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I don't understand what is the advantage of viewing a particular category as a category of modules over some ring. Can anybody tell me some application of Mitchell's embedding theorem so that I can understand the fruitfulness of this theorem? Thanks.

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Module categories are very well-studied. By embedding another category into some $R-\text{Mod}$, you can apply the knowledge and techniques you know about and for $R$-modules to the study of other categories.

e.g. you can make diagram chasing arguments without having to develop a full-blown theory of generalized elements (or deal with their quirks)

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  • $\begingroup$ thanks. I understand your point. can you tell me in what way module category is well studied? can you give me some particular example? thanks. $\endgroup$ – GA316 Apr 14 '13 at 16:41
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    $\begingroup$ e.g. if you grab an elementary abstract algebra textbook, it is very likely to have one or more chapters on modules over rings. Or a homological algebra textbook might talk only about (co)homology in the setting of modules over rings. It's much easier to find a lot more information about Abelian groups and modules over rings than it is to find information about abstract Abelian categories. (and the information found is generally a lot less complicated) $\endgroup$ – user14972 Apr 14 '13 at 17:29
  • $\begingroup$ Sorry. I thing, I dont tell point clearly. I have read modules and its structure theorem in Algebra book by Lang. So I know about category of R-modules. can you please tell me some examples of categories that show the use of Mitchell's theorem transparently? $\endgroup$ – GA316 Apr 14 '13 at 17:34
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    $\begingroup$ I think that a lot of the time the Mitchell embedding theorem isn't used for a given category per se. A more common use is in the case where one wants to prove a statement for an arbitrary abelian category and an proof using elements of modules is shorter and simpler than a general diagram-theoretic proof (but such a diagram proof is almost always possible). Weibel does this a couple times in his Homological Algebra book. I can't recall any use of the theorem for a specific abelian category, but I'm sure examples must exist. $\endgroup$ – tharris Apr 14 '13 at 17:47
  • $\begingroup$ ok thanks. I will refer Weibel's book. can you remember which chapter particularly. $\endgroup$ – GA316 Apr 15 '13 at 8:35
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The embedding theorem by Freyd-Mitchell (FM) is interesting in its own right. It offers a local classification of abelian categories. I write local here, because FM only refers to small abelian categories, and many interesting abelian categories are not essentially small. But this local classification may be a little bit overrated:

In my opinion it is an overkill to prove all the usual diagram lemmas using FM, because there are more direct and accessible proofs working with generalized elements, which I find more elegant because we use the abstract structure directly, and we don't have to deal with the case of modules separately. Instead we observe that the proof for modules works exactly the same way for every abelian category. The same remarks apply to further aspects of the theory, for example derived functors and spectral sequences. When reading through texts on homological algebra such as the book by Weibel, often element calculations are justified with FM. But instead we should really understand that abelian categories are precisely those abstract categories which work the same way as module categories. We don't need to use FM for that. When working with specific abelian categories, it is important to understand and manipulate its objects and morphisms directly.

One of the main points of category theory is to ignore the whole element business coming from (in my opinion old-fashioned) foundations such as ZFC and work with morphisms instead. This reveals the real nature of mathematical objects. The consequent usage of universal properties often offers one-line proofs for what else would need a whole page of clumsy element calculations (a typical example being the right exactness of the tensor product, still not treated concisely in most textbooks). The philosophy usually derived by FM ("every abelian category may be seen as a category of modules, and modules are sets with extra structure, so that we may use elements") is a little bit misleading, because there are lots of abelian categories whose embedding into any category of modules would be artificial and useless (for example the category of abelian sheaves on a space). Finally, although the proof of FM explicitly produces the ring $R$ for an abelian category $\mathcal{A}$ and the embedding $\mathcal{A} \hookrightarrow \mathsf{Mod}(R)$, this ring $R$ is terribly large and useless in most cases.

Besides, there are lots of interesting additional structures on abelian categories, and it is unclear or even wrong if there is some embedding result for them. See also MO/32173 and MO/47342. There is a preprint arXiv:math/0004160 which has an embedding theorem for abelian tensor categories, but this has some serious errors.

I doubt that there is any interesting result about abelian categories whose proof really needs FM. At least this is my impression.

Finally let me sketch what is so special (or not) about module categories: If $R$ is a ring, then $R \in \mathsf{Mod}(R)$ is a distinguished object with three nice properties: 1) It is a generator, meaning that $\hom(R,-)$ is faithful. 2) It is projective, meaning that $\hom(R,-)$ is exact. 3) It is compact, meaning that $\hom(R,-)$ preserves (infinite) coproducts. Conversely, every abelian category with coproducts and a projective compact generator $G$ is canonically equivalent to $\mathsf{Mod}(R)$, where $R:=\mathrm{End}(G)$. Although many interesting abelian categories are not equivalent to module categories, they have at least some generator $G$. It follows that every object $M$ fits into an exact sequence $G^{\oplus I} \to G^{\oplus J} \to M \to 0$. This generalizes the idea of free resolutions. In fact Grothendieck categories provide a nice framework for homological algebra.

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  • $\begingroup$ While I agree with the general spirit of this answer, I also feel obliged to admit that there is (arguably) merit in having a short path for those who have need to use Abelian categories, but do not have much category theoretic background and do not plan to go very deeply into the theory. $\endgroup$ – user14972 Apr 15 '13 at 8:39
  • $\begingroup$ since, I dont have clear idea about category theory, I can understand your last paragraph completely. any way thanks. I will read concern part in category theory first. $\endgroup$ – GA316 Apr 15 '13 at 13:36

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