The Stacks project

Lemma 100.11.12. Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$. Let $\mathcal{Z}$ be an algebraic stack satisfying the equivalent conditions of Lemma 100.11.3 and let $\mathcal{Z} \to \mathcal{X}$ be a monomorphism such that the image of $|\mathcal{Z}| \to |\mathcal{X}|$ is $x$. Then the residual gerbe $\mathcal{Z}_ x$ of $\mathcal{X}$ at $x$ exists and $\mathcal{Z} \to \mathcal{X}$ factors as $\mathcal{Z} \to \mathcal{Z}_ x \to \mathcal{X}$ where the first arrow is an equivalence.

Proof. Let $\mathcal{Z}_ x \subset \mathcal{X}$ be the full subcategory corresponding to the essential image of the functor $\mathcal{Z} \to \mathcal{X}$. Then $\mathcal{Z} \to \mathcal{Z}_ x$ is an equivalence, hence $\mathcal{Z}_ x$ is an algebraic stack, see Algebraic Stacks, Lemma 94.12.4. Since $\mathcal{Z}_ x$ inherits all the properties of $\mathcal{Z}$ from this equivalence it is clear from the uniqueness in Lemma 100.11.7 that $\mathcal{Z}_ x$ is the residual gerbe of $\mathcal{X}$ at $x$. $\square$

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