Grothendieck toposes

E621112

category-theoretic structure mathematical concept

Grothendieck toposes are highly structured categories that generalize topological spaces and serve as a unifying framework for geometry, logic, and cohomology in modern mathematics.

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Observed surface forms (3)

Surface form	Occurrences
Grothendieck topos	2
Grothendieck topos theory	1
Topos Theory	1

Statements (49)

Predicate	Object
instanceOf	category-theoretic structure ⓘ mathematical concept ⓘ
appearsIn	SGA 4 NERFINISHED ⓘ Séminaire de Géométrie Algébrique (SGA) NERFINISHED ⓘ
definedAs	categories equivalent to categories of sheaves on a site ⓘ categories of sheaves of sets on a Grothendieck site ⓘ
developedBy	Alexander Grothendieck NERFINISHED ⓘ
developedIn	1960s ⓘ
enables	comparison of different cohomology theories ⓘ relative cohomology theories ⓘ
field	algebraic geometry ⓘ category theory ⓘ cohomology theory ⓘ homological algebra ⓘ mathematical logic ⓘ topos theory ⓘ
generalizes	categories of sheaves ⓘ sites ⓘ topological spaces ⓘ
hasExample	the category of sheaves on a topological space ⓘ the classifying topos of a theory ⓘ the topos of sheaves on a site of open sets ⓘ the étale topos of a scheme ⓘ
hasProperty	has a subobject classifier ⓘ has all small colimits ⓘ is a category with finite limits ⓘ is cartesian closed ⓘ is exact ⓘ is extensive ⓘ supports internal logic ⓘ supports intuitionistic higher-order logic ⓘ
namedAfter	Alexander Grothendieck NERFINISHED ⓘ
relatedTo	Grothendieck topologies NERFINISHED ⓘ classifying toposes ⓘ elementary toposes ⓘ geometric morphisms ⓘ sheaf theory ⓘ sites ⓘ syntactic toposes ⓘ étale toposes ⓘ
supports	cohomology with values in sheaves ⓘ derived functor cohomology ⓘ geometric morphisms between toposes ⓘ
usedFor	defining cohomology theories ⓘ interpreting higher-level set theory ⓘ interpreting intuitionistic logic ⓘ studying descent theory ⓘ studying étale cohomology ⓘ unifying geometry and logic ⓘ

Referenced by (5)

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

Sheaves in Geometry and Logic → subject → Grothendieck toposes ⓘ

Sheaves in Geometry and Logic → relatedTo → Grothendieck toposes ⓘ

this entity surface form: Topos Theory

Grothendieck topology → hasKeyNotion → Grothendieck toposes ⓘ

this entity surface form: Grothendieck topos

Grothendieck topology → centralTo → Grothendieck toposes ⓘ

this entity surface form: Grothendieck topos theory

Grothendieck category → relatedTo → Grothendieck toposes ⓘ

this entity surface form: Grothendieck topos