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 cummutativity and exatness of diagrams. Category theory gets much more concrete by this embedding theorem.
Proof
The proof is constructive and uses the Yoneda Lemma. First an abelian category is embedded in the category of left exact functors by using the functor that sends an object to the covariant hom-functor. This embedding is fully faithful because of the Yoneda Lemma. The exactness of this embedding is complicated and uses localization theory.
The category of left exact functors is abelian, has a generator and enough injective objects, so its dual is also abelian, has a cogenerator and enough projective objects. By defining a ring by taking morphisms from the cogenerator to itself, we obtain the category of R-modules. The embedding into the category of R-modules is also exact and fully faithful.
References
- R. G. Swan (1968). Lecture Notes in Mathematics 76. Springer.
- Peter Freyd (19684). Abelian categories. Harper and Row.
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- 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.