Definition 7.42.1. Let $\mathcal{C}$ be a site. We say an object $U$ of $\mathcal{C}$ is sheaf theoretically empty if $\emptyset ^\# \to h_ U^\# $ is an isomorphism of sheaves.
7.42 Almost cocontinuous functors
Let $\mathcal{C}$ be a site. The category $\textit{PSh}(\mathcal{C})$ has an initial object, namely the presheaf which assigns the empty set to each object of $\mathcal{C}$. Let us denote this presheaf by $\emptyset $. It follows from the properties of sheafification that the sheafification $\emptyset ^\# $ of $\emptyset $ is an initial object of the category $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ of sheaves on $\mathcal{C}$.
The following lemma makes this notion more explicit.
Lemma 7.42.2. Let $\mathcal{C}$ be a site. Let $U$ be an object of $\mathcal{C}$. The following are equivalent:
$U$ is sheaf theoretically empty,
$\mathcal{F}(U)$ is a singleton for each sheaf $\mathcal{F}$,
$\emptyset ^\# (U)$ is a singleton,
$\emptyset ^\# (U)$ is nonempty, and
the empty family is a covering of $U$ in $\mathcal{C}$.
Moreover, if $U$ is sheaf theoretically empty, then for any morphism $U' \to U$ of $\mathcal{C}$ the object $U'$ is sheaf theoretically empty.
Proof. For any sheaf $\mathcal{F}$ we have $\mathcal{F}(U) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(h_ U^\# , \mathcal{F})$. Hence, we see that (1) and (2) are equivalent. It is clear that (2) implies (3) implies (4). If every covering of $U$ is given by a nonempty family, then $\emptyset ^+(U)$ is empty by definition of the plus construction. Note that $\emptyset ^+ = \emptyset ^\# $ as $\emptyset $ is a separated presheaf, see Theorem 7.10.10. Thus we see that (4) implies (5). If (5) holds, then $\mathcal{F}(U)$ is a singleton for every sheaf $\mathcal{F}$ by the sheaf condition for $\mathcal{F}$, see Remark 7.7.2. Thus (5) implies (2) and (1) – (5) are equivalent. The final assertion of the lemma follows from Axiom (3) of Definition 7.6.2 applied the empty covering of $U$. $\square$
Definition 7.42.3. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. We say $u$ is almost cocontinuous if for every object $U$ of $\mathcal{C}$ and every covering $\{ V_ j \to u(U)\} _{j \in J}$ there exists a covering $\{ U_ i \to U\} _{i \in I}$ in $\mathcal{C}$ such that for each $i$ in $I$ we have at least one of the following two conditions
$u(U_ i)$ is sheaf theoretically empty, or
the morphism $u(U_ i) \to u(U)$ factors through $V_ j$ for some $j \in J$.
The motivation for this definition comes from a closed immersion $i : Z \to X$ of topological spaces. As discussed in Example 7.21.9 the continuous functor $X_{Zar} \to Z_{Zar}$, $U \mapsto Z \cap U$ is not cocontinuous. But it is almost cocontinuous in the sense defined above. We know that $i_*$ while not exact on sheaves of sets, is exact on sheaves of abelian groups, see Sheaves, Remark 6.32.5. And this holds in general for continuous and almost cocontinuous functors.
Lemma 7.42.4. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume that $u$ is continuous and almost cocontinuous. Let $\mathcal{G}$ be a presheaf on $\mathcal{D}$ such that $\mathcal{G}(V)$ is a singleton whenever $V$ is sheaf theoretically empty. Then $(u^ p\mathcal{G})^\# = u^ p(\mathcal{G}^\# )$.
Proof. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. We have to show that $(u^ p\mathcal{G})^\# (U) = u^ p(\mathcal{G}^\# )(U)$. It suffices to show that $(u^ p\mathcal{G})^+(U) = u^ p(\mathcal{G}^+)(U)$ since $\mathcal{G}^+$ is another presheaf for which the assumption of the lemma holds. We have
\[ u^ p(\mathcal{G}^+)(U) = \mathcal{G}^+(u(U)) = \mathop{\mathrm{colim}}\nolimits _\mathcal {V} \check H^0(\mathcal{V}, \mathcal{G}) \]
where the colimit is over the coverings $\mathcal{V}$ of $u(U)$ in $\mathcal{D}$. On the other hand, we see that
\[ u^ p(\mathcal{G})^+(U) = \mathop{\mathrm{colim}}\nolimits _\mathcal {U} \check H^0(u(\mathcal{U}), \mathcal{G}) \]
where the colimit is over the category of coverings $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ of $U$ in $\mathcal{C}$ and $u(\mathcal{U}) = \{ u(U_ i) \to u(U)\} _{i \in I}$. The condition that $u$ is continuous means that each $u(\mathcal{U})$ is a covering. Write $I = I_1 \amalg I_2$, where
\[ I_2 = \{ i \in I \mid u(U_ i)\text{ is sheaf theoretically empty}\} \]
Then $u(\mathcal{U})' = \{ u(U_ i) \to u(U)\} _{i \in I_1}$ is still a covering of because each of the other pieces can be covered by the empty family and hence can be dropped by Axiom (2) of Definition 7.6.2. Moreover, $\check H^0(u(\mathcal{U}), \mathcal{G}) = \check H^0(u(\mathcal{U})', \mathcal{G})$ by our assumption on $\mathcal{G}$. Finally, the condition that $u$ is almost cocontinuous implies that for every covering $\mathcal{V}$ of $u(U)$ there exists a covering $\mathcal{U}$ of $U$ such that $u(\mathcal{U})'$ refines $\mathcal{V}$. It follows that the two colimits displayed above have the same value as desired. $\square$
Lemma 7.42.5. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume that $u$ is continuous and almost cocontinuous. Then $u^ s = u^ p : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ commutes with pushouts and coequalizers (and more generally finite connected colimits).
Proof. Let $\mathcal{I}$ be a finite connected index category. Let $\mathcal{I} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$, $i \mapsto \mathcal{G}_ i$ by a diagram. We know that the colimit of this diagram is the sheafification of the colimit in the category of presheaves, see Lemma 7.10.13. Denote $\mathop{\mathrm{colim}}\nolimits ^{Psh}$ the colimit in the category of presheaves. Since $\mathcal{I}$ is finite and connected we see that $\mathop{\mathrm{colim}}\nolimits ^{Psh}_ i \mathcal{G}_ i$ is a presheaf satisfying the assumptions of Lemma 7.42.4 (because a finite connected colimit of singleton sets is a singleton). Hence that lemma gives
\begin{align*} u^ s(\mathop{\mathrm{colim}}\nolimits _ i \mathcal{G}_ i) & = u^ s((\mathop{\mathrm{colim}}\nolimits ^{Psh}_ i \mathcal{G}_ i)^\# ) \\ & = (u^ p(\mathop{\mathrm{colim}}\nolimits ^{Psh}_ i \mathcal{G}_ i))^\# \\ & = (\mathop{\mathrm{colim}}\nolimits ^{PSh}_ i u^ p(\mathcal{G}_ i))^\# \\ & = \mathop{\mathrm{colim}}\nolimits _ i u^ s(\mathcal{G}_ i) \end{align*}
as desired. $\square$
Lemma 7.42.6. Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites associated to the continuous functor $u : \mathcal{C} \to \mathcal{D}$. If $u$ is almost cocontinuous then $f_*$ commutes with pushouts and coequalizers (and more generally finite connected colimits).
Proof. This is a special case of Lemma 7.42.5. $\square$
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