---
tag: GGE4KYWCNT
kind: definition
name: Grothendieck topos
msc: [18F10, 18B25]
defines: S3YVE24AQK
uses: [NYE0Z94MME, RT8A8FJ566, K7A04ZBD92, AND2WHZC8P]
sources:
  - url: https://en.wikipedia.org/wiki/Topos
---

## Definition

A *Grothendieck topos* is a category $\mathcal{E}$ that is [[equivalent|K7A04ZBD92]] to the category $\mathrm{Sh}(\mathcal{C}, J)$ of [[sheaves|RT8A8FJ566]] of sets on some small [[site|NYE0Z94MME]] $(\mathcal{C}, J)$.

## Remarks

Example: for a topological space $X$, the category $\mathrm{Sh}(X)$ of sheaves on the site of open sets of $X$ is a Grothendieck topos.
Example: for a small category $\mathcal{C}$, the category of [[presheaves|AND2WHZC8P]] on $\mathcal{C}$ is a Grothendieck topos, for the topology in which only maximal sieves cover.
Non-example: the category of finite sets is an elementary topos but not a Grothendieck topos, since it lacks infinite coproducts, which every Grothendieck topos has.
Every Grothendieck topos is an elementary topos and is complete and cocomplete.
Giraud's theorem gives an intrinsic characterization.