The Stacks project

Proof. There exists a $j_0 \in J$ such that $f$ is defined by $f' : V_{j_0} \to W$. For $j \geq j_0$ we set $V_{j, k} = V_ j \times _{f'\circ f_{j j_0}, W} W_ k$. Then $\{ V_{j, k} \to V_ j\} $ is a finite covering in the site $\mathcal{C}$. Hence $\mathcal{F}(V_ j) \subset \prod _ k \mathcal{F}(V_{j, k})$. By Categories, Lemma 4.19.2 once again we see that

is injective. Hence there exists a $k$ such that $s$ and $s'$ have distinct image in $\mathop{\mathrm{colim}}\nolimits _ j \mathcal{F}(V_{j, k})$. Let $J_0 = \{ j \in J, j \geq j_0\} $ and $I = J \amalg J_0$. We order $I$ so that no element of the second summand is smaller than any element of the first, but otherwise using the ordering on $J$. If $j \in I$ is in the first summand then we use $V_ j$ and if $j \in I$ is in the second summand then we use $V_{j, k}$. We omit the definition of the transition maps of the inverse system. By the above it follows that $s, s'$ have distinct image in $\mathcal{F}_ p$. Moreover, the restriction of $f'$ to $V_{j, k}$ factors through $W_ k$ by construction. $\square$

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

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. $\square$

Proof. We have to show that given any sheaf $\mathcal{F}$ on $\mathcal{C}$, any $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and any distinct sections $s, s' \in \mathcal{F}(U)$, there exists a point $p$ such that $s, s'$ have distinct image in $\mathcal{F}_ p$. See Lemma 7.38.3. Consider the system $(J, \geq , V_ j, g_{jj'})$ with $J = \{ 1\} $, $V_1 = U$, $g_{11} = \text{id}_ U$. Apply Lemma 7.39.2. By the result of that lemma we get a system $(I, \geq , U_ i, f_{ii'})$ refining our system such that $s_ p \not= s'_ p$ and such that moreover for every finite covering $\{ W_ k \to W\} $ of the site $\mathcal{C}$ the map $\coprod _ k u(W_ k) \to u(W)$ is surjective. Since every covering of $\mathcal{C}$ can be refined by a finite covering we conclude that $\coprod _ k u(W_ k) \to u(W)$ is surjective for any covering $\{ W_ k \to W\} $ of the site $\mathcal{C}$. This implies that $u = p$ is a point, see Proposition 7.33.3 (and the discussion at the beginning of this section which guarantees that $u$ commutes with finite limits). $\square$

Proof. By assumption (2) and the proposition $\mathcal{C}/U_ i$ has enough points. The points of $\mathcal{C}/U_ i$ give points of $\mathcal{C}$ via the procedure of Lemma 7.34.2. Thus it suffices to show: if $\phi : \mathcal{F} \to \mathcal{G}$ is a map of sheaves on $\mathcal{C}$ such that $\phi |_{\mathcal{C}/U_ i}$ is an isomorphism for all $i$, then $\phi $ is an isomorphism. By assumption (1) for every object $W$ of $\mathcal{C}$ there is a covering $\{ W_ j \to W\} _{j \in J}$ such that for $j \in J$ there is an $i \in I$ and a morphism $f_ j : W_ j \to U_ i$. Then the maps $\mathcal{F}(W_ j) \to \mathcal{G}(W_ j)$ are bijective and similarly for $\mathcal{F}(W_ j \times _ W W_{j'}) \to \mathcal{G}(W_ j \times _ W W_{j'})$. The sheaf condition tells us that $\mathcal{F}(W) \to \mathcal{G}(W)$ is bijective as desired. $\square$