# 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 $\ds \bigcup_{\alpha \mathop \in A} u_\alpha \in \mathbb U$

This page has been identified as a candidate for refactoring of medium complexity.In particular: Extract the above into a page in the "Axioms" namespaceUntil this has been finished, please leave
`{{Refactor}}` in the code.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Refactor}}` from the code. |

## 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 |

Work In ProgressIn particular: The table is to be completed and restructured for presentational aesthetics.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{WIP}}` from the code. |

## 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 $\S \text{II}$