A homomorphism $f:X\to Y$ of schemes is a **closed immersion** if it is tracked by a homeomorphism of the underlying topological space $X$ onto a closed subspace of $Y$, and the comorphism $f^\sharp:\mathcal{O}_Y\to f_*\mathcal{O}_X$ is an epimorphism of sheaves on $Y$.

More generally, let us consider some category of spaces, i.e. sheaves of sets on $C = Aff$ equipped with a subcanonical Grothendieck topology. Then a morphism $F\to G$ of spaces is said to be **closed immersion** if it is representable by a strict monomorphism.

Discussion in the context of higher geometry/higher algebra is in

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