Definition:Grothendieck Universe

From ProofWiki
Jump to navigation Jump to search

Definition

A Grothendieck universe is a set (not a class) which has the properties expected of the universe $\mathbb U$ of sets in the sense of the Zermelo-Fraenkel axioms with the following properties:

$(1): \quad \mathbb U$ is a transitive set: If $u \in \mathbb U$ and $x \in u$ then $x \in \mathbb U$
$(2): \quad$ If $ u, v \in \mathbb U$ then $\set {u, v} \in \mathbb U$
$(3): \quad$ If $u \in \mathbb U$ then the power set $\powerset u \in \mathbb U$
$(4): \quad$ If $A \in \mathbb U$, and $\set {u_\alpha: \alpha \in A}$ is a family of elements $u_\alpha \in \mathbb U$ indexed by $A$, then $\ds \bigcup_{\alpha \mathop \in A} u_\alpha \in \mathbb U$


Motivation

A Grothendieck universe allows us to work with something "like" the set of all sets without having to consider classes, which helped Grothendieck in his studies of algebraic geometry.

One can check that if $u, v \in \mathbb U$ and $f: u \to v$ is a mapping, then $f \in \mathbb U$, and similarly the Cartesian product $u \times v \in \mathbb U$, and so on.

In other words, it is closed under the algebra of sets.

A Grothendieck universe $\mathbb U$ is closed under many set-theoretical operations, some of them listed below.

Operation Result
Formation of mappings with source and target in $\mathbb U$ Grothendieck Universe is Closed under Mappings
Binary union Grothendieck Universe is Closed under Binary Union
Finite union Grothendieck Universe is Closed under Finite Union
Finite Cartesian Product in Kuratowski formalization
Subset Grothendieck Universe is Closed under Subset
Arbitrary intersection
If $\mathbb U \ne \O$, then $\O \in \mathbb U$ Empty Set is Element of Nonempty Grothendieck Universe
If $\mathbb U \ne \O$, then $\mathbb N \subseteq \mathbb U$ Nonempty Grothendieck Universe contains Von Neumann Natural Numbers




Also defined as

Some authors require additionally that $\mathbb U$ is not empty.


Also see

  • Results about Grothendieck universes can be found here.


Source of Name

This entry was named for Alexander Grothendieck.


Sources