Mitchell's embedding theorem

Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem, is a result stating 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 exactness 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 easily 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 prove that $\mathcal{L}$ is abelian by using localization theory (also Swan). $\mathcal{L}$ also has enough injective 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 construct 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 (1964). Abelian categories. Harper and Row.
Barry Mitchell (1964). The full imbedding theorem. The Johns Hopkins University Press.
Charles A. Weibel (1993). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics.