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 non-empty 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.

Let X be a topological space, and A a set. The sections of the constant sheaf A over an open set U may be interpreted as the continuous functions U → A, where A is given the discrete topology. If U is connected, then these locally constant functions are constant. If f: X → {pt} is the unique map to the one-point space and A is considered as a sheaf on {pt}, then the inverse image f⁻¹A is the constant sheaf A on X. The sheaf space of A is the projection map X × A → X (where A is given the discrete topology).

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