Constant sheaf

In mathematics, the constant sheaf on a topological space $$X$$ associated to a set $$A$$ is a sheaf of sets on $$X$$ whose stalks are all equal to $$A$$. It is denoted by $$\underline{A}$$ or $$A_X$$. The constant presheaf with value $$A$$ is the presheaf that assigns to each open subset of $$X$$ the value $$A$$, and all of whose restriction maps are the identity map $$A\to A$$. The constant sheaf associated to $$A$$ is the sheafification of the constant presheaf associated to $$A$$. This sheaf identifies with the sheaf of locally constant $$A$$-valued functions on $$X$$.

In certain cases, the set $$A$$ may be replaced with an object $$A$$ in some category $$\textbf{C}$$ (e.g. when $$\textbf{C}$$ is the category of abelian groups, or commutative rings).

Constant sheaves of abelian groups appear in particular as coefficients in sheaf cohomology.

Basics
Let $$X$$ be a topological space, and $$A$$ a set. The sections of the constant sheaf $$\underline{A}$$ over an open set $$U$$ may be interpreted as the continuous functions $$U\to A$$, where $$A$$ is given the discrete topology. If $$U$$ is connected, then these locally constant functions are constant. If $$f:X\to\{\text{pt}\}$$ is the unique map to the one-point space and $$A$$ is considered as a sheaf on $$\{\text{pt}\}$$, then the inverse image $$f^{-1}A$$ is the constant sheaf $$\underline{A}$$ on $$X$$. The sheaf space of $$\underline{A}$$ is the projection map $$A$$ (where $$X\times A\to X$$ is given the discrete topology).

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

A presheaf on $$X$$ chooses a set for each of the four open sets of $$X$$ and a restriction map for each of the inclusions (with identity map for $$U\subset U$$). The constant presheaf with value $$\textbf{Z}$$, denoted $$F$$, is the presheaf where all four sets are $$\textbf{Z}$$, the integers, and all restriction maps are the identity. $$F$$ 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, $$\varnothing = \bigcup\nolimits_{U\in\{\}} U  $$, and vacuously, any two sections in $$F(\varnothing)   $$ are equal when restricted to any set in the empty family $$\{\}  $$. The local identity axiom would therefore imply that any two sections in $$F(\varnothing)  $$ are equal, which is false.

To modify this into a presheaf $$G$$ that satisfies the local identity axiom, let $$G(\varnothing)=0$$, a one-element set, and give $$G$$ the value $$\textbf{Z}$$ 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 $$G(\varnothing)=0$$ is forced by the local identity axiom.

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



Modifying further to satisfy the gluing axiom, let "$H(\{p,q\}) = \mathrm{Fun}(\{p,q\},\mathbf{Z})\cong \Z\otimes\Z $,"the $$\mathbf Z $$-valued functions on $$\{p,q\}$$, and define the restriction maps of $$H$$ to be natural restriction of functions to $$\{p\}$$ and $$\{q\}$$, with the zero map restricting to $$\varnothing   $$. Then $$H$$ is a sheaf, called the constant sheaf on $$X$$ with value $$\textbf{Z}$$. Since all restriction maps are ring homomorphisms, $$H$$ is a sheaf of commutative rings.