---
tag: 7QM2KNZ358
kind: definition
name: Sheaf
msc: [54B40, 18F20, 14F06]
defines: TB6TM6R4MD
uses: [0CQHFPGJ45, 02BH3MCBQB, 082GHNPJ53, 0310REJJY8, 0VGFHTM75Y, KB76923A1M]
sources:
  - url: https://en.wikipedia.org/wiki/Sheaf_(mathematics)
---

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
A *presheaf* $\mathcal F$ of sets on $X$ consists of the following data.
For each [[open set|02BH3MCBQB]] $U \subseteq X$ there is a set $\mathcal F(U)$.
For each pair of open sets $V \subseteq U$ there is a map $\rho_{UV} \colon \mathcal F(U) \to \mathcal F(V)$, written $s \mapsto s|_V$.
These satisfy $\rho_{UU} = \id$ and $\rho_{VW} \circ \rho_{UV} = \rho_{UW}$ whenever $W \subseteq V \subseteq U$.
The elements of $\mathcal F(U)$ are the *sections* of $\mathcal F$ over $U$.
A presheaf $\mathcal F$ is a *sheaf* if, for every open set $U$ and every open cover $U = \bigcup_{i \in I} U_i$:

1. (Locality) if $s, t \in \mathcal F(U)$ satisfy $s|_{U_i} = t|_{U_i}$ for all $i$, then $s = t$;
2. (Gluing) if $s_i \in \mathcal F(U_i)$ satisfy $s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}$ for all $i, j$, then there is $s \in \mathcal F(U)$ with $s|_{U_i} = s_i$ for all $i$.

A sheaf of [[abelian groups|082GHNPJ53]] is a sheaf in which each $\mathcal F(U)$ is an abelian group and each restriction map is a homomorphism.
A sheaf of [[commutative rings|0310REJJY8]] is defined in the same way.

## Remarks

Applied to the empty cover of $U = \emptyset$, the two conditions say that $\mathcal F(\emptyset)$ has exactly one element.

Example: for $U \subseteq \R$ open, let $\mathcal C(U)$ be the ring of [[continuous|0VGFHTM75Y]] functions $U \to \R$, with the usual restriction.
Locality holds because a function is determined by its values.
Gluing holds because the glued function agrees near each point with some continuous $s_i$, so it is continuous.
So $\mathcal C$ is a sheaf of commutative rings.

Non-example: let $\mathcal B(U)$ be the set of bounded continuous functions on $U$.
Gluing fails for $U = \R$ with the cover by the intervals $U_m = (-m, m)$ and the sections $s_m(x) = x$ on $U_m$: they agree on overlaps, but their only possible gluing, $x \mapsto x$, is not bounded.

Equivalently, conditions 1 and 2 say that $\mathcal F(U) \to \prod_i \mathcal F(U_i) \rightrightarrows \prod_{i, j} \mathcal F(U_i \cap U_j)$ is an equalizer diagram.
Sheaves of abelian groups are the coefficients of [[sheaf cohomology|KB76923A1M]].