 Direct image functor

For a continuous map $f:X\to Y$, the direct-image functor sends a sheaf $\mathcal{F}$ on $X$ to the sheaf $V\mapsto \mathcal{F}(f^{-1}V)$ on $Y$, and sends sheaf morphisms to their induced maps on sections. Its output is a direct image sheaf.

In category theory, a **direct image functor** is a concept that arises in the context of functors between categories, particularly when dealing with the theories of sheaves, topology, or algebraic geometry.