Constant sheaf

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\times\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.

References

• Section II.1 of
• Section 2.4.6 of {{Citation

References

1. "Does the extension by zero sheaf of the constant sheaf have some nice description?".