nLab constant sheaf

Definition

A constant sheaf is a sheaf (on some site C) that is isomorphic to the sheafification of a constant presheaf (a constant functor).

Together with the global section functor, the constant sheaf functor is a geometric morphism

$$\Gamma : Sh(C) \stackrel{\leftarrow}{\to} Set : const$$

from the sheaf topos to the topos Set.

Examples

• The natural numbers object in any Grothendieck topos is the constant sheaf on the set $\mathbb{N}$ of natural numbers.

Pattern

• A locally constant function is a section of a constant sheaf;

• a locally constant sheaf is a section of a constant stack;

• a locally constant stack is a section of (… and so on…)

• a locally constant ∞-stack is a section of a constant ∞-stack.

A locally constant sheaf / $\infty$-stack is also called a local system.

See also at constant presheaf.

category: sheaf theory