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.