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 (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 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.

  • Swan, R. G. (1968). Algebraic K-theory, Lecture Notes in Mathematics 76. Springer. doi:. ISBN 978-3-540-04245-7.
  • Freyd, Peter (1964). . Harper and Row. reprinted with a forward as . Reprints in Theory and Applications of Categories. 3: 23–164. 2003.
  • Mitchell, Barry (July 1964). "The Full Imbedding Theorem". American Journal of Mathematics. 86 (3). The Johns Hopkins University Press: 619–637. doi:. JSTOR .
  • Weibel, Charles A. (1993). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. doi:. ISBN 9781139644136.