Freyd–Mitchell embedding theorem

E621113

mathematical theorem theorem in category theory

The Freyd–Mitchell embedding theorem is a fundamental result in category theory stating that every small abelian category can be faithfully represented as a full subcategory of a module category, thereby allowing the use of element-wise methods in abstract settings.

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Statements (50)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in category theory ⓘ
allows	interpretation of morphisms of an abelian category as module homomorphisms ⓘ interpretation of objects of an abelian category as modules ⓘ use of element-wise arguments in abelian categories ⓘ
appliesTo	small abelian category ⓘ
assumptionOnCategory	abelian structure of the category ⓘ smallness of the abelian category ⓘ
conclusion	abelian categories can be represented as categories of modules up to full embedding ⓘ every small abelian category embeds into a module category ⓘ every small abelian category is equivalent to a full subcategory of a module category ⓘ
context	abelian category ⓘ module category ⓘ
field	category theory ⓘ
guarantees	existence of a faithful exact functor from a small abelian category to a module category ⓘ existence of a full and faithful exact embedding into a module category ⓘ
historicalPeriod	20th century mathematics ⓘ
implies	abelian categories behave like module categories for homological algebra ⓘ
involvesConcept	Ab-enriched category ⓘ Yoneda embedding NERFINISHED ⓘ additive functor ⓘ category of left modules ⓘ category of right modules ⓘ cokernel ⓘ exact functor ⓘ exact sequence ⓘ faithful functor ⓘ full functor ⓘ kernel ⓘ representable functor ⓘ ring with identity ⓘ short exact sequence ⓘ
namedAfter	Barry Mitchell NERFINISHED ⓘ Peter Freyd NERFINISHED ⓘ
propertyPreserved	cokernels ⓘ exactness of sequences ⓘ finite colimits ⓘ finite limits ⓘ kernels ⓘ
relatedTo	Gabriel–Popescu theorem NERFINISHED ⓘ Yoneda lemma NERFINISHED ⓘ embedding theorems in category theory ⓘ
strengthens	view of abelian categories as generalized module categories ⓘ
typicalTargetCategory	Mod-R NERFINISHED ⓘ category of modules over a ring ⓘ
usedIn	cohomology theories ⓘ derived categories ⓘ homological algebra ⓘ representation theory ⓘ sheaf theory ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Peter Freyd → notableWork → Freyd–Mitchell embedding theorem ⓘ

Peter Freyd → knownFor → Freyd–Mitchell embedding theorem ⓘ