Mitchell's embedding theorem

Mitchell's embedding theorem, also known as the Freyd-Mitchell theorem, is a result saying that every abelian category admits a full and exact embedding into the category of R-modules. This allows one to use element-wise diagram chasing proofs in arbitrary abelian categories.

Applications

Using the theorem we can treat every abelian category as if it is the category of R-modules concerning theorems about existence of morphisms in a diagram and commutativity and exatness of diagrams. Category theory gets much more concrete by this embedding theorem.

Sketch of the proof

First we construct an embedding from an abelian category ${\mathcal {A}}$ to the category ${\mathcal {L}}=L({\mathcal {A}},Ab)\subset Fun({\mathcal {A}},Ab)$ of left exact functors from the abelian category ${\mathcal {A}}$ to the category of abelian groups $Ab$ through the functor $H$ by $H(A)=h_{A}$ for all $A\in {\mathcal {A}}$, where $h_{A}$ is the covariant hom-functor. The Yoneda Lemma states that $H$ is fully faithful and we also get the left exactness very easy because $h_{A}$ is already left exact. The proof of the right exactness is harder and can be read in Swan, Lecture notes on mathematics 76.

After that we proo that ${\mathcal {L}}$ is abelian by using localization theory (also Swan). ${\mathcal {L}}$ also has enough projective objects and a generator. This follows easily from $Fun({\mathcal {A}},Ab)$ having these properties.

By taking the dual category of ${\mathcal {L}}$ which we call ${\mathcal {L}}^{op}$ we get an exact and fully faithful embedding from our category ${\mathcal {C}}$ to an abelian category which has enough projective objects and a cogenerator.

We can then cunstruct a projective cogenerator $P$ which leads us via $R:=Hom_{{\mathcal {L}}^{op}}(P,P)$ to the ring we need for the category of R-modules.

By $T(B)=Hom_{{\mathcal {L}}^{op}}(P,B)$ we get an exact and fully faithful embedding from ${\mathcal {L}}^{op}$ to the category of R-modules.

References

• R. G. Swan (1968). Lecture Notes in Mathematics 76. Springer.

• Peter Freyd (19684). Abelian categories. Harper and Row. {{cite book}}: Check date values in: |year= (help)CS1 maint: year (link)

• Barry Mitchell (1964). The full imbedding theorem. The John Hopkins University Press.

• Charles A. Weibel (1993). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics.