# Definition:Grothendieck Universe

## 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 |

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## 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

- 1972: Nicolas Bourbaki:
*Séminaire de Géométrie Algébrique du Bois Marie*(SGA4, vol. 1): Appendix $\S \text{II}$