract level, things like functors in a sense capture the notion of "structure perservation", as devices which convert commutative diagrams from one category to another.

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