Mitchell's embedding theorem

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.

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

After that we prove that ${\displaystyle {\mathcal {L}}}$ 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 ${\displaystyle {\mathcal {L}}}$ is an AB5 category with a generator ${\displaystyle \bigoplus _{A\in {\mathcal {A}}}h^{A}}$. In other words it is a Grothendieck category and therefore has an injective cogenerator ${\displaystyle I}$.

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

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