# Mitchell's embedding theorem

__: Abelian categories, while abstractly defined, are in fact concrete categories of modules__

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

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

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

By [math]\displaystyle{ G(B) = \operatorname{Hom}_{\mathcal{L}} (B,I) }[/math] we get another contravariant, exact and fully faithful embedding [math]\displaystyle{ G:\mathcal{L}\to R\operatorname{-Mod}. }[/math] The composition [math]\displaystyle{ GH:\mathcal{A}\to R\operatorname{-Mod} }[/math] 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).
*Algebraic K-theory, Lecture Notes in Mathematics 76*. Springer. - Peter Freyd (1964).
*Abelian Categories: An Introduction to the Theory of Functors*. Harper and Row. https://archive.org/details/abeliancategorie00frey. reprinted with a forward as "Abelian Categories".*Reprints in Theory and Applications of Categories***3**: 23-164. 2003. http://www.emis.de/journals/TAC/reprints/articles/3/tr3abs.html. - Mitchell, Barry (July 1964). "The Full Imbedding Theorem".
*American Journal of Mathematics*(The Johns Hopkins University Press)**86**(3): 619-637. doi:10.2307/2373027. - Charles A. Weibel (1993).
*An introduction to homological algebra*. Cambridge Studies in Advanced Mathematics.

Original source: https://en.wikipedia.org/wiki/Mitchell's embedding theorem.
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