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• Mitchells_embedding_theorem abstract "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 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 objects of A can be thought of as R-modules, and the morphisms as R-linear maps, with kernels, cokernels, exact sequences and sums of morphisms being determined as in the case of modules. However, projective and injective objects in A do not necessarily correspond to projective and injective R-modules.".
• Mitchells_embedding_theorem wikiPageID "453755".
• Mitchells_embedding_theorem wikiPageLength "4066".
• Mitchells_embedding_theorem wikiPageOutDegree "29".
• Mitchells_embedding_theorem wikiPageRevisionID "635474609".
• Mitchells_embedding_theorem wikiPageWikiLink Abelian_category.
• Mitchells_embedding_theorem wikiPageWikiLink Category:Additive_categories.
• Mitchells_embedding_theorem wikiPageWikiLink Category:Module_theory.
• Mitchells_embedding_theorem wikiPageWikiLink Category:Theorems_in_algebra.
• Mitchells_embedding_theorem wikiPageWikiLink Category_of_abelian_groups.
• Mitchells_embedding_theorem wikiPageWikiLink Cokernel.
• Mitchells_embedding_theorem wikiPageWikiLink Commutative_diagram.
• Mitchells_embedding_theorem wikiPageWikiLink Concrete_category.
• Mitchells_embedding_theorem wikiPageWikiLink Endomorphism_ring.
• Mitchells_embedding_theorem wikiPageWikiLink Equivalence_of_categories.
• Mitchells_embedding_theorem wikiPageWikiLink Exact_category.
• Mitchells_embedding_theorem wikiPageWikiLink Exact_functor.
• Mitchells_embedding_theorem wikiPageWikiLink Exact_sequence.
• Mitchells_embedding_theorem wikiPageWikiLink Full_and_faithful_functors.
• Mitchells_embedding_theorem wikiPageWikiLink Functor.
• Mitchells_embedding_theorem wikiPageWikiLink Gabriel-Quillen_embedding_theorem.
• Mitchells_embedding_theorem wikiPageWikiLink Injective_cogenerator.
• Mitchells_embedding_theorem wikiPageWikiLink Injective_object.
• Mitchells_embedding_theorem wikiPageWikiLink Kernel_(category_theory).
• Mitchells_embedding_theorem wikiPageWikiLink Module_(mathematics).
• Mitchells_embedding_theorem wikiPageWikiLink Preadditive_category.
• Mitchells_embedding_theorem wikiPageWikiLink Projective_object.
• Mitchells_embedding_theorem wikiPageWikiLink Ring_(mathematics).
• Mitchells_embedding_theorem wikiPageWikiLink Subcategory.
• Mitchells_embedding_theorem wikiPageWikiLink Yoneda_lemma.
• Mitchells_embedding_theorem wikiPageWikiLinkText "Freyd–Mitchell theorem".
• Mitchells_embedding_theorem wikiPageWikiLinkText "Mitchell embedding theorem".
• Mitchells_embedding_theorem wikiPageWikiLinkText "Mitchell's embedding theorem".
• Mitchells_embedding_theorem wikiPageWikiLinkText "Mitchell–Freyd embedding theorem".
• Mitchells_embedding_theorem wikiPageWikiLinkText "theorem by Barry Mitchell".
• Mitchells_embedding_theorem wikiPageUsesTemplate Template:Cite_book.
• Mitchells_embedding_theorem wikiPageUsesTemplate Template:Refbegin.
• Mitchells_embedding_theorem wikiPageUsesTemplate Template:Refend.
• Mitchells_embedding_theorem subject Category:Additive_categories.
• Mitchells_embedding_theorem subject Category:Module_theory.
• Mitchells_embedding_theorem subject Category:Theorems_in_algebra.
• Mitchells_embedding_theorem hypernym Result.
• Mitchells_embedding_theorem type Theorem.
• Mitchells_embedding_theorem comment "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.".
• Mitchells_embedding_theorem label "Mitchell's embedding theorem".
• Mitchells_embedding_theorem sameAs Q1148215.
• Mitchells_embedding_theorem sameAs Einbettungssatz_von_Mitchell.
• Mitchells_embedding_theorem sameAs Théorème_de_plongement_de_Mitchell.
• Mitchells_embedding_theorem sameAs m.02bdr1.
• Mitchells_embedding_theorem sameAs Q1148215.
• Mitchells_embedding_theorem wasDerivedFrom Mitchells_embedding_theorem?oldid=635474609.
• Mitchells_embedding_theorem isPrimaryTopicOf Mitchells_embedding_theorem.