Lemma 7.29.1. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume that

1. $u$ is cocontinuous,

2. $u$ is continuous,

3. given $a, b : U' \to U$ in $\mathcal{C}$ such that $u(a) = u(b)$, then there exists a covering $\{ f_ i : U'_ i \to U'\}$ in $\mathcal{C}$ such that $a \circ f_ i = b \circ f_ i$,

4. given $U', U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and a morphism $c : u(U') \to u(U)$ in $\mathcal{D}$ there exists a covering $\{ f_ i : U_ i' \to U'\}$ in $\mathcal{C}$ and morphisms $c_ i : U_ i' \to U$ such that $u(c_ i) = c \circ u(f_ i)$, and

5. given $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ there exists a covering of $V$ in $\mathcal{D}$ of the form $\{ u(U_ i) \to V\} _{i \in I}$.

Then the morphism of topoi

$g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$

associated to the cocontinuous functor $u$ by Lemma 7.21.1 is an equivalence.

Proof. Assume $u$ satisfies properties (1) – (5). We will show that the adjunction mappings

$\mathcal{G} \longrightarrow g_*g^{-1}\mathcal{G} \quad \text{and}\quad g^{-1}g_*\mathcal{F} \longrightarrow \mathcal{F}$

are isomorphisms.

Note that Lemma 7.21.5 applies and we have $g^{-1}\mathcal{G}(U) = \mathcal{G}(u(U))$ for any sheaf $\mathcal{G}$ on $\mathcal{D}$. Next, let $\mathcal{F}$ be a sheaf on $\mathcal{C}$, and let $V$ be an object of $\mathcal{D}$. By definition we have $g_*\mathcal{F}(V) = \mathop{\mathrm{lim}}\nolimits _{u(U) \to V} \mathcal{F}(U)$. Hence

$g^{-1}g_*\mathcal{F}(U) = \mathop{\mathrm{lim}}\nolimits _{U', u(U') \to u(U)} \mathcal{F}(U')$

where the morphisms $\psi : u(U') \to u(U)$ need not be of the form $u(\alpha )$. The category of such pairs $(U', \psi )$ has a final object, namely $(U, \text{id})$, which gives rise to the map from the limit into $\mathcal{F}(U)$. Let $(s_{(U', \psi )})$ be an element of the limit. We want to show that $s_{(U', \psi )}$ is uniquely determined by the value $s_{(U, \text{id})} \in \mathcal{F}(U)$. By property (4) given any $(U', \psi )$ there exists a covering $\{ U'_ i \to U'\}$ such that the compositions $u(U'_ i) \to u(U') \to u(U)$ are of the form $u(c_ i)$ for some $c_ i : U'_ i \to U$ in $\mathcal{C}$. Hence

$s_{(U', \psi )}|_{U'_ i} = c_ i^*(s_{(U, \text{id})}).$

Since $\mathcal{F}$ is a sheaf it follows that indeed $s_{(U', \psi )}$ is determined by $s_{(U, \text{id})}$. This proves uniqueness. For existence, assume given any $s \in \mathcal{F}(U)$, $\psi : u(U') \to u(U)$, $\{ f_ i : U_ i' \to U'\}$ and $c_ i : U_ i' \to U$ such that $\psi \circ u(f_ i) = u(c_ i)$ as above. We claim there exists a (unique) element $s_{(U', \psi )} \in \mathcal{F}(U')$ such that

$s_{(U', \psi )}|_{U'_ i} = c_ i^*(s).$

Namely, a priori it is not clear the elements $c_ i^*(s)|_{U_ i' \times _{U'} U_ j'}$ and $c_ j^*(s)|_{U_ i' \times _{U'} U_ j'}$ agree, since the diagram

$\xymatrix{ U_ i' \times _{U'} U_ j' \ar[r]_-{\text{pr}_2} \ar[d]_{\text{pr}_1} & U_ j' \ar[d]^{c_ j} \\ U_ i' \ar[r]^{c_ i} & U}$

need not commute. But condition (3) of the lemma guarantees that there exist coverings $\{ f_{ijk} : U'_{ijk} \to U_ i' \times _{U'} U_ j'\} _{k \in K_{ij}}$ such that $c_ i \circ \text{pr}_1 \circ f_{ijk} = c_ j \circ \text{pr}_2 \circ f_{ijk}$. Hence

$f_{ijk}^* \left(c_ i^*s|_{U_ i' \times _{U'} U_ j'}\right) = f_{ijk}^* \left(c_ j^*s|_{U_ i' \times _{U'} U_ j'}\right)$

Hence $c_ i^*(s)|_{U_ i' \times _{U'} U_ j'} = c_ j^*(s)|_{U_ i' \times _{U'} U_ j'}$ by the sheaf condition for $\mathcal{F}$ and hence the existence of $s_{(U', \psi )}$ also by the sheaf condition for $\mathcal{F}$. The uniqueness guarantees that the collection $(s_{(U', \psi )})$ so obtained is an element of the limit with $s_{(U, \psi )} = s$. This proves that $g^{-1}g_*\mathcal{F} \to \mathcal{F}$ is an isomorphism.

Let $\mathcal{G}$ be a sheaf on $\mathcal{D}$. Let $V$ be an object of $\mathcal{D}$. Then we see that

$g_*g^{-1}\mathcal{G}(V) = \mathop{\mathrm{lim}}\nolimits _{U, \psi : u(U) \to V} \mathcal{G}(u(U))$

By the preceding paragraph we see that the value of the sheaf $g_*g^{-1}\mathcal{G}$ on an object $V$ of the form $V = u(U)$ is equal to $\mathcal{G}(u(U))$. (Formally, this holds because we have $g^{-1}g_*g^{-1} \cong g^{-1}$, and the description of $g^{-1}$ given at the beginning of the proof; informally just by comparing limits here and above.) Hence the adjunction mapping $\mathcal{G} \to g_*g^{-1}\mathcal{G}$ has the property that it is a bijection on sections over any object of the form $u(U)$. Since by axiom (5) there exists a covering of $V$ by objects of the form $u(U)$ we see easily that the adjunction map is an isomorphism. $\square$

There are also:

• 7 comment(s) on Section 7.29: Morphisms of topoi

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).