The Stacks project

Lemma 67.10.2. Let $S$ be a scheme. Let $j : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

  1. $j$ is a monomorphism (as in Definition 67.10.1),

  2. $j$ is a monomorphism in the category of algebraic spaces over $S$, and

  3. the diagonal morphism $\Delta _{X/Y} : X \to X \times _ Y X$ is an isomorphism.

Proof. Note that $X \times _ Y X$ is both the fibre product in the category of sheaves on $(\mathit{Sch}/S)_{fppf}$ and the fibre product in the category of algebraic spaces over $S$, see Spaces, Lemma 65.7.3. The equivalence of (1) and (3) is a general characterization of injective maps of sheaves on any site. The equivalence of (2) and (3) is a characterization of monomorphisms in any category with fibre products. $\square$

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