Theorem 7.49.3. Let $\mathcal{C}$ be a category. Let $J$ be a topology on $\mathcal{C}$. Let $\mathcal{F}$ be a presheaf of sets.
The presheaf $L\mathcal{F}$ is separated.
If $\mathcal{F}$ is separated, then $L\mathcal{F}$ is a sheaf and the map of presheaves $\mathcal{F} \to L\mathcal{F}$ is injective.
If $\mathcal{F}$ is a sheaf, then $\mathcal{F} \to L\mathcal{F}$ is an isomorphism.
The presheaf $LL\mathcal{F}$ is always a sheaf.
Proof. Part (3) is trivial from the definition of $L$ and the definition of a sheaf (Definition 7.47.10). Part (4) follows formally from the others.
We sketch the proof of (1). Suppose $S$ is a covering sieve of the object $U$. Suppose that $\varphi _ i \in L\mathcal{F}(U)$, $i = 1, 2$ map to the same element in $\mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S, L\mathcal{F})$. We may find a single covering sieve $S'$ on $U$ such that both $\varphi _ i$ are represented by elements $\varphi _ i \in \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S', \mathcal{F})$. We may assume that $S' = S$ by replacing both $S$ and $S'$ by $S' \cap S$ which is also a covering sieve, see Lemma 7.47.2. Suppose $V\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and $\alpha : V \to U$ in $S(V)$. Then we have $S \times _ U V = h_ V$, see Lemma 7.47.5. Thus the restrictions of $\varphi _ i$ via $V \to U$ correspond to sections $s_{i, V, \alpha }$ of $\mathcal{F}$ over $V$. The assumption is that there exist a covering sieve $S_{V, \alpha }$ of $V$ such that $s_{i, V, \alpha }$ restrict to the same element of $\mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S_{V, \alpha }, \mathcal{F})$. Consider the sieve $S''$ on $U$ defined by the rule
By axiom (2) of a topology we see that $S''$ is a covering sieve on $U$. By construction we see that $\varphi _1$ and $\varphi _2$ restrict to the same element of $\mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S'', L\mathcal{F})$ as desired.
We sketch the proof of (2). Assume that $\mathcal{F}$ is a separated presheaf of sets on $\mathcal{C}$ with respect to the topology $J$. Let $S$ be a covering sieve of the object $U$ of $\mathcal{C}$. Suppose that $\varphi \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(S, L\mathcal{F})$. We have to find an element $s \in L\mathcal{F}(U)$ restricting to $\varphi $. Suppose $V\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and $\alpha : V \to U$ in $S(V)$. The value $\varphi (\alpha ) \in L\mathcal{F}(V)$ is given by a covering sieve $S_{V, \alpha }$ of $V$ and a morphism of presheaves $\varphi _{V, \alpha } : S_{V, \alpha } \to \mathcal{F}$. As in the proof above, define a covering sieve $S''$ on $U$ by Equation (7.49.3.1). We define
by the following simple rule: For every $f : T \to U$, $f \in S''(T)$ choose $V, \alpha , g$ as in Equation (7.49.3.1). Then set
We claim this is independent of the choice of $V, \alpha , g$. Consider a second such choice$ V', \alpha ', g'$. The restrictions of $\varphi _{V, \alpha }$ and $\varphi _{V', \alpha '}$ to the intersection of the following covering sieves on $T$
agree. Namely, these restrictions both correspond to the restriction of $\varphi $ to $T$ (via $f$) and the desired equality follows because $\mathcal{F}$ is separated. Denote the common restriction $\psi $. The independence of choice follows because $\varphi _{V, \alpha }(g) = \psi (\text{id}_ T) = \varphi _{V', \alpha '}(g')$. OK, so now $\varphi ''$ gives an element $s \in L\mathcal{F}(U)$. We leave it to the reader to check that $s$ restricts to $\varphi $. $\square$
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