The Stacks project

Lemma 8.6.10. Let $\mathcal{C}$ be a site. Let

\[ \xymatrix{ \mathcal{T}_2 \ar[r] \ar[d] & \mathcal{T}_1 \ar[d]^ G \\ \mathcal{S}_2 \ar[r]^ F & \mathcal{S}_1 } \]

be a $2$-cartesian diagram of stacks in groupoids over $\mathcal{C}$. If

  1. $F : \mathcal{S}_2 \to \mathcal{S}_1$ is fully faithful,

  2. for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $x \in \mathop{\mathrm{Ob}}\nolimits ((\mathcal{S}_1)_ U)$ there exists a covering $\{ U_ i \to U\} $ such that $x|_{U_ i}$ is in the essential image of $F : (\mathcal{S}_2)_{U_ i} \to (\mathcal{S}_1)_{U_ i}$, and

  3. $\mathcal{T}_2$ is a stack in setoids.

then $\mathcal{T}_1$ is a stack in setoids.

Proof. We may assume that $\mathcal{T}_2$ is the category $\mathcal{S}_2 \times _{\mathcal{S}_1} \mathcal{T}_1$ described in Categories, Lemma 4.32.3. Pick $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $y \in \mathop{\mathrm{Ob}}\nolimits ((\mathcal{T}_1)_ U)$. We have to show that the sheaf $\mathit{Aut}(y)$ on $\mathcal{C}/U$ is trivial. To to this we may replace $U$ by the members of a covering of $U$. Hence by assumption (2) we may assume that there exists an object $x \in \mathop{\mathrm{Ob}}\nolimits ((\mathcal{S}_2)_ U)$ and an isomorphism $f : F(x) \to G(y)$. Then $y' = (U, x, y, f)$ is an object of $\mathcal{T}_2$ over $U$ which is mapped to $y$ under the projection $\mathcal{T}_2 \to \mathcal{T}_1$. Because $F$ is fully faithful by (1) the map $\mathit{Aut}(y') \to \mathit{Aut}(y)$ is surjective, use the explicit description of morphisms in $\mathcal{T}_2$ in Categories, Lemma 4.32.3. Since by (3) the sheaf $\mathit{Aut}(y')$ is trivial we get the result of the lemma. $\square$

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