## 7.45 Pullback maps

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

$\Gamma (\mathcal{C}, \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(*, \mathcal{F})$

where $*$ is the final object in the category of presheaves on $\mathcal{C}$, i.e., the presheaf which associates to every object a singleton.

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

$\xymatrix{ h_ V^\# \ar@<1ex>[r] \ar@<-1ex>[r] & h_ U^\# \ar[r] & {*} }$

is a coequalizer in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. Then $\Gamma (\mathcal{C}, \mathcal{F})$ is the equalizer of $a^*, b^* : \mathcal{F}(U) \to \mathcal{F}(V)$.

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

$f^{-1} : \Gamma (\mathcal{C}, \mathcal{F}) \longrightarrow \Gamma (\mathcal{D}, f^{-1}\mathcal{F})$

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

$\Gamma (\mathcal{C}, \mathcal{F}) \longrightarrow \Gamma (\mathcal{D}, \mathcal{G})$

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

$\xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \ar[rd]_ h \ar[rr]_ f & & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[ld]^ g \\ & \mathop{\mathit{Sh}}\nolimits (\mathcal{B}) }$

Next, suppose that we have a sheaf $\mathcal{F}$ on $\mathcal{C}$. Then there is a pullback map

$f^{-1} : g_*\mathcal{F} \longrightarrow h_*f^{-1}\mathcal{F}$

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

$g_*\mathcal{F} \longrightarrow h_*\mathcal{G}$

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

$\xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \ar[d]_ h \ar[r]_ f & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[d]^ g \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{B}) \ar[r]^ e & \mathop{\mathit{Sh}}\nolimits (\mathcal{A}) }$

and a sheaf $\mathcal{G}$ on $\mathcal{C}$. Then there is a base change map

$e^{-1}g_*\mathcal{G} \longrightarrow h_*f^{-1}\mathcal{G}.$

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

$\xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{B}') \ar[r]_ k \ar[d]_{f'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{B}) \ar[d]^ f \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') \ar[r]^ l \ar[d]_{g'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[d]^ g \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D}') \ar[r]^ m & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) }$

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

\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*}

is the base change map for the rectangle. We omit the verification.

Remark 7.45.4. Consider a commutative diagram

$\xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}'') \ar[r]_{g'} \ar[d]_{f''} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') \ar[r]_ g \ar[d]_{f'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[d]^ f \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D}'') \ar[r]^{h'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}') \ar[r]^ h & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) }$

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

\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*}

is the base change map for the rectangle. We omit the verification.

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