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Mitchell's Embedding Theorem
Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem or the full embedding theorem, is a result about abelian categories; it essentially states that these categories, while rather abstractly defined, are in fact concrete categories of modules. This allows one to use element-wise diagram chasing proofs in these categories. The theorem is named after Barry Mitchell and Peter Freyd. Details The precise statement is as follows: if A is a small abelian category, then there exists a ring ''R'' (with 1, not necessarily commutative) and a full, faithful and exact functor ''F'': A → ''R''-Mod (where the latter denotes the category of all left ''R''-modules). The functor ''F'' yields an equivalence between A and a full 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. Such an equivalence is necessarily additive. The theorem thus essentially says that the objec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 prototypical example of an abelian category is the category of abelian groups, Ab. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very ''stable'' categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories are na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |