Grothendieck category
E254135
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Grothendieck category canonical | 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 |
Éléments de géométrie algébrique
ⓘ
surface form:
EGA (Éléments de Géométrie Algébrique)
Séminaire de Géométrie Algébrique du Bois Marie ⓘ
surface form:
SGA (Séminaire de Géométrie Algébrique)
|
| 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 toposes ⓘ
surface form:
Grothendieck topos
locally presentable category ⓘ |
| usedIn |
algebraic geometry
ⓘ
cohomology theories ⓘ derived category theory ⓘ homological algebra ⓘ representation 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: Grothendieck category Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.