---
tag: S3YVE24AQK
kind: concept
name: Grothendieck topos
msc: [18F10, 18B25]
specializes: [6837RS7N82]
sources:
  - url: https://en.wikipedia.org/wiki/Topos
---

## Summary

A Grothendieck topos is a category equivalent to the category of [[sheaves|RT8A8FJ566]] of sets on a small [[site|NYE0Z94MME]].
Grothendieck introduced these categories as generalized spaces.
A topological space is studied through its category of sheaves, and many sites that are not spaces still give such categories.
Every Grothendieck topos is an elementary [[topos|6837RS7N82]], and [[Giraud's theorem|57BXX32WF0]] characterizes Grothendieck toposes by intrinsic properties.
Different sites can give equivalent toposes, so the topos is the invariant object.

## Explanations

### L1 human

A topological space can be studied through the sheaves on it, rather than through its points.
A Grothendieck topos is any category that looks like "all sheaves on some space", where the space may be a generalized one given by a site.
The category of sets is the topos of sheaves on a one-point space.
For a group $G$, the category of sets with a $G$-action is the topos of presheaves on $G$ viewed as a one-object category, so it is also a Grothendieck topos.