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==Definition==
{{Images of sheaves}}
Let ''f'': ''X'' → ''Y'' be a [[continuous mapping]] of [[topological space]]s, and ''Sh''(–) the [[category (mathematics)|category]] of sheaves of [[abelian group]]s on a topological space. The '''direct image with compact (or proper) support'''
:''f''<sub>!</sub>: ''Sh''(''X'') → ''Sh''(''Y'')
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