The Stacks project

Lemma 8.10.5. Let $\mathcal{C}$ be a site. Let $p : \mathcal{X} \to \mathcal{C}$ and $q : \mathcal{Y} \to \mathcal{C}$ be stacks in groupoids. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories over $\mathcal{C}$. If $F$ turns $\mathcal{X}$ into a category fibred in groupoids over $\mathcal{Y}$, then $\mathcal{X}$ is a stack in groupoids over $\mathcal{Y}$ (with topology inherited from $\mathcal{C}$).

Proof. Let us prove descent for objects. Let $\{ y_ i \to y\} $ be a covering of $\mathcal{Y}$. Let $(x_ i, \varphi _{ij})$ be a descent datum in $\mathcal{X}$ with respect to this covering. Then $(x_ i, \varphi _{ij})$ is also a descent datum with respect to the covering $\{ q(y_ i) \to q(y)\} $ of $\mathcal{C}$. As $\mathcal{X}$ is a stack in groupoids we obtain an object $x$ over $q(y)$ and isomorphisms $\psi _ i : x|_{q(y_ i)} \to x_ i$ over $q(y_ i)$ compatible with the $\varphi _{ij}$, i.e., such that

\[ \varphi _{ij} = \psi _ j|_{q(y_ i) \times _{q(y)} q(y_ j)} \circ \psi _ i^{-1}|_{q(y_ i) \times _{q(y)} q(y_ j)}. \]

Consider the sheaf $\mathit{I} = \mathit{Isom}_\mathcal {Y}(F(x), y)$ on $\mathcal{C}/p(x)$. Note that $s_ i = F(\psi _ i) \in \mathit{I}(q(x_ i))$ because $F(x_ i) = y_ i$. Because $F(\varphi _{ij}) = \text{id}$ (as we started with a descent datum over $\{ y_ i \to y\} $) the displayed formula shows that $s_ i|_{q(y_ i) \times _{q(y)} q(y_ j)} = s_ j|_{q(y_ i) \times _{q(y)} q(y_ j)}$. Hence the local sections $s_ i$ glue to $s : F(x) \to y$. As $F$ is fibred in groupoids we see that $x$ is isomorphic to an object $x'$ with $F(x') = y$. We omit the verification that $x'$ in the fibre category of $\mathcal{X}$ over $y$ is a solution to the problem of descent posed by the descent datum $(x_ i, \varphi _{ij})$. We also omit the proof of the sheaf property of the $\mathit{Isom}$-presheaves of $\mathcal{X}/\mathcal{Y}$. $\square$

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