---
tag: BAH89SW049
kind: definition
name: Sheaf on a site
msc: [18F20, 18F10]
defines: RT8A8FJ566
uses: [NYE0Z94MME, GDRFSJ8E2B, CNCXNWKC6R, AND2WHZC8P]
sources:
  - url: https://en.wikipedia.org/wiki/Sheaf_(mathematics)
---

## Definition

Let $(\mathcal{C}, J)$ be a [[site|NYE0Z94MME]] and let $P \colon \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$ be a [[presheaf|AND2WHZC8P]].
For $x \in P(c)$ and $f \colon d \to c$, write $x|_f = P(f)(x)$.
Let $S$ be a [[sieve|CNCXNWKC6R]] on $c$.
A *matching family* for $S$ is a choice of an element $x_f \in P(d)$ for each $f \colon d \to c$ in $S$ such that $x_{f \circ g} = x_f|_g$ for every morphism $g \colon e \to d$.
An *amalgamation* of the family is an element $x \in P(c)$ with $x|_f = x_f$ for every $f \in S$.
The presheaf $P$ is a *sheaf* for $J$ when the following holds.
For every object $c$ and every covering sieve $S \in J(c)$, each matching family for $S$ has exactly one amalgamation.

## Remarks

Equivalently, $P$ is a sheaf when, for every covering sieve $S \subseteq \Hom(-, c)$, every natural transformation $S \to P$ extends uniquely to $\Hom(-, c) \to P$.
A morphism of sheaves is a natural transformation; sheaves on $(\mathcal{C}, J)$ form a full subcategory $\mathrm{Sh}(\mathcal{C}, J)$ of presheaves.
Example: on the site of open sets of a space $X$, the presheaf $P(U) = C(U, \R)$ of continuous functions is a sheaf.
Functions on the members of a covering sieve that agree on overlaps define one function on $U$, which is continuous because continuity is local.
Non-example: on the same site, the constant presheaf $P(U) = \set{0, 1}$ fails, because the empty cover of $U = \emptyset$ has a unique (empty) matching family but two amalgamations.
For this site the definition gives back the [[sheaves on a topological space|TB6TM6R4MD]].