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 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
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.
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.
for any two objects A, B in the two categories being adjoint by F and G.
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