Grothendieck category

E254135

abelian category category-theoretic structure mathematical concept

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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All labels observed (4)

Label	Occurrences
Grothendieck categories	3
Grothendieck category canonical	2
Grothendieck abelian categories	1
Grothendieck category has a generator	1

How this entity was disambiguated

This entity first appeared as the object of triple T2290656 — 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: Grothendieck category
Context triple: [Alexander Grothendieck, notableConcept, Grothendieck category]

A. 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.
B. Categories for the Working Mathematician
Categories for the Working Mathematician is a foundational textbook in category theory that systematically develops the subject and its applications for professional mathematicians.
C. Sheaves in Geometry and Logic
Sheaves in Geometry and Logic is a foundational monograph that develops the theory of sheaves and topos theory and explores their deep connections to geometry, logic, and the foundations of mathematics.
D. Alexandrov–Čech cohomology
Alexandrov–Čech cohomology is a topological cohomology theory that computes invariants of spaces using inverse limits over open covers, closely related to and often coinciding with sheaf cohomology.
E. 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.
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: Grothendieck category
Target entity description: 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.

A. 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.
B. Categories for the Working Mathematician
Categories for the Working Mathematician is a foundational textbook in category theory that systematically develops the subject and its applications for professional mathematicians.
C. Sheaves in Geometry and Logic
Sheaves in Geometry and Logic is a foundational monograph that develops the theory of sheaves and topos theory and explores their deep connections to geometry, logic, and the foundations of mathematics.
D. Alexandrov–Čech cohomology
Alexandrov–Čech cohomology is a topological cohomology theory that computes invariants of spaces using inverse limits over open covers, closely related to and often coinciding with sheaf cohomology.
E. 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.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	abelian category ⓘ category-theoretic structure ⓘ mathematical concept ⓘ
appearsIn	EGA (Éléments de Géométrie Algébrique) ⓘ linked to: Éléments de géométrie algébrique SGA (Séminaire de Géométrie Algébrique) ⓘ linked to: Séminaire de Géométrie Algébrique du Bois Marie
definitionCondition	every object is a quotient of a coproduct of copies of a generator ⓘ filtered colimits of exact sequences are exact ⓘ is an abelian category with exact filtered colimits and a generator ⓘ
generalizes	category of modules over a ring ⓘ category of presheaves of abelian groups ⓘ category of quasi-coherent sheaves on a scheme ⓘ category of sheaves of abelian groups on a site ⓘ
hasFeature	enables definition of derived functors ⓘ exactness of direct limits of short exact sequences ⓘ existence of enough colimits for homological constructions ⓘ supports derived categories ⓘ supports injective resolutions ⓘ supports spectral sequences ⓘ
hasProperty	AB5 category ⓘ exactness of filtered colimits ⓘ has a generator ⓘ has arbitrary coproducts ⓘ has enough injectives ⓘ has exact filtered colimits ⓘ has small colimits ⓘ is AB3 category ⓘ is AB4 category ⓘ is AB5 category with generator ⓘ is cocomplete ⓘ is complete with respect to small limits ⓘ is well-powered in subobjects ⓘ local presentability (under mild set-theoretic assumptions) ⓘ
implies	existence of derived functors of left exact functors ⓘ existence of enough injective objects ⓘ existence of injective envelopes (under mild conditions) ⓘ
isCharacterizedBy	Gabriel–Popescu theorem ⓘ
isSubClassOf	AB5 category with generator ⓘ cocomplete abelian category ⓘ
namedAfter	Alexander Grothendieck ⓘ
relatedTo	AB5 abelian category ⓘ Gabriel localization theory ⓘ Grothendieck topos ⓘ linked to: Grothendieck toposes locally presentable category ⓘ
usedIn	algebraic geometry ⓘ cohomology theories ⓘ derived category theory ⓘ homological algebra ⓘ representation theory ⓘ

How these facts were elicited