The Stacks project

Lemma 95.6.3. A morphism $f : \mathcal{X} \to \mathcal{Y}$ of algebraic stacks is

  1. a monomorphism if and only if $\Delta _{f, 1}$ is an isomorphism, and

  2. representable by algebraic spaces if and only if $\Delta _{f, 1}$ is a monomorphism.

Moreover, the second diagonal $\Delta _{f, 2}$ is always a monomorphism.

Proof. Recall from Properties of Stacks, Lemma 94.8.4 that a morphism of algebraic stacks is a monomorphism if and only if its diagonal is an isomorphism of stacks. Thus Lemma 95.6.2 can be rephrased as saying that a morphism is representable by algebraic spaces if the diagonal is a monomorphism. In particular, it shows that condition (3) of Lemma 95.3.4 is actually an if and only if, i.e., a morphism of algebraic stacks is representable by algebraic spaces if and only if its diagonal is a monomorphism. $\square$


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