Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem or the full embedding theorem, is a result about small abelian categories; it states that these categories, while abstractly defined, can be represented as concrete categories whose objects are modules. In particular, the result allows one to use element-wise diagram chasing proofs in abelian 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 (with1, 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

be the category of left exact functors to the category of abelian groups. First we construct a contravariant. The Yoneda Lemma is harder and can be read in Swan, Lecture Notes in Mathematics 76.

is an abelian category by using localization theory (also Swan). This is the hard part of the proof.

is an AB5 category with a generator. In other words it is a Grothendieck category.

The endomorphism ring is the ring we need for the category of R-modules.

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

• reprinted with a forward as