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 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 → A. The constant sheaf associated to A is the sheafification of the constant presheaf associated to A.

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

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