Constant sheaf

A detailed example

Let ${\displaystyle X}$ be the topological space consisting of two points ${\displaystyle p}$ and ${\displaystyle q}$ with the discrete topology. ${\displaystyle X}$ has four open sets: ${\displaystyle \varnothing ,\{p\},\{q\},\{p,q\}}$. The five non-trivial inclusions of the open sets of ${\displaystyle X}$ are shown in the chart.

A presheaf on ${\displaystyle X}$ chooses a set for each of the four open sets of ${\displaystyle X}$ and a restriction map for each of the inclusions (with identity map for ${\displaystyle U\subset U}$). The constant presheaf with value ${\displaystyle \mathbf{\text{Z}}}$, denoted ${\displaystyle F}$, is the presheaf where all four sets are ${\displaystyle \mathbf{\text{Z}}}$, the integers, and all restriction maps are the identity. ${\displaystyle F}$ is a functor on the diagram of inclusions (a presheaf), because it is constant. It satisfies the gluing axiom, but is not a sheaf because it fails the local identity axiom on the empty set. This is because the empty set is covered by the empty family of sets, ${\displaystyle \varnothing ={\bigcup }_{U\in \{\}}U}$, and vacuously, any two sections in ${\displaystyle F(\varnothing )}$ are equal when restricted to any set in the empty family ${\displaystyle \{\}}$. The local identity axiom would therefore imply that any two sections in ${\displaystyle F(\varnothing )}$ are equal, which is false.

To modify this into a presheaf ${\displaystyle G}$ that satisfies the local identity axiom, let ${\displaystyle G(\varnothing )=0}$, a one-element set, and give ${\displaystyle G}$ the value ${\displaystyle \mathbf{\text{Z}}}$ on all non-empty sets. For each inclusion of open sets, let the restriction be the unique map to 0 if the smaller set is empty, or the identity map otherwise. Note that ${\displaystyle G(\varnothing )=0}$ is forced by the local identity axiom.

Now ${\displaystyle G}$ is a separated presheaf (satisfies local identity), but unlike ${\displaystyle F}$ it fails the gluing axiom. Indeed, ${\displaystyle \{p,q\}}$ is disconnected, covered by non-intersecting open sets ${\displaystyle \{p\}}$ and ${\displaystyle \{q\}}$. Choose distinct sections ${\displaystyle m\ne n}$ in ${\displaystyle \mathbf{Z}}$ over ${\displaystyle \{p\}}$ and ${\displaystyle \{q\}}$ respectively. Because ${\displaystyle m}$ and ${\displaystyle n}$ restrict to the same element 0 over ${\displaystyle \varnothing }$, the gluing axiom would guarantee the existence of a unique section ${\displaystyle s}$ on ${\displaystyle G(\{p,q\})}$ that restricts to ${\displaystyle m}$ on ${\displaystyle \{p\}}$ and ${\displaystyle n}$ on ${\displaystyle \{q\}}$; but the restriction maps are the identity, giving ${\displaystyle m=s=n}$, which is false. Intuitively, ${\displaystyle G(\{p,q\})}$ is too small to carry information about both connected components ${\displaystyle \{p\}}$ and ${\displaystyle \{q\}}$.

Modifying further to satisfy the gluing axiom, let

${\displaystyle H(\{p,q\})=\mathrm{F}\mathrm{u}\mathrm{n}(\{p,q\},\mathbf{Z})\cong \mathbb{Z}\times \mathbb{Z}}$,

the ${\displaystyle \mathbf{Z}}$-valued functions on ${\displaystyle \{p,q\}}$, and define the restriction maps of ${\displaystyle H}$ to be natural restriction of functions to ${\displaystyle \{p\}}$ and ${\displaystyle \{q\}}$, with the zero map restricting to ${\displaystyle \varnothing }$. Then ${\displaystyle H}$ is a sheaf, called the constant sheaf on ${\displaystyle X}$ with value ${\displaystyle \mathbf{\text{Z}}}$. Since all restriction maps are ring homomorphisms, ${\displaystyle H}$ is a sheaf of commutative rings.