Let $u : \mathcal{C} \to \mathcal{D}$ be a functor between sites. Assume that
$u$ is cocontinuous, and
$u$ is continuous.
Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be the morphism of topoi associated with $u$, see Sites, Lemma 7.21.1. Recall that $g^{-1} = u^ p$, i.e., $g^{-1}$ is given by the simple formula $(g^{-1}\mathcal{G})(U) = \mathcal{G}(u(U))$, see Sites, Lemma 7.21.5. We would like to show that $g^{-1} : \textit{Ab}(\mathcal{D}) \to \textit{Ab}(\mathcal{C})$ has a left adjoint $g_!$. By Sites, Lemma 7.21.5 the functor $g^{Sh}_! = (u_ p\ )^\# $ is a left adjoint on sheaves of sets. Moreover, we know that $g^{Sh}_!\mathcal{F}$ is the sheaf associated to the presheaf
where the colimit is over $(\mathcal{I}_ V^ u)^{opp}$ and is taken in the category of sets. Hence the following definition is natural.
Definition 18.16.1. With $u : \mathcal{C} \to \mathcal{D}$ satisfying (a), (b) above. For $\mathcal{F} \in \textit{PAb}(\mathcal{C})$ we define $g_{p!}\mathcal{F}$ as the presheaf with colimits over $(\mathcal{I}_ V^ u)^{opp}$ taken in $\textit{Ab}$. For $\mathcal{F} \in \textit{PAb}(\mathcal{C})$ we set $g_!\mathcal{F} = (g_{p!}\mathcal{F})^\# $.
\[ V \longmapsto \mathop{\mathrm{colim}}\nolimits _{V \to u(U)} \mathcal{F}(U) \]
The reason for being so explicit with this is that the functors $g^{Sh}_!$ and $g_!$ are different. Whenever we use both we have to be careful to make the distinction clear.
Lemma 18.16.2. The functor $g_{p!}$ is a left adjoint to the functor $u^ p$. The functor $g_!$ is a left adjoint to the functor $g^{-1}$. In other words the formulas hold bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.
\begin{align*} \mathop{\mathrm{Mor}}\nolimits _{\textit{PAb}(\mathcal{C})}(\mathcal{F}, u^ p\mathcal{G}) & = \mathop{\mathrm{Mor}}\nolimits _{\textit{PAb}(\mathcal{D})}(g_{p!}\mathcal{F}, \mathcal{G}), \\ \mathop{\mathrm{Mor}}\nolimits _{\textit{Ab}(\mathcal{C})}(\mathcal{F}, g^{-1}\mathcal{G}) & = \mathop{\mathrm{Mor}}\nolimits _{\textit{Ab}(\mathcal{D})}(g_!\mathcal{F}, \mathcal{G}) \end{align*}
Proof. The second formula follows formally from the first, since if $\mathcal{F}$ and $\mathcal{G}$ are abelian sheaves then
by the universal property of sheafification.
To prove the first formula, let $\mathcal{F}$, $\mathcal{G}$ be abelian presheaves. To prove the lemma we will construct maps from the group on the left to the group on the right and omit the verification that these are mutually inverse.
Note that there is a canonical map of abelian presheaves $\mathcal{F} \to u^ pg_{p!}\mathcal{F}$ which on sections over $U$ is the natural map $\mathcal{F}(U) \to \mathop{\mathrm{colim}}\nolimits _{u(U) \to u(U')} \mathcal{F}(U')$, see Sites, Lemma 7.5.3. Given a map $\alpha : g_{p!}\mathcal{F} \to \mathcal{G}$ we get $u^ p\alpha : u^ pg_{p!}\mathcal{F} \to u^ p\mathcal{G}$. which we can precompose by the map $\mathcal{F} \to u^ pg_{p!}\mathcal{F}$.
Note that there is a canonical map of abelian presheaves $g_{p!}u^ p\mathcal{G} \to \mathcal{G}$ which on sections over $V$ is the natural map $\mathop{\mathrm{colim}}\nolimits _{V \to u(U)} \mathcal{G}(u(U)) \to \mathcal{G}(V)$. It maps a section $s \in u(U)$ in the summand corresponding to $t : V \to u(U)$ to $t^*s \in \mathcal{G}(V)$. Hence, given a map $\beta : \mathcal{F} \to u^ p\mathcal{G}$ we get a map $g_{p!}\beta : g_{p!}\mathcal{F} \to g_{p!}u^ p\mathcal{G}$ which we can postcompose with the map $g_{p!}u^ p\mathcal{G} \to \mathcal{G}$ above. $\square$
Lemma 18.16.3. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume that
$u$ is cocontinuous,
$u$ is continuous, and
fibre products and equalizers exist in $\mathcal{C}$ and $u$ commutes with them.
In this case the functor $g_! : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})$ is exact.
Proof. Compare with Sites, Lemma 7.21.6. Assume (a), (b), and (c). We already know that $g_!$ is right exact as it is a left adjoint, see Categories, Lemma 4.24.6 and Homology, Section 12.7. We have $g_! = (g_{p!}\ )^\# $. We have to show that $g_!$ transforms injective maps of abelian sheaves into injective maps of abelian presheaves. Recall that sheafification of abelian presheaves is exact, see Lemma 18.3.2. Thus it suffices to show that $g_{p!}$ transforms injective maps of abelian presheaves into injective maps of abelian presheaves. To do this it suffices that colimits over the categories $(\mathcal{I}_ V^ u)^{opp}$ of Sites, Section 7.5 transform injective maps between diagrams into injections. This follows from Sites, Lemma 7.5.1 and Algebra, Lemma 10.8.10. $\square$
Lemma 18.16.4. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume that
$u$ is cocontinuous,
$u$ is continuous, and
$u$ is fully faithful.
For $g_!, g^{-1}, g_*$ as above the canonical maps $\mathcal{F} \to g^{-1}g_!\mathcal{F}$ and $g^{-1}g_*\mathcal{F} \to \mathcal{F}$ are isomorphisms for all abelian sheaves $\mathcal{F}$ on $\mathcal{C}$.
Proof. The map $g^{-1}g_*\mathcal{F} \to \mathcal{F}$ is an isomorphism by Sites, Lemma 7.21.7 and the fact that pullback and pushforward of abelian sheaves agrees with pullback and pushforward on the underlying sheaves of sets.
Pick $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. We will show that $g^{-1}g_!\mathcal{F}(U) = \mathcal{F}(U)$. First, note that $g^{-1}g_!\mathcal{F}(U) = g_!\mathcal{F}(u(U))$. Hence it suffices to show that $g_!\mathcal{F}(u(U)) = \mathcal{F}(U)$. We know that $g_!\mathcal{F}$ is the (abelian) sheaf associated to the presheaf $g_{p!}\mathcal{F}$ which is defined by the rule
with colimit taken in $\textit{Ab}$. If $V = u(U)$, then, as $u$ is fully faithful this colimit is over $U \to U'$. Hence we conclude that $g_{p!}\mathcal{F}(u(U) = \mathcal{F}(U)$. Since $u$ is cocontinuous and continuous any covering of $u(U)$ in $\mathcal{D}$ can be refined by a covering (!) $\{ u(U_ i) \to u(U)\} $ of $\mathcal{D}$ where $\{ U_ i \to U\} $ is a covering in $\mathcal{C}$. This implies that $(g_{p!}\mathcal{F})^+(u(U)) = \mathcal{F}(U)$ also, since in the colimit defining the value of $(g_{p!}\mathcal{F})^+$ on $u(U)$ we may restrict to the cofinal system of coverings $\{ u(U_ i) \to u(U)\} $ as above. Hence we see that $(g_{p!}\mathcal{F})^+(u(U)) = \mathcal{F}(U)$ for all objects $U$ of $\mathcal{C}$ as well. Repeating this argument one more time gives the equality $(g_{p!}\mathcal{F})^\# (u(U)) = \mathcal{F}(U)$ for all objects $U$ of $\mathcal{C}$. This produces the desired equality $g^{-1}g_!\mathcal{F} = \mathcal{F}$. $\square$
Remark 18.16.5. In general the functor $g_!$ cannot be extended to categories of modules in case $g$ is (part of) a morphism of ringed topoi. Namely, given any ring map $A \to B$ the functor $M \mapsto B \otimes _ A M$ has a right adjoint (restriction) but not in general a left adjoint (because its existence would imply that $A \to B$ is flat). We will see in Section 18.19 below that it is possible to define $j_!$ on sheaves of modules in the case of a localization of sites. We will discuss this in greater generality in Section 18.41 below.
Lemma 18.16.6. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be the morphism of topoi associated to a continuous and cocontinuous functor $u : \mathcal{C} \to \mathcal{D}$.
If $u$ has a left adjoint $w$, then $g_!$ agrees with $g_!^{\mathop{\mathit{Sh}}\nolimits }$ on underlying sheaves of sets and $g_!$ is exact.
If in addition $w$ is cocontinuous, then $g_! = h^{-1}$ and $g^{-1} = h_*$ where $h : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is the morphism of topoi associated to $w$.
Proof. This Lemma is the analogue of Sites, Lemma 7.23.1. From Sites, Lemma 7.19.3 we see that the categories $\mathcal{I}_ V^ u$ have an initial object. Thus the underlying set of a colimit of a system of abelian groups over $(\mathcal{I}_ V^ u)^{opp}$ is the colimit of the underlying sets. Whence the agreement of $g_!^{\mathop{\mathit{Sh}}\nolimits }$ and $g_!$ by our construction of $g_!$ in Definition 18.16.1. The exactness and (2) follow immediately from the corresponding statements of Sites, Lemma 7.23.1. $\square$
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