Freyd–Mitchell embedding theorem
E621113
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Freyd–Mitchell embedding theorem canonical | 2 |
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]
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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.
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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.
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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.
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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.
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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 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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Freyd–Mitchell embedding theorem Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.