The Stacks project

Lemma 8.10.6. Let $\mathcal{C}$ be a site. Let $p : \mathcal{X} \to \mathcal{C}$ be a stack. Endow $\mathcal{X}$ with the topology inherited from $\mathcal{C}$ and let $q : \mathcal{Y} \to \mathcal{X}$ be a stack. Then $\mathcal{Y}$ is a stack over $\mathcal{C}$. If $p$ and $q$ define stacks in groupoids, then $\mathcal{Y}$ is a stack in groupoids over $\mathcal{C}$.

Proof. We check the three conditions in Definition 8.4.1 to prove that $\mathcal{Y}$ is a stack over $\mathcal{C}$. By Categories, Lemma 4.32.12 we find that $\mathcal{Y}$ is a fibred category over $\mathcal{C}$. Thus condition (1) holds.

Let $U$ be an object of $\mathcal{C}$ and let $y_1, y_2$ be objects of $\mathcal{Y}$ over $U$. Denote $x_ i = q(y_ i)$ in $\mathcal{X}$. Consider the map of presheaves

\[ q : \mathit{Mor}_{\mathcal{Y}/\mathcal{C}}(y_1, y_2) \longrightarrow \mathit{Mor}_{\mathcal{X}/\mathcal{C}}(x_1, x_2) \]

on $\mathcal{C}/U$, see Lemma 8.2.3. Let $\{ U_ i \to U\} $ be a covering and let $\varphi _ i$ be a section of the presheaf on the left over $U_ i$ such that $\varphi _ i$ and $\varphi _ j$ restrict to the same section over $U_ i \times _ U U_ j$. We have to find a morphism $\varphi : x_1 \to x_2$ restricting to $\varphi _ i$. Note that $q(\varphi _ i) = \psi |_{U_ i}$ for some morphism $\psi : x_1 \to x_2$ over $U$ because the second presheaf is a sheaf (by assumption). Let $y_{12} \to y_2$ be the stronly $\mathcal{X}$-cartesian morphism of $\mathcal{Y}$ lying over $\psi $. Then $\varphi _ i$ corresponds to a morphism $\varphi '_ i : y_1|_{U_ i} \to y_{12}|_{U_ i}$ over $x_1|_{U_ i}$. In other words, $\varphi '_ i$ now define local sections of the presheaf

\[ \mathit{Mor}_{\mathcal{Y}/\mathcal{X}}(y_1, y_{12}) \]

over the members of the covering $\{ x_1|_{U_ i} \to x_1\} $. By assumption these glue to a unique morphism $y_1 \to y_{12}$ which composed with the given morphism $y_{12} \to y_2$ produces the desired morphism $y_1 \to y_2$.

Finally, we show that descent data are effective. Let $\{ f_ i : U_ i \to U\} $ be a covering of $\mathcal{C}$ and let $(y_ i, \varphi _{ij})$ be a descent datum relative to this covering (Definition 8.3.1). Setting $x_ i = q(y_ i)$ and $\psi _{ij} = q(\varphi _{ij})$ we obtain a descent datum $(x_ i, \psi _{ij})$ for the covering in $\mathcal{X}$. By assumption on $\mathcal{X}$ we may assume $x_ i = x|_{U_ i}$ and the $\psi _{ij}$ equal to the canonical descent datum (Definition 8.3.5). In this case $\{ x|_{U_ i} \to x\} $ is a covering and we can view $(y_ i, \varphi _{ij})$ as a descent datum relative to this covering. By our assumption that $\mathcal{Y}$ is a stack over $\mathcal{C}$ we see that it is effective which finishes the proof of condition (3).

The final assertion follows because $\mathcal{Y}$ is a stack over $\mathcal{C}$ and is fibred in groupoids by Categories, Lemma 4.34.13. $\square$


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