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.

Try in SPARQL Jump to: Surface forms Statements Referenced by

All labels observed (1)

Label	Occurrences
Grothendieck category canonical	1

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.

Alexander Grothendieck → notableConcept → Grothendieck category ⓘ