Mitchell's embedding theorem

Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem, states that to every small abelian category ${\mathcal {A}}$ there exists a ring R (associative, with 1) and a a full, faithful and exact embedding of ${\mathcal {A}}$ into the category of R-modules.

The theorem essentially says that the objects of ${\mathcal {A}}$ can be thought of as R-modules, and the morphisms as R-linear maps, with kernels and cokernels being computed as in the case of modules. However, projective and injective objects in ${\mathcal {A}}$ do not necessarily correspond to projective and injective R-modules.

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. The theorem allows one to use element-wise diagram chasing proofs in arbitrary abelian categories. Category theory gets much more concrete by this embedding theorem.

Sketch of the proof

Let ${\mathcal {L}}\subset \operatorname {Fun} ({\mathcal {A}},Ab)$ be the category of left exact functors from the abelian category ${\mathcal {A}}$ to the category of abelian groups $Ab$. First we construct a contravariant embedding $H:{\mathcal {A}}\to {\mathcal {L}}$ by $H(A)=h_{A}$ for all $A\in {\mathcal {A}}$, where $h_{A}$ is the covariant hom-functor, $h_{A}(X)=\operatorname {Hom} _{\mathcal {A}}(A,X)$. 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 $\operatorname {Fun} ({\mathcal {A}},Ab)$ 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 ${\mathcal {A}}$ to an abelian category which has enough projective objects and a cogenerator.

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

By $T(B)=\operatorname {Hom} _{{\mathcal {L}}^{op}}(P,B)$ 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.