Mitchell's embedding theorem

Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem or the full embedding theorem, is a result about abelian categories; it essentially states that these categories, while rather abstractly defined, are in fact concrete categories of modules. This allows one to use element-wise diagram chasing proofs in these categories. The theorem is named after Barry Mitchell and Peter Freyd.

Details
The precise statement is as follows: if A is a small abelian category, then there exists a ring R (with 1, not necessarily commutative) and a full, faithful and exact functor F: A → R-Mod (where the latter denotes the category of all left R-modules).

The functor F yields an equivalence between A and a full subcategory of R-Mod in such a way that kernels and cokernels computed in A correspond to the ordinary kernels and cokernels computed in R-Mod. Such an equivalence is necessarily additive. The theorem thus essentially says that the objects of A can be thought of as R-modules, and the morphisms as R-linear maps, with kernels, cokernels, exact sequences and sums of morphisms being determined as in the case of modules. However, projective and injective objects in A do not necessarily correspond to projective and injective R-modules.

Sketch of the proof
Let $$\mathcal{L} \subset \operatorname{Fun}(\mathcal{A}, Ab)$$ be the category of left exact functors from the abelian category $$\mathcal{A}$$ to the category of abelian groups $$Ab$$. First we construct a contravariant embedding $$H:\mathcal{A}\to\mathcal{L}$$ by $$H(A) = h^A$$ for all $$A\in\mathcal{A}$$, where $$h^A$$ is the covariant hom-functor, $$h^A(X)=\operatorname{Hom}_\mathcal{A}(A,X)$$. The Yoneda Lemma states that $$H$$ is fully faithful and we also get the left exactness of $$H$$ very easily because $$h^A$$ is already left exact. The proof of the right exactness of $$H$$ is harder and can be read in Swan, Lecture Notes in Mathematics 76.

After that we prove that $$\mathcal{L}$$ is an abelian category by using localization theory (also Swan). This is the hard part of the proof.

It is easy to check that the abelian category $$\mathcal{L}$$ is an AB5 category with a generator $$\bigoplus_{A\in\mathcal{A}} h^A$$. In other words it is a Grothendieck category and therefore has an injective cogenerator $$I$$.

The endomorphism ring $$R := \operatorname{Hom}_{\mathcal{L}} (I,I)$$ is the ring we need for the category of R-modules.

By $$G(B) = \operatorname{Hom}_{\mathcal{L}} (B,I)$$ we get another contravariant, exact and fully faithful embedding $$G:\mathcal{L}\to R\operatorname{-Mod}.$$ The composition $$GH:\mathcal{A}\to R\operatorname{-Mod}$$ is the desired covariant exact and fully faithful embedding.

Note that the proof of the Gabriel–Quillen embedding theorem for exact categories is almost identical.