Gabriel–Popescu theorem
E884933
The Gabriel–Popescu theorem is a fundamental result in category theory that characterizes Grothendieck categories as exact reflective localizations of module categories.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Gabriel–Popescu theorem canonical | 3 |
| Gabriel’s theorem on abelian categories | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10773403 — 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: Gabriel–Popescu theorem Context triple: [Grothendieck category, isCharacterizedBy, Gabriel–Popescu theorem]
-
A.
Freyd–Mitchell embedding theorem
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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B.
Freyd adjoint functor theorem
The Freyd adjoint functor theorem is a fundamental result in category theory that provides general conditions under which a functor admits a left or right adjoint, linking completeness and solution-set conditions to the existence of adjoint functors.
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C.
“Abelian Categories: An Introduction to the Theory of Functors”
“Abelian Categories: An Introduction to the Theory of Functors” is a foundational monograph in category theory that systematically develops the theory of abelian categories and functors, significantly shaping modern homological algebra.
-
D.
Grothendieck toposes
Grothendieck toposes are highly structured categories that generalize topological spaces and serve as a unifying framework for geometry, logic, and cohomology in modern mathematics.
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E.
Yoneda lemma
The Yoneda lemma is a fundamental result in category theory that characterizes objects by their sets of morphisms into them, providing a powerful bridge between abstract categories and concrete set-valued functors.
- 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: Gabriel–Popescu theorem Target entity description: The Gabriel–Popescu theorem is a fundamental result in category theory that characterizes Grothendieck categories as exact reflective localizations of module categories.
-
A.
Freyd–Mitchell embedding theorem
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.
-
B.
Freyd adjoint functor theorem
The Freyd adjoint functor theorem is a fundamental result in category theory that provides general conditions under which a functor admits a left or right adjoint, linking completeness and solution-set conditions to the existence of adjoint functors.
-
C.
“Abelian Categories: An Introduction to the Theory of Functors”
“Abelian Categories: An Introduction to the Theory of Functors” is a foundational monograph in category theory that systematically develops the theory of abelian categories and functors, significantly shaping modern homological algebra.
-
D.
Grothendieck toposes
Grothendieck toposes are highly structured categories that generalize topological spaces and serve as a unifying framework for geometry, logic, and cohomology in modern mathematics.
-
E.
Yoneda lemma
The Yoneda lemma is a fundamental result in category theory that characterizes objects by their sets of morphisms into them, providing a powerful bridge between abstract categories and concrete set-valued functors.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf | mathematical theorem ⓘ |
| appliesTo |
Grothendieck abelian categories
ⓘ
linked to:
Grothendieck category
|
| assumes |
Grothendieck category has a generator
ⓘ
linked to:
Grothendieck category
Grothendieck category satisfies AB5 ⓘ |
| characterizes |
Grothendieck categories
ⓘ
linked to:
Grothendieck category
|
| characterizesAs | exact reflective localizations of module categories ⓘ |
| consequence | Grothendieck categories can be studied via module-theoretic methods ⓘ |
| domain |
abelian categories
ⓘ
module categories ⓘ |
| field |
category theory
ⓘ
homological algebra ⓘ |
| hasGeneralization | results on localization of Grothendieck categories ⓘ |
| hasImpactOn |
modern homological algebra
ⓘ
theory of abelian categories ⓘ |
| involvesConcept |
AB5 condition
ⓘ
Grothendieck category ⓘ Serre subcategory ⓘ adjoint functors ⓘ exact embedding ⓘ exact functor ⓘ filtered colimits ⓘ fully faithful functor ⓘ generator of an abelian category ⓘ left exact functor ⓘ localization of categories ⓘ localizing subcategory ⓘ quotient of an abelian category ⓘ reflective subcategory ⓘ |
| languageOfFormulation | category-theoretic language ⓘ |
| namedAfter |
Nicolae Popescu
ⓘ
Pierre Gabriel ⓘ |
| provedBy |
Nicolae Popescu
ⓘ
Pierre Gabriel ⓘ |
| provides | representation of Grothendieck categories as localizations of module categories ⓘ |
| publishedIn |
Publications Mathématiques de l’IHÉS
ⓘ
linked to:
Publ. Math. IHÉS
|
| relatedTo |
Freyd–Mitchell embedding theorem
ⓘ
Gabriel’s theorem on abelian categories ⓘ
linked to:
Gabriel–Popescu theorem
localization of abelian categories ⓘ |
| statesThat |
a Grothendieck category is equivalent to the category of modules over a ring localized at a localizing subcategory
ⓘ
every Grothendieck abelian category is a localization of a module category ⓘ every Grothendieck category is equivalent to a reflective localization of a module category ⓘ there exists a fully faithful exact functor from a Grothendieck category into a module category ⓘ |
| usedFor | reducing problems in Grothendieck categories to problems in module categories ⓘ |
| usedIn |
algebraic geometry
ⓘ
homological algebra ⓘ representation theory ⓘ |
| yearProved | 1964 ⓘ |
How these facts were elicited
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Input
Subject: Gabriel–Popescu theorem Description of subject: The Gabriel–Popescu theorem is a fundamental result in category theory that characterizes Grothendieck categories as exact reflective localizations of module categories.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Gabriel–Popescu theorem