Lemma 7.32.8. Let $\mathcal{C}$ be a site. Let $p$ be a point of $\mathcal{C}$ given by $u : \mathcal{C} \to \textit{Sets}$. Let $S_0$ be an infinite set such that $u(U) \subset S_0$ for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $\mathcal{S}$ be the site constructed out of the powerset $S = \mathcal{P}(S_0)$ in Remark 7.15.3. Then

1. there is an equivalence $i : \mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{S})$,

2. the functor $u : \mathcal{C} \to \mathcal{S}$ induces a morphism of sites $f : \mathcal{S} \to \mathcal{C}$, and

3. the composition

$\mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{S}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$

is the morphism of topoi $(p_*, p^{-1})$ of Lemma 7.32.7.

Proof. Part (1) we saw in Remark 7.15.3. Moreover, recall that the equivalence associates to the set $E$ the sheaf $i_*E$ on $\mathcal{S}$ defined by the rule $V \mapsto \mathop{Mor}\nolimits _{\textit{Sets}}(V, E)$. Part (2) is clear from the definition of a point of $\mathcal{C}$ (Definition 7.32.2) and the definition of a morphism of sites (Definition 7.14.1). Finally, consider $f_*i_*E$. By construction we have

$f_*i_*E(U) = i_*E(u(U)) = \mathop{Mor}\nolimits _{\textit{Sets}}(u(U), E)$

which is equal to $p_*E(U)$, see Equation (7.32.3.1). This proves (3). $\square$

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