Lemma 7.39.2. Let $\mathcal{C}$ be a site. Let $(J, \geq , V_ j, g_{jj'})$ be a system as above with associated pair of functors $(u', p')$. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Let $s, s' \in \mathcal{F}_{p'}$ be distinct elements. There exists a refinement $(I, \geq , U_ i, f_{ii'})$ of $(J, \geq , V_ j, g_{jj'})$ such that $s, s'$ map to distinct elements of $\mathcal{F}_ p$ and such that for every finite covering $\{ W_ k \to W\}$ of the site $\mathcal{C}$, and any $f \in u'(W)$ the image of $f$ in $u(W)$ is in the image of one of the $u(W_ k)$.

Proof. Let $E$ be the set of pairs $(\{ W_ k \to W\} , f\in u'(W))$. Consider pairs $(E' \subset E, (I, \geq , U_ i, f_{ii'}))$ such that

1. $(I, \geq , U_ i, g_{ii'})$ is a refinement of $(J, \geq , V_ j, g_{jj'})$,

2. $s, s'$ map to distinct elements of $\mathcal{F}_ p$, and

3. for every pair $(\{ W_ k \to W\} , f\in u'(W)) \in E'$ we have that the image of $f$ in $u(W)$ is in the image of one of the $u(W_ k)$.

We order such pairs by inclusion in the first factor and by refinement in the second. Denote $\mathcal{S}$ the class of all pairs $(E' \subset E, (I, \geq , U_ i, f_{ii'}))$ as above. We claim that the hypothesis of Zorn's lemma holds for $\mathcal{S}$. Namely, suppose that $(E'_ a, (I_ a, \geq , U_ i, f_{ii'}))_{a \in A}$ is a totally ordered subset of $\mathcal{S}$. Then we can define $E' = \bigcup _{a \in A} E'_ a$ and we can set $I = \bigcup _{a \in A} I_ a$. We claim that the corresponding pair $(E' , (I, \geq , U_ i, f_{ii'}))$ is an element of $\mathcal{S}$. Conditions (1) and (3) are clear. For condition (2) you note that

$u = \mathop{\mathrm{colim}}\nolimits _{a \in A} u_ a \text{ and correspondingly } \mathcal{F}_ p = \mathop{\mathrm{colim}}\nolimits _{a \in A} \mathcal{F}_{p_ a}$

The distinctness of the images of $s, s'$ in this stalk follows from the description of a directed colimit of sets, see Categories, Section 4.19. We will simply write $(E', (I, \ldots )) = \bigcup _{a \in A}(E'_ a, (I_ a, \ldots ))$ in this situation.

OK, so Zorn's Lemma would apply if $\mathcal{S}$ was a set, and this would, combined with Lemma 7.39.1 above easily prove the lemma. It doesn't since $\mathcal{S}$ is a class. In order to circumvent this we choose a well ordering on $E$. For $e \in E$ set $E'_ e = \{ e' \in E \mid e' \leq e\}$. Using transfinite recursion we construct pairs $(E'_ e, (I_ e, \ldots )) \in \mathcal{S}$ such that $e_1 \leq e_2 \Rightarrow (E'_{e_1}, (I_{e_1}, \ldots )) \leq (E'_{e_2}, (I_{e_2}, \ldots ))$. Let $e \in E$, say $e = (\{ W_ k \to W\} , f\in u'(W))$. If $e$ has a predecessor $e - 1$, then we let $(I_ e, \ldots )$ be a refinement of $(I_{e - 1}, \ldots )$ as in Lemma 7.39.1 with respect to the system $e = (\{ W_ k \to W\} , f\in u'(W))$. If $e$ does not have a predecessor, then we let $(I_ e, \ldots )$ be a refinement of $\bigcup _{e' < e} (I_{e'}, \ldots )$ with respect to the system $e = (\{ W_ k \to W\} , f\in u'(W))$. Finally, the union $\bigcup _{e \in E} I_ e$ will be a solution to the problem posed in the lemma. $\square$

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