Mitchell's embedding theorem

The embedding theorem of Mitchell is a mathematical result on abelian categories. It says that these categories initially very abstractly defined can be regarded as quite specific categories of modules. As a result thereof may be about the method of proof used by element-wise diagram hunting in any abelian categories.

Statement of the theorem

The exact statement is: Let A be a small abelian category. Then there is a ring R and a fully faithful and exact functor F: A → R- Mod of A in the category R- Mod of left modules over R.

The functor F induces an equivalence between A and a subcategory of R -Mod. In A computed kernels and cokernels correspond via this equivalence to ordinary nuclei and Kokernen in R -Mod.

Idea of ​​proof

The idea of ​​the proof is based on the Yoneda lemma. Suppose A is already lay in R -Mod. Then provides each object X is a left exact functor HomA (X, -): A → Ab The mapping X → HomA (X, - ) then provides a duality between R- Mod and the category of left exact functors from A to Ab To R recover from A, one is therefore as follows: In the category D of the left exact functors from From a to one constructs a certain injective cogenerator H whose endomorphism ring is chosen as R. By X in each A F ( X) = HomD ( HomA (X, - ), H) is, to a functor F is then obtained with the desired properties.

Apply to broad categories

Immediately the embedding theorem of Mitchell seems to justify the method of the diagram only hunting for any small abelian categories. However, is a diagram given to any abelian category A, then, consider the smallest abelian full subcategory B from A, which contains all the objects appearing in the chart. This is a small abelian category. Clearly formulated, take the amount (!) Of the objects used in the diagram as objects of B and then adds repeated lack nuclei and cokernels of morphisms and Biprodukte of objects added.