Inverse image functor

In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map ${\displaystyle f:X\to Y}$, the inverse image functor is a functor from the category of sheaves on Y to the category of sheaves on X. The direct image functor is the primary operation on sheaves, with the simplest definition. The inverse image exhibits some relatively subtle features.

Image functors for sheaves
direct image f∗
inverse image f∗
direct image with compact support f!
exceptional inverse image Rf!
${\displaystyle f^{*}\leftrightarrows f_{*}}$
${\displaystyle (R)f_{!}\leftrightarrows (R)f^{!}}$
Base change theorems