Mitchell embedding theorem

The embedding theorem of Mitchell is a mathematical result on Abelian categories . It states that these initially very abstract categories can be understood as concrete categories of modules . As a consequence of this, the proof procedure by element-by-element diagram hunting may be used in any Abelian categories. The set is named after Barry Mitchell .

Statement of the sentence

The exact statement is: Be a small Abelian category. Then there is a ring and a fully faithful and exact functor from over into the category of links modules . ${\displaystyle \mathbf {A} }$ ${\displaystyle R}$ ${\displaystyle F\colon \mathbf {A} \to R{\mbox{-}}Mod}$ ${\displaystyle \mathbf {A} }$ ${\displaystyle R{\mbox{-}}Mod}$ ${\displaystyle R}$

The functor induces an equivalence between and a subcategory of . In calculated cores and coke cores correspond to the ordinary cores and coke cores in . ${\displaystyle F}$ ${\displaystyle \mathbf {A} }$ ${\displaystyle R{\mbox{-}}Mod}$ ${\displaystyle \mathbf {A} }$ ${\displaystyle R{\mbox{-}}Mod}$

Proof idea

The idea of ​​proof is based on the Yoneda lemma . Assumed would already be in . Then every object delivers a left exact functor . The assignment then provides a duality between and the category of left exact functors from to . In order to recover from , one proceeds as follows: In the category of the left exact functors from to one constructs a certain injective cogenerator , whose endomorphism ring one chooses as. By for in each set, then obtained a functor with the desired properties. ${\displaystyle \mathbf {A} }$ ${\displaystyle R{\mbox{-}}Mod}$ ${\displaystyle X}$ ${\displaystyle \mathrm {Hom} _{\mathbf {A} }(X,-):\mathbf {A} \to \mathbf {Ab} }$ ${\displaystyle X\to \mathrm {Hom} _{\mathbf {A} }(X,-)}$ ${\displaystyle R{\mbox{-}}Mod}$ ${\displaystyle \mathbf {A} }$ ${\displaystyle \mathbf {Ab} }$ ${\displaystyle R}$ ${\displaystyle \mathbf {A} }$ ${\displaystyle {\mathbf {D} }}$ ${\displaystyle \mathbf {A} }$ ${\displaystyle \mathbf {Ab} }$ ${\displaystyle H}$ ${\displaystyle R}$ ${\displaystyle X}$ ${\displaystyle \mathbf {A} }$ ${\displaystyle F(X)=\mathrm {Hom} _{\mathbf {D} }(\mathrm {Hom} _{\mathbf {A} }(X,-),H)}$ ${\displaystyle F}$

Application to large categories

Immediately, Mitchell's embedding theorem seems to justify the chart hunt procedure only for all small Abelian categories. If, however, a diagram is given for an arbitrary Abelian category , consider the smallest Abelian full sub-category of , which contains all objects appearing in the diagram. This is a small Abelian category. To put it clearly, one takes the amount (!) Of objects used in the diagram as objects and then repeatedly adds missing cores and coke cores of morphisms and biproducts of objects. ${\displaystyle \mathbf {A} }$ ${\displaystyle {\mathbf {B} }}$ ${\displaystyle \mathbf {A} }$ ${\displaystyle \mathbf {A} }$

literature

• Mitchell's embedding theorem. In: PlanetMath. Retrieved October 10, 2010 . 
• B. Mitchell: The Full Embedding Theorem . In: American Journal of Math . tape 86 , 1964, pp. 619-637 (English).