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 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 $\mathcal{L}$ has an injective cogenerator

I=\prod_{A\in\mathcal{A}} h_A.

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.

References

R. G. Swan (1968). Lecture Notes in Mathematics 76. Springer.
Peter Freyd (1964). Abelian categories. Harper and Row.
Barry Mitchell (1964). The full imbedding theorem. The Johns Hopkins University Press.
Charles A. Weibel (1993). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics.

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