Sajun.org
In
mathematics, '''Mitchell's embedding theorem''' is an important result about
abelian categories; it states that these categories, while rather abstractly defined, are all quite
concrete categories of
modules. This allows one to use element-wise
diagram chasing proofs in arbitrary abelian categories.
The precise statement is as follows: if '''A''' is a small abelian category, then there exists a
ring ''R'' and a
full,
faithful and
exact functor ''F'' : '''A''' → ''R''-Mod (where the latter describes the abelian category of all left modules over ''R'').
The functor ''F'' identifies '''A''' with a
subcategory of ''R''-Mod: ''F'' yields an
equivalence between '''A''' and a subcategory of ''R''-Mod in such a way that
kernels and
cokernels computed in '''A''' correspond to the ordinary kernels and cokernels computed in ''R''-Mod.
The proof idea is suggested by the
Yoneda lemma. Let's assume '''A''' sits inside ''R''-Mod. Then every module ''X'' in ''R''-Mod yields a
left exact functor Hom
'''A'''(''X'',-) : '''A''' → '''Ab''', and assigning ''X'' to Hom
'''A'''(''X'',-) yields a
duality between ''R''-Mod and a subcategory of the category of all left exact functors from '''A''' to '''Ab'''. To recover ''R''-Mod from '''A''', we therefore proceed as follows:
in the category '''D''' of all left-exact functors from '''A''' to '''Ab''' we can construct a certain
injective cogenerator ''H'' whose endomorphism ring we call ''R''. Then for every ''A'' in '''A''' we can define ''F''(''A'') = Hom
'''D'''(Hom
'''A'''(''A'',-),''H''), and ''F'' is a functor from '''A''' to ''R''-Mod with the required properties.
