Grothendieck universe
E98601
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.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
set-theoretic construct ⓘ tool in category theory ⓘ |
| advantage | allows working inside a single set that behaves like a miniature universe of sets ⓘ |
| alternativeTo |
Morse–Kelley set theory by class–set distinction
ⓘ
surface form:
Morse–Kelley set theory
von Neumann–Bernays–Gödel set theory ⓘ |
| assumption | existence is not provable in ZFC without large cardinal axioms ⓘ |
| contains |
all usual small mathematical objects used in a given context
ⓘ
empty set ⓘ finite sets of its elements ⓘ natural numbers ⓘ power sets of its elements ⓘ unions of families of its elements indexed by its elements ⓘ |
| correspondsTo | strongly inaccessible cardinal ⓘ |
| definition |
a set U such that if (x_i)_{i∈I} is a family of elements of U indexed by I ∈ U then ⋃_{i∈I} x_i ∈ U
ⓘ
a set U such that if x ∈ U and y ∈ x then y ∈ U ⓘ a set U such that if x ∈ U then P(x) ∈ U ⓘ a set U such that if x,y ∈ U then {x,y} ∈ U ⓘ |
| equivalentCondition | U is a Grothendieck universe iff U = V_κ for some strongly inaccessible κ (in ZFC + inaccessibles) ⓘ |
| field |
category theory
ⓘ
set theory ⓘ |
| formalization | often added to ZFC as an extra axiom scheme ⓘ |
| hasModel | V_κ for an inaccessible cardinal κ ⓘ |
| implies | if x ∈ U then the transitive closure of x is contained in U ⓘ |
| limitation | existence of nontrivial universes is independent of ZFC ⓘ |
| namedAfter | Alexander Grothendieck ⓘ |
| philosophicalRole | provides a relative notion of smallness and largeness in mathematics ⓘ |
| property |
closed under pairing
ⓘ
closed under power set ⓘ closed under unions of families indexed by elements of the universe ⓘ transitive set ⓘ |
| relatedConcept |
cumulative hierarchy
ⓘ
inaccessible cardinal ⓘ large cardinal ⓘ universe axiom ⓘ |
| requires | inaccessible cardinal axiom for nontrivial examples ⓘ |
| role | separates size issues from structural arguments in category theory ⓘ |
| usedFor |
avoiding set-theoretic paradoxes
ⓘ
formalizing universes of discourse in mathematics ⓘ foundations of category theory ⓘ handling large categories ⓘ |
| usedIn |
algebraic geometry
ⓘ
higher category theory ⓘ homological algebra ⓘ topos theory ⓘ Éléments de géométrie algébrique ⓘ |
| usedToDefine |
locally small category
ⓘ
small category ⓘ universe of sets for a category ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.