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. History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... In mathematics, an abelian category is a certain kind of category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ... In mathematics, a concrete category is a category in which, roughly speaking, all objects are sets possibly carrying some additional structure, all morphisms are functions between those sets, and the composition of morphisms is the composition of functions. ... In abstract algebra, a module is a generalization of a vector space. ...

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). In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In category theory, a full functor is a functor which is surjective when restricted to each set of morphisms with a given source and target. ... In category theory, a faithful functor is a functor which is injective when restricted to each set of morphisms with a given source and target. ... In homological algebra, an exact functor is one which preserves exact sequences. ... In category theory, a functor is a special type of mapping between categories. ...

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. A subcategory in Wikipedia is a category that depends on another category. ... In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are essentially the same. There are numerous examples of categorical equivalences from many areas of mathematics. ... In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. ... In abstract algebra, the cokernel of a homomorphism f : X → Y is the quotient of Y by the image of f. ...

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. In mathematics, the Yoneda lemma in category theory is an abstract result on functors of the type morphisms into a fixed object. ... In homological algebra, an exact functor is one which preserves exact sequences. ... The word duality has a variety of different meanings in different contexts: In mathematics, see duality (mathematics). ... In category theory, the concept of an injective cogenerator is drawn from examples such as Pontryagin duality. ...