Mitchell's embedding theorem

In mathematics, Mitchell's embedding theorem is an important result about abelian categories; it states that these categories, while rather abstractly defined, are all quite concrete categories of modules. This allows to use element-wise diagram chasing proofs also in arbitrary abelian categories.

The precise statement is as follows: if A is a small abelian category, then there exists a ring R and a full, faithful and exact functor F: A → R-Mod (where the latter describes the abelian category of all left modules over R).

The functor F allows to identify A with a subcategory of R-Mod: F yields an equivalence between A and a 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.

The proof idea is suggested by the Yoneda lemma. Let's assume A sits inside R-Mod. Then every module X in R-Mod yields a left exact functor Hom_A(X,-) : A → Ab, and assigning X to Hom_A(X,-) yields a duality between R-Mod and a subcategory of the category of all left exact functors from A to Ab. To recover R-Mod from A, we therefore proceed as follows: in the category D of all left-exact functors from A to Ab we can construct a certain injective cogenerator H whose endomorphism ring we call R. Then for every A in A we can define F(A) = Hom_D(Hom_A(A,-),H), and F is a functor from A to R-Mod with the required properties.