The Stacks project

Definition 61.26.1. Let $j : U \to X$ be a weakly étale morphism of schemes.

1. The restriction functor $j^{-1} : \mathop{\mathit{Sh}}\nolimits (X_{pro\text{-}\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (U_{pro\text{-}\acute{e}tale})$ has a left adjoint $j_!^{Sh} : \mathop{\mathit{Sh}}\nolimits (X_{pro\text{-}\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (U_{pro\text{-}\acute{e}tale})$.

2. The restriction functor $j^{-1} : \textit{Ab}(X_{pro\text{-}\acute{e}tale}) \to \textit{Ab}(U_{pro\text{-}\acute{e}tale})$ has a left adjoint which is denoted $j_! : \textit{Ab}(U_{pro\text{-}\acute{e}tale}) \to \textit{Ab}(X_{pro\text{-}\acute{e}tale})$ and called extension by zero.

3. Let $\Lambda$ be a ring. The functor $j^{-1} : \textit{Mod}(X_{pro\text{-}\acute{e}tale}, \Lambda ) \to \textit{Mod}(U_{pro\text{-}\acute{e}tale}, \Lambda )$ has a left adjoint $j_! : \textit{Mod}(U_{pro\text{-}\acute{e}tale}, \Lambda ) \to \textit{Mod}(X_{pro\text{-}\acute{e}tale}, \Lambda )$ and called extension by zero.