---
tag: NYE0Z94MME
kind: concept
name: Site
msc: [18F10, 14F20]
sources:
  - url: https://en.wikipedia.org/wiki/Grothendieck_topology
---

## Summary

A site is a [[category|A2DE66937D]] together with a [[Grothendieck topology|GDRFSJ8E2B]] on it.
It is the setting in which [[sheaves|RT8A8FJ566]], sheafification and sheaf cohomology make sense beyond topological spaces.
The open sets of a topological space form a site, and so do the étale maps into a scheme, which give étale cohomology.
The category of sheaves on a small site is a [[Grothendieck topos|S3YVE24AQK]].

## Explanations

### L1 human

A site is a "generalized space" described by its pieces and by the rule saying which families of pieces cover a piece.
The pieces need not be subsets; they can be maps from other objects.
For example, the open sets of the real line, with the usual covers, form a site, and its sheaves are the usual sheaves on $\R$.