The Stacks project

Definition 7.43.7. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a morphism of topoi.

1. We say $f$ is an open immersion if $f$ is an embedding and the essential image of $f_*$ is an open subtopos.

2. We say $f$ is a closed immersion if $f$ is an embedding and the essential image of $f_*$ is a closed subtopos.