Constant sheaf

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In mathematics, the constant sheaf on a topological space ${\displaystyle X}$ associated to a set ${\displaystyle A}$ is a sheaf of sets on ${\displaystyle X}$ whose stalks are all equal to ${\displaystyle A}$. It is denoted by ${\displaystyle {\underline {A}}}$ or ${\displaystyle A_{X}}$. The constant presheaf with value ${\displaystyle A}$ is the presheaf that assigns to each open subset of ${\displaystyle X}$ the value ${\displaystyle A}$, and all of whose restriction maps are the identity map ${\displaystyle A\to A}$. The constant sheaf associated to ${\displaystyle A}$ is the sheafification of the constant presheaf associated to ${\displaystyle A}$. This sheaf identifies with the sheaf of locally constant ${\displaystyle A}$-valued functions on ${\displaystyle X}$.^[1]

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

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