Sheaf

Definition

A sheaf on a topological space $X$ is a mathematical structure that systematically assigns data to the open sets of $X$ in a way that respects “local-to-global” principles. Formally, a sheaf $\mathcal{F}$ consists of:

1. For each open set $U\subseteq X$, a set $\mathcal{F}(U)$ (called the sections of $\mathcal{F}$ over $U$)
2. For each inclusion $V\subseteq U$ of open sets, a restriction map ${\rho }_{UV}:\mathcal{F}(U)\to \mathcal{F}(V)$

These must satisfy the sheaf axioms:

Identity: ${\rho }_{UU}={\text{id}}_{\mathcal{F}(U)}$ for all open sets $U$

Composition: If $W\subseteq V\subseteq U$, then ${\rho }_{UW}={\rho }_{VW}\circ {\rho }_{UV}$

Locality: If $\{U_i{\}}_{i\in I}$ is an open cover of $U$ and $s,t\in \mathcal{F}(U)$ such that ${\rho }_{U{U}_i}(s)={\rho }_{U{U}_i}(t)$ for all $i$, then $s=t$

Gluing: If $\{U_i{\}}_{i\in I}$ is an open cover of $U$ and $s_i\in \mathcal{F}(U_i)$ such that ${\rho }_{U_i{U}_i\cap U_j}(s_i)={\rho }_{U_j{U}_i\cap U_j}(s_j)$ for all $i,j$, then there exists a unique $s\in \mathcal{F}(U)$ with ${\rho }_{U{U}_i}(s)=s_i$ for all $i$

Presheaves

A presheaf satisfies only the first two conditions (identity and composition). A sheaf is a presheaf that additionally satisfies locality and gluing.

Examples

Constant sheaf: For any set $S$, the constant sheaf $\underline{S}$ assigns to each connected open set $U$ the set $S^{{\pi }_0(U)}$, where ${\pi }_0(U)$ is the set of connected components of $U$.

Sheaf of continuous functions: ${\mathcal{C}}^0(U)=\{f:U\to \mathbb{R}\mid f\text{ is continuous}\}$ with restriction given by function restriction.

Sheaf of smooth functions: On a smooth manifold, ${\mathcal{C}}^{\infty }(U)$ consists of smooth real-valued functions on $U$.

Structure sheaf: On an algebraic variety or scheme, the structure sheaf assigns to each open set the ring of regular functions on that set.

Stalks

For a point $x\in X$, the stalk ${\mathcal{F}}_x$ is the direct limit: ${\mathcal{F}}_x={\underrightarrow{\lim }}_{U\ni x}\mathcal{F}(U)$ where the limit is taken over all open neighborhoods $U$ of $x$. Elements of stalks are called germs.

Morphisms

A morphism of sheaves $\varphi :\mathcal{F}\to \mathcal{G}$ consists of maps ${\varphi }_U:\mathcal{F}(U)\to \mathcal{G}(U)$ for each open set $U$, compatible with restriction maps.

Sheafification

Every presheaf $\mathcal{P}$ has an associated sheaf ${\mathcal{P}}^{+}$ called its sheafification, obtained by forcing the locality and gluing axioms to hold.

Related Concepts

• Topos
• Sheaf Model
• Grothendieck Topology
• Étale Space