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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All labels observed (1)

Label	Occurrences
Freyd–Mitchell embedding theorem canonical	5

How this entity was disambiguated

This entity first appeared as the object of triple T6834094 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Freyd–Mitchell embedding theorem
Context triple: [Peter Freyd, notableWork, Freyd–Mitchell embedding theorem]

A. Grothendieck category
A Grothendieck category is an abelian category with exact filtered colimits and a generator, providing a highly general framework that extends the properties of module and sheaf categories in homological algebra.
B. Grothendieck spectral sequence
The Grothendieck spectral sequence is a fundamental tool in homological algebra that relates the derived functors of a composite functor to the derived functors of its components, enabling efficient computation of cohomology.
C. Auslander–Buchsbaum formula
The Auslander–Buchsbaum formula is a fundamental result in commutative algebra that relates the projective dimension of a finitely generated module over a Noetherian local ring to the depth of the module and the depth of the ring.
D. Grothendieck universe
A Grothendieck universe is a set-theoretic construct large enough to contain all the usual objects and operations of mathematics, used to rigorously handle "large" categories while avoiding paradoxes.
E. Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Freyd–Mitchell embedding theorem
Target entity description: 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.

A. Grothendieck category
A Grothendieck category is an abelian category with exact filtered colimits and a generator, providing a highly general framework that extends the properties of module and sheaf categories in homological algebra.
B. Grothendieck spectral sequence
The Grothendieck spectral sequence is a fundamental tool in homological algebra that relates the derived functors of a composite functor to the derived functors of its components, enabling efficient computation of cohomology.
C. Auslander–Buchsbaum formula
The Auslander–Buchsbaum formula is a fundamental result in commutative algebra that relates the projective dimension of a finitely generated module over a Noetherian local ring to the depth of the module and the depth of the ring.
D. Grothendieck universe
A Grothendieck universe is a set-theoretic construct large enough to contain all the usual objects and operations of mathematics, used to rigorously handle "large" categories while avoiding paradoxes.
E. Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
F. None of above. chosen

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 ⓘ 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 ⓘ Peter Freyd ⓘ
propertyPreserved	cokernels ⓘ exactness of sequences ⓘ finite colimits ⓘ finite limits ⓘ kernels ⓘ
relatedTo	Gabriel–Popescu theorem ⓘ Yoneda lemma ⓘ embedding theorems in category theory ⓘ
strengthens	view of abelian categories as generalized module categories ⓘ
typicalTargetCategory	Mod-R ⓘ category of modules over a ring ⓘ
usedIn	cohomology theories ⓘ derived categories ⓘ homological algebra ⓘ representation theory ⓘ sheaf theory ⓘ

How these facts were elicited

Referenced by (5)

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 ⓘ

Peter Freyd → notableWork → Freyd–Mitchell embedding theorem ⓘ

subject linked to: Freyd