In mathematics, the constant sheaf on a topological space associated to a set is a sheaf of sets whose stalks. The constant presheaf is the presheaf that assigns to each open subset is the sheafification.

may be replaced with an object in some category is the category of abelian groups, or commutative rings).

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

Basics

is given the discrete topology is connected is the unique map, then the inverse image. The sheaf space is given the discrete topology).

A detailed example

with the discrete topology are shown in the chart.

and a restriction map for each of the inclusions (with identity map). The constant presheaf 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 are equal, which is false.

is forced by the local identity axiom.

is disconnected.

Modifying further to satisfy the gluing axiom, let , is a sheaf, called the constant sheaf is a sheaf of commutative rings.

See also

• Locally constant sheaf

References

• Section II.1 of

• Section 2.4.6 of