The Stacks project

Lemma 8.10.1. Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. Let $\text{Cov}(\mathcal{S})$ be the set of families $\{ x_ i \to x\} _{i \in I}$ of morphisms in $\mathcal{S}$ with fixed target such that (a) each $x_ i \to x$ is strongly cartesian, and (b) $\{ p(x_ i) \to p(x)\} _{i \in I}$ is a covering of $\mathcal{C}$. Then $(\mathcal{S}, \text{Cov}(\mathcal{S}))$ is a site.

Proof. We have to check the three conditions of Sites, Definition 7.6.2.

  1. If $x \to y$ is an isomorphism of $\mathcal{S}$, then it is strongly cartesian by Categories, Lemma 4.33.2 and $p(x) \to p(y)$ is an isomorphism of $\mathcal{C}$. Thus $\{ p(x) \to p(y)\} $ is a covering of $\mathcal{C}$ whence $\{ x \to y\} \in \text{Cov}(\mathcal{S})$.

  2. If $\{ x_ i \to x\} _{i\in I} \in \text{Cov}(\mathcal{S})$ and for each $i$ we have $\{ y_{ij} \to x_ i\} _{j\in J_ i} \in \text{Cov}(\mathcal{S})$, then each composition $p(y_{ij}) \to p(x)$ is strongly cartesian by Categories, Lemma 4.33.2 and $\{ p(y_{ij}) \to p(x)\} _{i \in I, j\in J_ i} \in \text{Cov}(\mathcal{C})$. Hence also $\{ y_{ij} \to x\} _{i \in I, j\in J_ i} \in \text{Cov}(\mathcal{S})$.

  3. Suppose $\{ x_ i \to x\} _{i\in I}\in \text{Cov}(\mathcal{S})$ and $y \to x$ is a morphism of $\mathcal{S}$. As $\{ p(x_ i) \to p(x)\} $ is a covering of $\mathcal{C}$ we see that $p(x_ i) \times _{p(x)} p(y)$ exists. Hence Categories, Lemma 4.33.13 implies that $x_ i \times _ x y$ exists, that $p(x_ i \times _ x y) = p(x_ i) \times _{p(x)} p(y)$, and that $x_ i \times _ x y \to y$ is strongly cartesian. Since also $\{ p(x_ i) \times _{p(x)} p(y) \to p(y) \} _{i\in I} \in \text{Cov}(\mathcal{C})$ we conclude that $\{ x_ i \times _ x y \to y \} _{i\in I} \in \text{Cov}(\mathcal{S})$

This finishes the proof. $\square$

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