It sometimes happens that a site $\mathcal{C}$ does not have a final object. In this case we define the global section functor as follows.
Definition 7.45.1. The global sections of a presheaf of sets $\mathcal{F}$ over a site $\mathcal{C}$ is the set where $*$ is the final object in the category of presheaves on $\mathcal{C}$, i.e., the presheaf which associates to every object a singleton.
\[ \Gamma (\mathcal{C}, \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(*, \mathcal{F}) \]
Of course the same definition applies to sheaves as well. Here is one way to compute global sections.
Lemma 7.45.2. Let $\mathcal{C}$ be a site. Let $a, b : V \to U$ be objects of $\mathcal{C}$ such that is a coequalizer in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. Then $\Gamma (\mathcal{C}, \mathcal{F})$ is the equalizer of $a^*, b^* : \mathcal{F}(U) \to \mathcal{F}(V)$.
\[ \xymatrix{ h_ V^\# \ar@<1ex>[r] \ar@<-1ex>[r] & h_ U^\# \ar[r] & {*} } \]
Proof. Since $\mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(h_ U^\# , \mathcal{F}) = \mathcal{F}(U)$ this is clear from the definitions. $\square$
Now, let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a morphism of topoi. Then for any sheaf $\mathcal{F}$ on $\mathcal{C}$ there is a pullback map
Namely, as $f^{-1}$ is exact it transforms $*$ into $*$. Hence a global section $s$ of $\mathcal{F}$ over $\mathcal{C}$, which is a map of sheaves $s : * \to \mathcal{F}$, can be pulled back to $f^{-1}s : * = f^{-1}* \to f^{-1}\mathcal{F}$.
We can generalize this a bit by considering a pair of sheaves $\mathcal{F}$, $\mathcal{G}$ on $\mathcal{C}$, $\mathcal{D}$ together with a map $f^{-1}\mathcal{F} \to \mathcal{G}$. Then we compose the construction above with the obvious map $\Gamma (\mathcal{D}, f^{-1}\mathcal{F}) \to \Gamma (\mathcal{D}, \mathcal{G})$ to get a map
This map is sometimes also called a pullback map.
A slightly more general construction which occurs frequently in nature is the following. Suppose that we have a commutative diagram of morphisms of topoi
Next, suppose that we have a sheaf $\mathcal{F}$ on $\mathcal{C}$. Then there is a pullback map
Namely, it is just the map coming from the identification $g_*f_*f^{-1}\mathcal{F} = h_*f^{-1}\mathcal{F}$ together with $g_*$ applied to the canonical map $\mathcal{F} \to f_*f^{-1}\mathcal{F}$. If $g$ is the identity, then this map on global sections agrees with the pullback map above.
In the situation of the previous paragraph, suppose we have a pair of sheaves $\mathcal{F}$, $\mathcal{G}$ on $\mathcal{C}$, $\mathcal{D}$ together with a map $f^{-1}\mathcal{F} \to \mathcal{G}$, then we compose the pullback map above with $h_*$ applied to $f^{-1}\mathcal{F} \to \mathcal{G}$ to get a map
Restricting to sections over an object of $\mathcal{B}$ one recovers the “pullback map” on global sections discussed above (with suitable choices of sites).
An even more general situation is where we have a commutative diagram of topoi
and a sheaf $\mathcal{G}$ on $\mathcal{C}$. Then there is a base change map
Namely, this map is adjoint to a map $g_*\mathcal{G} \to e_*h_*f^{-1}\mathcal{G} = (e \circ h)_*f^{-1}\mathcal{G}$ which is the pullback map just described.
Remark 7.45.3. Consider a commutative diagram of topoi. Then the base change maps for the two squares compose to give the base change map for the outer rectangle. More precisely, the composition is the base change map for the rectangle. We omit the verification.
\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{B}') \ar[r]_ k \ar[d]_{f'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{B}) \ar[d]^ f \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') \ar[r]^ l \ar[d]_{g'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[d]^ g \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D}') \ar[r]^ m & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) } \]
\begin{align*} m^{-1} \circ (g \circ f)_* & = m^{-1} \circ g_* \circ f_* \\ & \to g'_* \circ l^{-1} \circ f_* \\ & \to g'_* \circ f'_* \circ k^{-1} \\ & = (g' \circ f')_* \circ k^{-1} \end{align*}
Remark 7.45.4. Consider a commutative diagram of ringed topoi. Then the base change maps for the two squares compose to give the base change map for the outer rectangle. More precisely, the composition is the base change map for the rectangle. We omit the verification.
\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}'') \ar[r]_{g'} \ar[d]_{f''} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') \ar[r]_ g \ar[d]_{f'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[d]^ f \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D}'') \ar[r]^{h'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}') \ar[r]^ h & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) } \]
\begin{align*} (h \circ h')^{-1} \circ f_* & = (h')^{-1} \circ h^{-1} \circ f_* \\ & \to (h')^{-1} \circ f'_* \circ g^{-1} \\ & \to f''_* \circ (g')^{-1} \circ g^{-1} \\ & = f”_* \circ (g \circ g')^{-1} \end{align*}
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