Sheaf theory

Image functors for sheaves

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 that 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. (Wikipedia).

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Related pages

Topological space | Shriek map | Étale topology | Topology | Inverse image functor | Exact sequence | Complement (set theory) | Direct image with compact support | Six operations | Perverse sheaf | Mathematics | Algebraic geometry | Sheaf (mathematics) | Category (mathematics) | Morphism | Direct image functor | Scheme (mathematics) | Derived category | Exceptional inverse image functor | Verdier duality | Abelian group