The Stacks project

Lemma 8.11.6. Let $\mathcal{C}$ be a site. Let $F : \mathcal{X} \to \mathcal{Y}$ and $G : \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of stacks in groupoids over $\mathcal{C}$. If $\mathcal{X}$ is a gerbe over $\mathcal{Y}$ and $\mathcal{Y}$ is a gerbe over $\mathcal{Z}$, then $\mathcal{X}$ is a gerbe over $\mathcal{Z}$.

Proof. Let us prove properties (2)(a) and (2)(b) of Lemma 8.11.3 for $\mathcal{X} \to \mathcal{Z}$.

Let $z$ be an object of $\mathcal{Z}$ lying over the object $U$ of $\mathcal{C}$. By assumption on $G$ there exists a covering $\{ U_ i \to U\} $ of $U$ and objects $y_ i \in \mathcal{Y}_{U_ i}$ such that $G(y_ i) \cong z|_{U_ i}$. By assumption on $F$ there exist coverings $\{ U_{ij} \to U_ i\} $ and objects $x_{ij} \in \mathcal{X}_{U_{ij}}$ such that $F(x_{ij}) \cong y_ i|_{U_{ij}}$. Then $\{ U_{ij} \to U\} $ is a covering of $\mathcal{C}$ and $(G \circ F)(x_{ij}) \cong z|_{U_{ij}}$. Thus (2)(a) holds.

Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, let $x_1, x_2$ be objects of $\mathcal{X}$ over $U$, and let $c : (G \circ F)(x_1) \to (G \circ F)(x_2)$ be a morphism in $\mathcal{Z}_ U$. By assumption on $G$ there exists a covering $\{ U_ i \to U\} $ of $U$ and morphisms $b_ i : F(x_1)|_{U_ i} \to F(x_2)|_{U_ i}$ in $\mathcal{Y}_{U_ i}$ such that $G(b_ i) = c|_{U_ i}$. By assumption on $F$ there exist coverings $\{ U_{ij} \to U_ i\} $ and morphisms $a_{ij} : x_1|_{U_{ij}} \to x_2|_{U_{ij}}$ in $\mathcal{X}_{U_{ij}}$ such that $F(a_{ij}) = b_ i|_{U_{ij}}$. Then $\{ U_{ij} \to U\} $ is a covering of $\mathcal{C}$ and $(G \circ F)(a_{ij}) = c|_{U_{ij}}$ as required in (2)(b). $\square$


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