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.
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 L ⊂ ⊂ --> Fun --> ( A , A b ) {\displaystyle {\mathcal {L}}\subset \operatorname {Fun} ({\mathcal {A}},Ab)} be the category of left exact functors from the abelian category A {\displaystyle {\mathcal {A}}} to the category of abelian groups A b {\displaystyle Ab} . First we construct a contravariant embedding H : A → → --> L {\displaystyle H:{\mathcal {A}}\to {\mathcal {L}}} by H ( A ) = h A {\displaystyle H(A)=h^{A}} for all A ∈ ∈ --> A {\displaystyle A\in {\mathcal {A}}} , where h A {\displaystyle h^{A}} is the covariant hom-functor, h A ( X ) = Hom A --> ( A , X ) {\displaystyle h^{A}(X)=\operatorname {Hom} _{\mathcal {A}}(A,X)} . The Yoneda Lemma states that H {\displaystyle H} is fully faithful and we also get the left exactness of H {\displaystyle H} very easily because h A {\displaystyle h^{A}} is already left exact. The proof of the right exactness of H {\displaystyle H} is harder and can be read in Swan, Lecture Notes in Mathematics 76.
After that we prove that L {\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 L {\displaystyle {\mathcal {L}}} is an AB5 category with a generator ⨁ ⨁ --> A ∈ ∈ --> A h A {\displaystyle \bigoplus _{A\in {\mathcal {A}}}h^{A}} . In other words it is a Grothendieck category and therefore has an injective cogenerator I {\displaystyle I} .
The endomorphism ring R := Hom L --> ( I , I ) {\displaystyle R:=\operatorname {Hom} _{\mathcal {L}}(I,I)} is the ring we need for the category of R-modules.
By G ( B ) = Hom L --> ( B , I ) {\displaystyle G(B)=\operatorname {Hom} _{\mathcal {L}}(B,I)} we get another contravariant, exact and fully faithful embedding G : L → → --> R - M o d . {\displaystyle G:{\mathcal {L}}\to R\operatorname {-Mod} .} The composition G H : A → → --> R - M o d {\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.
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