The Stacks project

Lemma 7.20.2. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be cocontinuous. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Then ${}_ pu\mathcal{F}$ is a sheaf on $\mathcal{D}$, which we will denote ${}_ su\mathcal{F}$.

Proof. Let $\{ V_ j \to V\} _{j \in J}$ be a covering of the site $\mathcal{D}$. We have to show that

\[ \xymatrix{ & \ \phantom{}_ pu\mathcal{F}(V) \ar[r] & \prod {}_ pu\mathcal{F}(V_ j) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod {}_ pu\mathcal{F}(V_ j \times _ V V_{j'}) & } \]

is an equalizer diagram. Since ${}_ pu$ is right adjoint to $u^ p$ we have

\[ {}_ pu\mathcal{F}(V) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{D})}(h_ V, {}_ pu\mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(u^ ph_ V, \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}((u^ ph_ V)^\# , \mathcal{F}) \]

Hence it suffices to show that
\begin{equation} \label{sites-equation-coequalizer} \xymatrix{ \coprod u^ p h_{V_ j \times _ V V_{j'}} \ar@<1ex>[r] \ar@<-1ex>[r] & \coprod u^ p h_{V_ j} \ar[r] & u^ p h_ V } \end{equation}

becomes a coequalizer diagram after sheafification. (Recall that a coproduct in the category of sheaves is the sheafification of the coproduct in the category of presheaves, see Lemma 7.10.13.)

We first show that the second arrow of ( becomes surjective after sheafification. To do this we use Lemma 7.11.2. Thus it suffices to show a section $s$ of $u^ ph_ V$ over $U$ lifts to a section of $\coprod u^ p h_{V_ j}$ on the members of a covering of $U$. Note that $s$ is a morphism $s : u(U) \to V$. Then $\{ V_ j \times _{V, s} u(U) \to u(U)\} $ is a covering of $\mathcal{D}$. Hence, as $u$ is cocontinuous, there is a covering $\{ U_ i \to U\} $ such that $\{ u(U_ i) \to u(U)\} $ refines $\{ V_ j \times _{V, s} u(U) \to u(U)\} $. This means that each restriction $s|_{U_ i} : u(U_ i) \to V$ factors through a morphism $s_ i : u(U_ i) \to V_ j$ for some $j$, i.e., $s|_{U_ i}$ is in the image of $u^ ph_{V_ j}(U_ i) \to u^ ph_ V(U_ i)$ as desired.

Let $s, s' \in (\coprod u^ ph_{V_ j})^\# (U)$ map to the same element of $(u^ ph_ V)^\# (U)$. To finish the proof of the lemma we show that after replacing $U$ by the members of a covering that $s, s'$ are the image of the same section of $\coprod u^ p h_{V_ j \times _ V V_{j'}}$ by the two maps of ( We may first replace $U$ by the members of a covering and assume that $s \in u^ ph_{V_ j}(U)$ and $s' \in u^ ph_{V_{j'}}(U)$. A second such replacement guarantees that $s$ and $s'$ have the same image in $u^ ph_ V(U)$ instead of in the sheafification. Hence $s : u(U) \to V_ j$ and $s' : u(U) \to V_{j'}$ are morphisms of $\mathcal{D}$ such that

\[ \xymatrix{ u(U) \ar[r]_{s'} \ar[d]_ s & V_{j'} \ar[d] \\ V_ j \ar[r] & V } \]

is commutative. Thus we obtain $t = (s, s') : u(U) \to V_ j \times _ V V_{j'}$, i.e., a section $t \in u^ ph_{V_ j \times _ V V_{j'}}(U)$ which maps to $s, s'$ as desired. $\square$

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