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In mathematics, especially in sheaf theory, a domain applied in areas such as topology, logic and algebraic geometry, there are four image functors for sheaves which belong together in various senses. Given a continuous mapping ''f'': ''X'' → ''Y'' of topological spaces, and the category ''Sh''(–) of sheaves of abelian groups on a topological space. The functors in question are * direct image ''f''∗ : ''Sh''(''X'') → ''Sh''(''Y'') * inverse image ''f''∗ : ''Sh''(''Y'') → ''Sh''(''X'') * direct image with compact support ''f''! : ''Sh''(''X'') → ''Sh''(''Y'') * exceptional inverse image ''Rf''! : ''D''(''Sh''(''Y'')) → ''D''(''Sh''(''X'')). The exclamation mark is often pronounced "shriek" (slang for exclamation mark), and the maps called "''f'' shriek" or "''f'' lower shriek" and "''f'' upper shriek" – see also shriek map. The exceptional inverse image is in general defined on the level of derived categories only. Similar considerations apply to étale sheaves on schemes. ==Adjointness== The functors are adjoint to each other as depicted at the right, where, as usual, means that ''F'' is left adjoint to ''G'' (equivalently ''G'' right adjoint to ''F''), i.e. :''Hom''(''F''(''A''), ''B'') ≅ ''Hom''(''A'', ''G''(''B'')) for any two objects ''A'', ''B'' in the two categories being adjoint by ''F'' and ''G''. For example, ''f''∗ is the left adjoint of ''f'' *. By the standard reasoning with adjointness relations, there are natural unit and counit morphisms and for on ''Y'' and on ''X'', respectively. However, these are ''almost never'' isomorphisms - see the localization example below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Image functors for sheaves」の詳細全文を読む スポンサード リンク
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