Mitchell's embedding theorem
Encyclopedia
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.
to the category 
of left exact functors from the abelian category 
to the category of abelian groups 
through the functor 
by 
for all 
, where 
is the covariant hom-functor. The Yoneda Lemma states that 
is fully faithful and we also get the left exactness very easily because 
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
is abelian by using localization theory (also Swan). 
also has enough injective objects and a generator. This follows easily from 
having these properties.
By taking the dual category of
which we call 
we get an exact and fully faithful embedding from our category 
to an abelian category which has enough projective objects and a cogenerator.
We can then construct a projective cogenerator
which leads us via 
to the ring we need for the category of R-modules.
By
we get an exact and fully faithful embedding from 
to the category of R-modules.
Abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
admits a full and exact embedding into the category of R-modules
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
. 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 to the category of left exact functors from the abelian category to the category of abelian groups through the functor by for all , where is the covariant hom-functor. The Yoneda Lemma states that is fully faithful and we also get the left exactness very easily because 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
is abelian by using localization theory (also Swan). also has enough injective objects and a generator. This follows easily from having these properties.By taking the dual category of
which we call we get an exact and fully faithful embedding from our category to an abelian category which has enough projective objects and a cogenerator.We can then construct a projective cogenerator
which leads us via to the ring we need for the category of R-modules.By
we get an exact and fully faithful embedding from to the category of R-modules.