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

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 ⓘ

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 ⓘ

this entity surface form: étale topology