Grothendieck topology

E254130

categorical structure mathematical concept

A Grothendieck topology is an abstract framework in category theory that generalizes the notion of open covers in topology to define sheaves on arbitrary categories.

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All labels observed (5)

Label	Occurrences
Grothendieck topology canonical	2
étale topology	2
Grothendieck pretopology	1
Grothendieck topos	1
fppf topology	1

How this entity was disambiguated

This entity first appeared as the object of triple T2290649 — 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 topology
Context triple: [Alexander Grothendieck, notableConcept, Grothendieck topology]

A. 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.
B. 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.
C. 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.
D. Zariski topology
The Zariski topology is a fundamental topology in algebraic geometry, defined on the spectrum of a ring or an algebraic variety, whose closed sets correspond to solution sets of polynomial equations.
E. 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.
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 topology
Target entity description: A Grothendieck topology is an abstract framework in category theory that generalizes the notion of open covers in topology to define sheaves on arbitrary categories.

A. 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.
B. 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.
C. 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.
D. Zariski topology
The Zariski topology is a fundamental topology in algebraic geometry, defined on the spectrum of a ring or an algebraic variety, whose closed sets correspond to solution sets of polynomial equations.
E. 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.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	categorical structure ⓘ mathematical concept ⓘ
appearsIn	higher category theory ⓘ homotopical algebra ⓘ motivic homotopy theory ⓘ
appliesTo	arbitrary small category ⓘ category of manifolds ⓘ category of schemes ⓘ category of topological spaces ⓘ
centralTo	Grothendieck toposes ⓘ surface form: Grothendieck topos theory theory of sheaves on sites ⓘ
characterizedBy	assignment of covering sieves to each object of a category ⓘ local character of covering sieves ⓘ maximal sieve is covering ⓘ stability of covering sieves under pullback ⓘ
definedOn	category ⓘ
enables	construction of Grothendieck toposes ⓘ definition of sheaf cohomology on categories ⓘ
field	algebraic geometry ⓘ category theory ⓘ
formalizedAs	collection of covering sieves satisfying axioms ⓘ
generalizes	Nisnevich topology ⓘ Zariski topology ⓘ Grothendieck topology self-linksurface differs ⓘ surface form: fppf topology fpqc topology ⓘ open cover in topology ⓘ Grothendieck topology self-linksurface differs ⓘ surface form: étale topology
hasAlternativeFormulation	Grothendieck topology self-linksurface differs ⓘ surface form: Grothendieck pretopology
hasKeyNotion	Grothendieck toposes ⓘ surface form: Grothendieck topos covering sieve ⓘ sieve ⓘ site ⓘ
introducedInContextOf	foundations of algebraic geometry ⓘ Éléments de géométrie algébrique ⓘ
isAbstractionOf	notion of open cover ⓘ notion of open set ⓘ
namedAfter	Alexander Grothendieck ⓘ
relatedTo	presheaf ⓘ pretopology ⓘ sheaf ⓘ site of definition ⓘ topological space ⓘ
requires	locality axiom ⓘ pullbacks of covering families ⓘ transitivity axiom ⓘ
usedFor	defining sheaves on a category ⓘ defining sites ⓘ defining toposes ⓘ

How these facts were elicited

Referenced by (7)

Full triples — surface form annotated when it differs from this entity's canonical label.

Alexander Grothendieck → notableConcept → Grothendieck topology ⓘ

this entity surface form: Grothendieck topos

étale cohomology → definedUsing → Grothendieck topology ⓘ

Grothendieck topology → generalizes → Grothendieck topology self-linksurface differs ⓘ

this entity surface form: étale topology

Grothendieck topology → generalizes → Grothendieck topology self-linksurface differs ⓘ

this entity surface form: fppf topology

Grothendieck topology → hasAlternativeFormulation → Grothendieck topology self-linksurface differs ⓘ

this entity surface form: Grothendieck pretopology

Grothendieck–Ogg–Shafarevich formula → language → Grothendieck topology ⓘ