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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

from the sheaf topos to the topos Set.

Generalizations

The notion of constant sheaf sits in a hierarchy of related notions:

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

• A section of a constant stack is a locally constant sheaf.

For the full story see locally constant ∞-stack.

Examples

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