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 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,
By the standard reasoning with adjointness relations, there are natural unit and counit morphisms
Verdier duality gives another link between them: morally speaking, it exchanges "∗" and "!
", i.e. in the synopsis above it exchanges functors along the diagonals.
For example the direct image is dual to the direct image with compact support.
This phenomenon is studied and used in the theory of perverse sheaves.
Another useful property of the image functors is base change.
In the particular situation of a closed subspace i: Z ⊂ X and the complementary open subset j: U ⊂ X, the situation simplifies insofar that for j∗=j!
and i!=i∗ and for any sheaf F on X, one gets exact sequences Its Verdier dual reads a distinguished triangle in the derived category of sheaves on X.
The adjointness relations read in this case and