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I am self-studying Hungerford's book Algebra. He uses a whole section to talk about categories. In the next section (Direct Products and Direct Sums) he proves that the category of groups has a product, but I think I don't need categories in order to prove that a direct product of groups exists. I've looked in Rotman's book (about Group Theory) and he doesn't mention anything about categories. Here is my question: Why should someone study Category Theory in order to study Group Theory?

PS. I am a layman, but I would like to learn about this subject.

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I think it's a useful perspective to take when treating the free product and amalgamated product of groups. At some point, the constructions of various objects become somewhat unwieldy in comparison to their universal properties -- of course, this is not the case for the product of groups! –  Dylan Moreland Feb 28 '12 at 22:54
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Not really an answer to your question, but if you want to see what a treatment of algebra looks like that considers category theory a first-order concept, check out Aluffi's Algebra Chapter 0. Universal properties are introduced very early on and, indeed, provide context for the entire treatment. –  ItsNotObvious Feb 28 '12 at 23:28
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In short, you don't need it. However, it's a good time to learn category theory. If you move on to study homological algebra, algebraic topology, algebraic geometry, manifolds, etc. then the language of category theory is very convenient to express a lot of information that you already know. And it gives nice ideas of what structures correspond to what structures in different systems. For any field K, the category of TotallySplitKAlgebras is antiisomorphic to the category of of FiniteSets. That's cool. And it gives sense to why there are $m^n$ maps in one direction, vs $n^m$ in the other. Odd. –  mixedmath Feb 29 '12 at 4:29

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To complement the other answer and comments: the utility of ideas of category theory is not at all limited to "algebra", either. For example, the product topology on an infinite product of topological spaces had always struck me as disappointingly weak, and I wondered why that was "the definition". In fact, that construction is a construction of (a model for) the categorical product in the category of topological spaces. That is, instead of just being a definition we've inherited, it has functional properties. In fact, with this bit of hindsight, it seems to me perverse to "define" so many things without explaining what is supposed to be happening. A categorical characterization is often much more informative.

Another example, algebraic, but not "fancy": what is an "indeterminate", after all? A "variable"? Certainly there are heuristics that we'd tell beginners, and we know how to use "indeterminates x,y...", but what are they? One precise form is to say that $\mathbb Z[x]$ is the free ring with identity on one generator $x$... meaning that, given $r\in R$ in an arbitrary ring $R$ with identity, there is a unique ring homomorhism $\mathbb Z[x]$ to $R$ sending $x$ to $r$.

In a quite different direction, the topology on the space of test functions on $\mathbb R^n$, or even just on compactly-supported continuous functions, is a colimit. For continuous compactly-supported, it is the colimit of Banach spaces $C^o_K$ of continuous functions on $\mathbb R^n$ supported on compact $K$. In contrast, the "definition" given in Rudin's "Functional Analysis" for the test function topology is actually a construction, and the following section proves several mysterious lemmas which, I only realized later, were, in effect, verification of the colimit properties. (Indeed, Schwartz overtly used the notion of colimit c. 1950, but use of such notions had been out-of-style for U.S. analysts for many decades. Some of that may be anti-Bourbaki reaction, even though Bourbaki did not use category-theory ideas, either.)

In the short term, it is usually possible to "get along" without overt use of category-theory terms or ideas. However, the more things one finds reason to remember, the more imperative there is to organize them well, to eliminate redundancies and duplications and waste, etc. Category theoretic ideas are very helpful in this regard.

(This is not to say that "formal" category theory is necessarily as broadly useful, in the same way that, while set theory is undeniably useful, the utility of a highly developed formal or axiomatic set theory is probably not nearly as useful as the basic parts.)

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Category theory is a useful way to think about mathematical structures. You can learn group theory without knowing any category theory, but it is good to get some acquaintance with the basics of category theory as you learn group theory, because (1) it will broaden your perspective on what you know in group theory and (2) it will better prepare you to continue on as you go deeper into the subject. If you learn nothing else in category theory, learn the idea of defining objects in terms of universal properties, so that the object is unique (up to unique isomorphism) if it exists. This is a really helpful way of thinking about abstract algebraic structures.

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I totally agree with Michael Joyce! Wait till you hit subjects as rings and fields, linear algebra, topology or commutative algebra and you will see similarities in notions as homomorphisms, direct sums, direct products, and many, many more. Category theory simply gets you onto a higher abstraction level. I am not familiar with Hungerford's book, but maybe the Yoneda Lemma is treated. This generalizes the well-known Cayley Theorem from group theory (embedding an abstract group in a permutation group via actions on the abstract group in question). –  Nicky Hekster Feb 28 '12 at 23:22

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