---
tag: RT8A8FJ566
kind: concept
name: Sheaf on a site
msc: [18F20, 18F10]
specializes: [AND2WHZC8P]
sources:
  - url: https://en.wikipedia.org/wiki/Sheaf_(mathematics)
---

## Summary

A sheaf on a [[site|NYE0Z94MME]] $(\mathcal{C}, J)$ is a [[presheaf|AND2WHZC8P]] on $\mathcal{C}$ with a gluing property.
Compatible local data on every covering [[sieve|CNCXNWKC6R]] glue to a unique global datum.
It generalizes a [[sheaf on a topological space|TB6TM6R4MD]], which is the case of the site of open sets.
Sheaves on a small site form a [[Grothendieck topos|S3YVE24AQK]], and every presheaf has a best approximation by a sheaf, its [[sheafification|2HYK3KBQNA]].
Sheaves on the étale site of a scheme give étale cohomology.

## Explanations

### L1 human

A presheaf gives, for each piece of a space, a set of data, and a way to restrict data to smaller pieces.
It is a sheaf when data can be checked and built locally.
If pieces of data on a cover agree where they overlap, there is exactly one datum on the whole that restricts to them.
For example, continuous real functions on the open sets of a space form a sheaf, since continuity is a local property.
Constant functions do not: a function equal to $0$ on one component and $1$ on another is locally constant but not constant.