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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 or ''AX''. 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 . 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.
==Basics==
Let ''X'' be a topological space, and ''A'' a set. The sections of the constant sheaf over an open set ''U'' may be interpreted as the continuous functions , where ''A'' is given the discrete topology. If ''U'' is connected, then these locally constant functions are constant. If ''f'': ''X'' → is the unique map to the one-point space and ''A'' is considered as a sheaf on , then the inverse image ''f''−1''A'' is the constant sheaf on ''X''. The sheaf space of is the projection map ''X'' × ''A'' → ''X'' (where ''A'' is given the discrete topology).
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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