The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

7.29 Morphisms of topoi

In this section we show that any morphism of topoi is equivalent to a morphism of topoi which comes from a morphism of sites. Please compare with [Exposé IV, Proposition 4.9.4, SGA4].

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. $\square$

It will be convenient to give cocontinuous functors as in Lemma 7.29.1 a name.

Definition 7.29.2. Let $\mathcal{C}$, $\mathcal{D}$ be sites. A special cocontinuous functor $u$ from $\mathcal{C}$ to $\mathcal{D}$ is a cocontinuous functor $u : \mathcal{C} \to \mathcal{D}$ satisfying the assumptions and conclusions of Lemma 7.29.1.

Lemma 7.29.3. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a special cocontinuous functor. For every object $U$ of $\mathcal{C}$ we have a commutative diagram

\[ \xymatrix{ \mathcal{C}/U \ar[r]_{j_ U} \ar[d] & \mathcal{C} \ar[d]^ u \\ \mathcal{D}/u(U) \ar[r]^-{j_{u(U)}} & \mathcal{D} } \]

as in Lemma 7.28.4. The left vertical arrow is a special cocontinuous functor. Hence in the commutative diagram of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[r]_{j_ U} \ar[d] & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[d]^ u \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/u(U)) \ar[r]^-{j_{u(U)}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) } \]

the vertical arrows are equivalences.

Proof. We have seen the existence and commutativity of the diagrams in Lemma 7.28.4. We have to check hypotheses (1) – (5) of Lemma 7.29.1 for the induced functor $u : \mathcal{C}/U \to \mathcal{D}/u(U)$. This is completely mechanical.

Property (1). This is Lemma 7.28.4.

Property (2). Let $\{ U_ i'/U \to U'/U\} _{i \in I}$ be a covering of $U'/U$ in $\mathcal{C}/U$. Because $u$ is continuous we see that $\{ u(U_ i')/u(U) \to u(U')/u(U)\} _{i \in I}$ is a covering of $u(U')/u(U)$ in $\mathcal{D}/u(U)$. Hence (2) holds for $u : \mathcal{C}/U \to \mathcal{D}/u(U)$.

Property (3). Let $a, b : U''/U \to U'/U$ in $\mathcal{C}/U$ be morphisms such that $u(a) = u(b)$ in $\mathcal{D}/u(U)$. Because $u$ satisfies (3) we see there exists a covering $\{ f_ i : U''_ i \to U''\} $ in $\mathcal{C}$ such that $a \circ f_ i = b \circ f_ i$. This gives a covering $\{ f_ i : U''_ i/U \to U''/U\} $ in $\mathcal{C}/U$ such that $a \circ f_ i = b \circ f_ i$. Hence (3) holds for $u : \mathcal{C}/U \to \mathcal{D}/u(U)$.

Property (4). Let $U''/U, U'/U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}/U)$ and a morphism $c : u(U'')/u(U) \to u(U')/u(U)$ in $\mathcal{D}/u(U)$ be given. Because $u$ satisfies property (4) there exists a covering $\{ f_ i : U_ i'' \to U''\} $ in $\mathcal{C}$ and morphisms $c_ i : U_ i'' \to U'$ such that $u(c_ i) = c \circ u(f_ i)$. We think of $U_ i''$ as an object over $U$ via the composition $U_ i'' \to U'' \to U$. It may not be true that $c_ i$ is a morphism over $U$! But since $u(c_ i)$ is a morphism over $u(U)$ we may apply property (3) for $u$ and find coverings $\{ f_{ik} : U''_{ik} \to U''_ i\} $ such that $c_{ik} = c_ i \circ f_{ik} : U''_{ik} \to U'$ are morphisms over $U$. Hence $\{ f_ i \circ f_{ik} : U''_{ik}/U \to U''/U\} $ is a covering in $\mathcal{C}/U$ such that $u(c_{ik}) = c \circ u(f_{ik})$. Hence (4) holds for $u : \mathcal{C}/U \to \mathcal{D}/u(U)$.

Property (5). Let $h : V \to u(U)$ be an object of $\mathcal{D}/u(U)$. Because $u$ satisfies property (5) there exists a covering $\{ c_ i : u(U_ i) \to V\} $ in $\mathcal{D}$. By property (4) we can find coverings $\{ f_{ij} : U_{ij} \to U_ i\} $ and morphisms $c_{ij} : U_{ij} \to U$ such that $u(c_{ij}) = h \circ c_ i \circ u(f_{ij})$. Hence $\{ u(U_{ij})/u(U) \to V/u(U)\} $ is a covering in $\mathcal{D}/u(U)$ of the desired shape and we conclude that (5) holds for $u : \mathcal{C}/U \to \mathcal{D}/u(U)$. $\square$

Lemma 7.29.4. Let $\mathcal{C}$ be a site. Let $\mathcal{C}' \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a full subcategory (with a set of objects) such that

  1. $h_ U^\# \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}')$ for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and

  2. $\mathcal{C}'$ is preserved under fibre products in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

Declare a covering of $\mathcal{C}'$ to be any family $\{ \mathcal{F}_ i \to \mathcal{F}\} _{i \in I}$ of maps such that $\coprod _{i \in I} \mathcal{F}_ i \to \mathcal{F}$ is a surjective map of sheaves. Then

  1. $\mathcal{C}'$ is a site (after choosing a set of coverings, see Sets, Lemma 3.11.1),

  2. representable presheaves on $\mathcal{C}'$ are sheaves (i.e., the topology on $\mathcal{C}'$ is subcanonical, see Definition 7.12.2),

  3. the functor $v : \mathcal{C} \to \mathcal{C}'$, $U \mapsto h_ U^\# $ is a special cocontinuous functor, hence induces an equivalence $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}')$,

  4. for any $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}')$ we have $g^{-1}h_\mathcal {F} = \mathcal{F}$, and

  5. for any $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we have $g_*h_ U^\# = h_{v(U)} = h_{h_ U^\# }$.

Proof. Warning: Some of the statements above may look be a bit confusing at first; this is because objects of $\mathcal{C}'$ can also be viewed as sheaves on $\mathcal{C}$! We omit the proof that the coverings of $\mathcal{C}'$ as described in the lemma satisfy the conditions of Definition 7.6.2.

Suppose that $\{ \mathcal{F}_ i \to \mathcal{F}\} $ is a surjective family of morphisms of sheaves. Let $\mathcal{G}$ be another sheaf. Part (2) of the lemma says that the equalizer of

\[ \xymatrix{ \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}( \coprod _{i \in I} \mathcal{F}_ i, \mathcal{G}) \ar@<1ex>[r] \ar@<-1ex>[r] & \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}( \coprod _{(i_0, i_1) \in I \times I} \mathcal{F}_{i_0} \times _\mathcal {F} \mathcal{F}_{i_1}, \mathcal{G}) } \]

is $\mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{F}, \mathcal{G}).$ This is clear (for example use Lemma 7.11.3).

To prove (3) we have to check conditions (1) – (5) of Lemma 7.29.1. The fact that $v$ is cocontinuous is equivalent to the description of surjective maps of sheaves in Lemma 7.11.2. The functor $v$ is continuous because $U \mapsto h_ U^\# $ commutes with fibre products, and transforms coverings into coverings (see Lemma 7.10.14, and Lemma 7.12.4). Properties (3), (4) of Lemma 7.29.1 are statements about morphisms $f : h_{U'}^\# \to h_ U^\# $. Such a morphism is the same thing as an element of $h_ U^\# (U')$. Hence (3) and (4) are immediate from the construction of the sheafification. Property (5) of Lemma 7.29.1 is Lemma 7.12.5. Denote $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}')$ the equivalence of topoi associated with $v$ by Lemma 7.29.1.

Let $\mathcal{F}$ be as in part (4) of the lemma. For any $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we have

\[ g^{-1}h_\mathcal {F}(U) = h_\mathcal {F}(v(U)) = \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(h_ U^\# , \mathcal{F}) = \mathcal{F}(U) \]

The first equality by Lemma 7.21.5. Thus part (4) holds.

Let $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}')$. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Then

\begin{align*} g_*h_ U^\# (\mathcal{F}) & = \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}')}(h_\mathcal {F}, g_*h_ U^\# ) \\ & = \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(g^{-1}h_\mathcal {F}, h_ U^\# ) \\ & = \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{F}, h_ U^\# ) \\ & = \mathop{Mor}\nolimits _{\mathcal{C}'}(\mathcal{F}, h_ U^\# ) \end{align*}

as desired (where the third equality was shown above). $\square$

Using this we can massage any topos to live over a site having all finite limits.

Lemma 7.29.5. Let $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a topos. Let $\{ \mathcal{F}_ i\} _{i \in I}$ be a set of sheaves on $\mathcal{C}$. There exists an equivalence of topoi $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}')$ induced by a special cocontinuous functor $u : \mathcal{C} \to \mathcal{C}'$ of sites such that

  1. $\mathcal{C}'$ has a subcanonical topology,

  2. a family $\{ V_ j \to V\} $ of morphisms of $\mathcal{C}'$ is (combinatorially equivalent to) a covering of $\mathcal{C}'$ if and only if $\coprod h_{V_ j} \to h_ V$ is surjective,

  3. $\mathcal{C}'$ has fibre products and a final object (i.e., $\mathcal{C}'$ has all finite limits),

  4. every subsheaf of a representable sheaf on $\mathcal{C}'$ is representable, and

  5. each $g_*\mathcal{F}_ i$ is a representable sheaf.

Proof. Consider the full subcategory $\mathcal{C}_1 \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ consisting of all $h_ U^\# $ for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, the given sheaves $\mathcal{F}_ i$ and the final sheaf $*$ (see Example 7.10.2). We are going to inductively define full subcategories

\[ \mathcal{C}_1 \subset \mathcal{C}_2 \subset \mathcal{C}_2 \subset \ldots \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \]

Namely, given $\mathcal{C}_ n$ let $\mathcal{C}_{n + 1}$ be the full subcategory consisting of all fibre products and subsheaves of objects of $\mathcal{C}_ n$. (Note that $\mathcal{C}_{n + 1}$ has a set of objects.) Set $\mathcal{C}' = \bigcup _{n \geq 1} \mathcal{C}_ n$. A covering in $\mathcal{C}'$ is any family $\{ \mathcal{G}_ j \to \mathcal{G}\} _{j \in J}$ of morphisms of objects of $\mathcal{C}'$ such that $\coprod \mathcal{G}_ j \to \mathcal{G}$ is surjective as a map of sheaves on $\mathcal{C}$. The functor $v : \mathcal{C} \to \mathcal{C'}$ is given by $U \mapsto h_ U^\# $. Apply Lemma 7.29.4. $\square$

Here is the goal of the current section.


Lemma 7.29.6. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be a morphism of topoi. Then there exists a site $\mathcal{C}'$ and a diagram of functors

\[ \xymatrix{ \mathcal{C} \ar[r]_ v & \mathcal{C}' & \mathcal{D} \ar[l]^ u } \]

such that

  1. the functor $v$ is a special cocontinuous functor,

  2. the functor $u$ commutes with fibre products, is continuous and defines a morphism of sites $\mathcal{C}' \to \mathcal{D}$, and

  3. the morphism of topoi $f$ agrees with the composition of morphisms of topoi

    \[ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \]

    where the first arrow comes from $v$ via Lemma 7.29.1 and the second arrow from $u$ via Lemma 7.15.2.

Proof. Consider the full subcategory $\mathcal{C}_1 \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ consisting of all $h_ U^\# $ and all $f^{-1}h_ V^\# $ for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and all $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$. Let $\mathcal{C}_{n + 1}$ be a full subcategory consisting of all fibre products of objects of $\mathcal{C}_ n$. Set $\mathcal{C}' = \bigcup _{n \geq 1} \mathcal{C}_ n$. A covering in $\mathcal{C}'$ is any family $\{ \mathcal{F}_ i \to \mathcal{F}\} _{i \in I}$ such that $\coprod _{i \in I} \mathcal{F}_ i \to \mathcal{F}$ is surjective as a map of sheaves on $\mathcal{C}$. The functor $v : \mathcal{C} \to \mathcal{C'}$ is given by $U \mapsto h_ U^\# $. The functor $u : \mathcal{D} \to \mathcal{C'}$ is given by $V \mapsto f^{-1}h_ V^\# $.

Part (1) follows from Lemma 7.29.4.

Proof of (2) and (3) of the lemma. The functor $u$ commutes with fibre products as both $V \mapsto h_ V^\# $ and $f^{-1}$ do. Moreover, since $f^{-1}$ is exact and commutes with arbitrary colimits we see that it transforms a covering into a surjective family of morphisms of sheaves. Hence $u$ is continuous. To see that it defines a morphism of sites we still have to see that $u_ s$ is exact. In order to do this we will show that $g^{-1} \circ u_ s = f^{-1}$. Namely, then since $g^{-1}$ is an equivalence and $f^{-1}$ is exact we will conclude. Because $g^{-1}$ is adjoint to $g_*$, and $u_ s$ is adjoint to $u^ s$, and $f^{-1}$ is adjoint to $f_*$ it also suffices to prove that $u^ s \circ g_* = f_*$. Let $U$ be an object of $\mathcal{C}$ and let $V$ be an object of $\mathcal{D}$. Then

\begin{align*} (u^ sg_*h_ U^\# )(V) & = g_*h_ U^\# (f^{-1}h_ V^\# ) \\ & = \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(f^{-1}h_ V^\# , h_ U^\# ) \\ & = \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}(h_ V^\# , f_*h_ U^\# ) \\ & = f_*h_ U^\# (V) \end{align*}

The first equality because $u^ s = u^ p$. The second equality by Lemma 7.29.4 (5). The third equality by adjointness of $f_*$ and $f^{-1}$ and the final equality by properties of sheafification and the Yoneda lemma. We omit the verification that these identities are functorial in $U$ and $V$. Hence we see that we have $u^ s \circ g_* = f_*$ for sheaves of the form $h_ U^\# $. This implies that $u^ s \circ g_* = f_*$ and we win (some details omitted). $\square$

Remark 7.29.7. Notation and assumptions as in Lemma 7.29.6. If the site $\mathcal{D}$ has a final object and fibre products then the functor $u : \mathcal{D} \to \mathcal{C}'$ satisfies all the assumptions of Proposition 7.14.7. Namely, in addition to the properties mentioned in the lemma $u$ also transforms the final object of $\mathcal{D}$ into the final object of $\mathcal{C}'$. This is clear from the construction of $u$. Hence, if we first apply Lemmas 7.29.5 to $\mathcal{D}$ and then Lemma 7.29.6 to the resulting morphism of topoi $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D}')$ we obtain the following statement: Any morphism of topoi $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ fits into a commutative diagram

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

where the following properties hold:

  1. the morphisms $e$ and $g$ are equivalences given by special cocontinuous functors $\mathcal{C} \to \mathcal{C}'$ and $\mathcal{D} \to \mathcal{D}'$,

  2. the sites $\mathcal{C}'$ and $\mathcal{D}'$ have fibre products, final objects and have subcanonical topologies,

  3. the morphism $f' : \mathcal{C}' \to \mathcal{D}'$ comes from a morphism of sites corresponding to a functor $u : \mathcal{D}' \to \mathcal{C}'$ to which Proposition 7.14.7 applies, and

  4. given any set of sheaves $\mathcal{F}_ i$ (resp. $\mathcal{G}_ j$) on $\mathcal{C}$ (resp. $\mathcal{D}$) we may assume each of these is a representable sheaf on $\mathcal{C}'$ (resp. $\mathcal{D}'$).

It is often useful to replace $\mathcal{C}$ and $\mathcal{D}$ by $\mathcal{C}'$ and $\mathcal{D}'$.

Remark 7.29.8. Notation and assumptions as in Lemma 7.29.6. Suppose that in addition the original morphism of topoi $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ is an equivalence. Then the construction in the proof of Lemma 7.29.6 gives two functors

\[ \mathcal{C} \rightarrow \mathcal{C}' \leftarrow \mathcal{D} \]

which are both special cocontinuous functors. Hence in this case we can actually factor the morphism of topoi as a composition

\[ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \rightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') = \mathop{\mathit{Sh}}\nolimits (\mathcal{D}') \leftarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \]

as in Remark 7.29.7, but with the middle morphism an identity.

Comments (7)

Comment #1157 by Julia on

Could it be possible to know the reference where Lemma 7.28.6 is proven? Thank you in advance.

Comment #1175 by on

The proof is right here? On the other hand, in a slightly different language I am sure you can find this lemma in SGA 4. When you find it, please leave a comment with the precise reference so I can add it for future visitors. Thanks!

Comment #1182 by Julia on

Thanks for your reply! I found an equivalent statement to 7.28.6.(Tag 03A2), which at first glance, as predicted, has a quite different appearance, but in the end provides us the same result, as it is stated at the beginning of this Tag 039Z: "In this section we show that any morphism of topoi is equivalent to a morphism of topoi which comes from a morphism of sites."

The statement is Proposition 4.9.4. Exposé IV, SGA 4.

Comment #1186 by Julia on

Furthermore, in order to get the whole result from statement 7.28.6. (Tag 03A2), I think one should also use Remarque 4.7.4. in Exposé IV, SGA 4, Tome 1.

Comment #1202 by on

OK, wonderful! Added your references in this commit. If you want to be added to the list of contributors, leave your full name in your next comment.

Comment #1211 by Julia on

Thank you very much! My complete name is Julia Ramos González. It has been a pleasure to collaborate with these comments.

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