Mitchell 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.

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.

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