The Stacks project

Lemma 7.35.1. Let $\mathcal{C}$ be a site. Let $p$ be a point of $\mathcal{C}$ given by $u : \mathcal{C} \to \textit{Sets}$. Let $U$ be an object of $\mathcal{C}$ and let $x \in u(U)$. The functor

\[ v : \mathcal{C}/U \longrightarrow \textit{Sets}, \quad (\varphi : V \to U) \longmapsto \{ y \in u(V) \mid u(\varphi )(y) = x\} \]

defines a point $q$ of the site $\mathcal{C}/U$ such that the diagram

\[ \xymatrix{ & \mathop{\mathit{Sh}}\nolimits (pt) \ar[d]^ p \ar[ld]_ q \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[r]^{j_ U} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) } \]

commutes. In other words $\mathcal{F}_ p = (j_ U^{-1}\mathcal{F})_ q$ for any sheaf on $\mathcal{C}$.

Proof. Choose $S$ and $\mathcal{S}$ as in Lemma 7.32.8. We may identify $\mathop{\mathit{Sh}}\nolimits (pt) = \mathop{\mathit{Sh}}\nolimits (\mathcal{S})$ as in that lemma, and we may write $p = f : \mathop{\mathit{Sh}}\nolimits (\mathcal{S}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ for the morphism of topoi induced by $u$. By Lemma 7.28.1 we get a commutative diagram of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{S}/u(U)) \ar[r]_-{j_{u(U)}} \ar[d]_{p'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{S}) \ar[d]^ p \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[r]^{j_ U} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}), } \]

where $p'$ is given by the functor $u' : \mathcal{C}/U \to \mathcal{S}/u(U)$, $V/U \mapsto u(V)/u(U)$. Consider the functor $j_ x : \mathcal{S} \cong \mathcal{S}/x$ obtained by assigning to a set $E$ the set $E$ endowed with the constant map $E \to u(U)$ with value $x$. Then $j_ x$ is a fully faithful cocontinuous functor which has a continuous right adjoint $v_ x : (\psi : E \to u(U)) \mapsto \psi ^{-1}(\{ x\} )$. Note that $j_{u(U)} \circ j_ x = \text{id}_\mathcal {S}$, and $v_ x \circ u' = v$. These observations imply that we have the following commutative diagram of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{S}) \ar[rd]^ a \ar[rdd]_ q \ar `r[rrr] `d[dd]^ p [rrdd] & & & \\ & \mathop{\mathit{Sh}}\nolimits (\mathcal{S}/u(U)) \ar[r]_-{j_{u(U)}} \ar[d]^{p'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{S}) \ar[d]^ p & \\ & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[r]^{j_ U} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) & } \]


  1. The morphism $a : \mathop{\mathit{Sh}}\nolimits (\mathcal{S}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{S}/u(U))$ is the morphism of topoi associated to the cocontinuous functor $j_ x$, which equals the morphism associated to the continuous functor $v_ x$, see Lemma 7.21.1 and Section 7.22.

  2. The composition $p \circ j_{u(U)} \circ a = p$ since $j_{u(U)} \circ j_ x = \text{id}_\mathcal {S}$.

  3. The composition $p' \circ a$ gives a morphism of topoi. Moreover, it is the morphism of topoi associated to the continuous functor $v_ x \circ u' = v$. Hence $v$ does indeed define a point $q$ of $\mathcal{C}/U$ which fits into the diagram above by construction.

This ends the proof of the lemma. $\square$

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