The Stacks project

Lemma 7.18.2. In Situation 7.18.1 we can construct a site $(\mathcal{C}, \text{Cov}(\mathcal{C}))$ as follows

  1. as a category $\mathcal{C} = \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i$, and

  2. $\text{Cov}(\mathcal{C})$ is the union of the images of $\text{Cov}(\mathcal{C}_ i)$ by $u_ i : \mathcal{C}_ i \to \mathcal{C}$.

Proof. Our definition of composition of morphisms of sites implies that $u_ b \circ u_ a = u_ c$ whenever $c = a \circ b$ in $\mathcal{I}$. The formula $\mathcal{C} = \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i$ means that $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ i)$ and $\text{Arrows}(\mathcal{C}) = \mathop{\mathrm{colim}}\nolimits \text{Arrows}(\mathcal{C}_ i)$. Then source, target, and composition are inherited from the source, target, and composition on $\text{Arrows}(\mathcal{C}_ i)$. In this way we obtain a category. Denote $u_ i : \mathcal{C}_ i \to \mathcal{C}$ the obvious functor. Remark that given any finite diagram in $\mathcal{C}$ there exists an $i$ such that this diagram is the image of a diagram in $\mathcal{C}_ i$.

Let $\{ U^ t \to U\} $ be a covering of $\mathcal{C}$. We first prove that if $V \to U$ is a morphism of $\mathcal{C}$, then $U^ t \times _ U V$ exists. By our remark above and our definition of coverings, we can find an $i$, a covering $\{ U_ i^ t \to U_ i\} $ of $\mathcal{C}_ i$ and a morphism $V_ i \to U_ i$ whose image by $u_ i$ is the given data. We claim that $U^ t \times _ U V$ is the image of $U^ t_ i \times _{U_ i} V_ i$ by $u_ i$. Namely, for every $a : j \to i$ in $\mathcal{I}$ the functor $u_ a$ is continuous, hence $u_ a(U^ t_ i \times _{U_ i} V_ i) = u_ a(U^ t_ i) \times _{u_ a(U_ i)} u_ a(V_ i)$. In particular we can replace $i$ by $j$, if we so desire. Thus, if $W$ is another object of $\mathcal{C}$, then we may assume $W = u_ i(W_ i)$ and we see that

\begin{align*} & \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, u_ i(U^ t_ i \times _{U_ i} V_ i)) \\ & = \mathop{\mathrm{colim}}\nolimits _{a : j \to i} \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_ j}(u_ a(W_ i), u_ a(U^ t_ i \times _{U_ i} V_ i)) \\ & = \mathop{\mathrm{colim}}\nolimits _{a : j \to i} \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_ j}(u_ a(W_ i), u_ a(U^ t_ i)) \times _{\mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_ j}(u_ a(W_ i), u_ a(U_ i))} \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_ j}(u_ a(W_ i), u_ a(V_ i)) \\ & = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, U^ t) \times _{\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, U)} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, V) \end{align*}

as filtered colimits commute with finite limits (Categories, Lemma 4.19.2). It also follows that $\{ U^ t \times _ U V \to V\} $ is a covering in $\mathcal{C}$. In this way we see that axiom (3) of Definition 7.6.2 holds.

To verify axiom (2) of Definition 7.6.2 let $\{ U^ t \to U\} _{t \in T}$ be a covering of $\mathcal{C}$ and for each $t$ let $\{ U^{ts} \to U^ t\} $ be a covering of $\mathcal{C}$. Then we can find an $i$ and a covering $\{ U^ t_ i \to U_ i\} _{t \in T}$ of $\mathcal{C}_ i$ whose image by $u_ i$ is $\{ U^ t \to U\} $. Since $T$ is finite we may choose an $a : j \to i$ in $\mathcal{I}$ and coverings $\{ U^{ts}_ j \to u_ a(U^ t_ i)\} $ of $\mathcal{C}_ j$ whose image by $u_ j$ gives $\{ U^{ts} \to U^ t\} $. Then we conclude that $\{ U^{ts} \to U\} $ is a covering of $\mathcal{C}$ by an application of axiom (2) to the site $\mathcal{C}_ j$.

We omit the proof of axiom (1) of Definition 7.6.2. $\square$

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