---
tag: SMMK9E0MTM
kind: concept
name: Grothendieck universe
msc: ["03E55", "18A15"]
sources:
  - url: https://en.wikipedia.org/wiki/Grothendieck_universe
same_as:
  wikidata: Q1465090
  wikipedia: Grothendieck universe
---

## Summary

A Grothendieck universe is a set $U$ that is closed under the usual operations of set theory.
These are taking elements, pairs, [[power sets|00E0YZCERH]] and unions of families indexed by elements of $U$.
Inside a universe one can do ordinary mathematics without leaving $U$.
Category theorists use universes to speak of the category of all "small" sets or groups as a set-sized object.
Uncountable Grothendieck universes are exactly the sets $V_\kappa$ for [[inaccessible cardinals|NW4DREQ5DQ]] $\kappa$, so their existence goes beyond ZFC.

## Explanations

### L1 human

In category theory one wants to talk about "the category of all groups", but the collection of all groups is too big to be a set.
A universe is a large set that is closed under every construction one normally uses.
One then calls a set small when it lies in the universe, and the category of all small groups is a legitimate object.
The axiom that every set lies in some universe was used by Grothendieck in algebraic geometry.