The Stacks project

Lemma 8.6.9. Let $\mathcal{C}$ be a site. Let

\[ \xymatrix{ \mathcal{T}_2 \ar[r] \ar[d]_{G'} & \mathcal{T}_1 \ar[d]^ G \\ \mathcal{S}_2 \ar[r]^ F & \mathcal{S}_1 } \]

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

  1. 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

  2. $G'$ is faithful,

then $G$ is faithful.

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. By Categories, Lemma 4.35.8 the faithfulness of $G, G'$ can be checked on fibre categories. Suppose that $y, y'$ are objects of $\mathcal{T}_1$ over the object $U$ of $\mathcal{C}$. Let $\alpha , \beta : y \to y'$ be morphisms of $(\mathcal{T}_1)_ U$ such that $G(\alpha ) = G(\beta )$. Our object is to show that $\alpha = \beta $. Considering instead $\gamma = \alpha ^{-1} \circ \beta $ we see that $G(\gamma ) = \text{id}_{G(y)}$ and we have to show that $\gamma = \text{id}_ y$. By assumption we can find a covering $\{ U_ i \to U\} $ such that $G(y)|_{U_ i}$ is in the essential image of $F :(\mathcal{S}_2)_{U_ i} \to (\mathcal{S}_1)_{U_ i}$. Since it suffices to show that $\gamma |_{U_ i} = \text{id}$ for each $i$, we may therefore assume that we have $f : F(x) \to G(y)$ for some object $x$ of $\mathcal{S}_2$ over $U$ and morphisms $f$ of $(\mathcal{S}_1)_ U$. In this case we get a morphism

\[ (1, \gamma ) : (U, x, y, f) \longrightarrow (U, x, y, f) \]

in the fibre category of $\mathcal{S}_2 \times _{\mathcal{S}_1} \mathcal{T}_1$ over $U$ whose image under $G'$ in $\mathcal{S}_1$ is $\text{id}_ x$. As $G'$ is faithful we conclude that $\gamma = \text{id}_ y$ and we win. $\square$


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