Definition 34.4.1. Let $T$ be a scheme. An *étale covering of $T$* is a family of morphisms $\{ f_ i : T_ i \to T\} _{i \in I}$ of schemes such that each $f_ i$ is étale and such that $T = \bigcup f_ i(T_ i)$.

## 34.4 The étale topology

Let $S$ be a scheme. We would like to define the étale-topology on the category of schemes over $S$. According to our general principle we first introduce the notion of an étale covering.

Lemma 34.4.2. Any Zariski covering is an étale covering.

**Proof.**
This is clear from the definitions and the fact that an open immersion is an étale morphism, see Morphisms, Lemma 29.36.9.
$\square$

Next, we show that this notion satisfies the conditions of Sites, Definition 7.6.2.

Lemma 34.4.3. Let $T$ be a scheme.

If $T' \to T$ is an isomorphism then $\{ T' \to T\} $ is an étale covering of $T$.

If $\{ T_ i \to T\} _{i\in I}$ is an étale covering and for each $i$ we have an étale covering $\{ T_{ij} \to T_ i\} _{j\in J_ i}$, then $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is an étale covering.

If $\{ T_ i \to T\} _{i\in I}$ is an étale covering and $T' \to T$ is a morphism of schemes then $\{ T' \times _ T T_ i \to T'\} _{i\in I}$ is an étale covering.

**Proof.**
Omitted.
$\square$

Lemma 34.4.4. Let $T$ be an affine scheme. Let $\{ T_ i \to T\} _{i \in I}$ be an étale covering of $T$. Then there exists an étale covering $\{ U_ j \to T\} _{j = 1, \ldots , m}$ which is a refinement of $\{ T_ i \to T\} _{i \in I}$ such that each $U_ j$ is an affine scheme. Moreover, we may choose each $U_ j$ to be open affine in one of the $T_ i$.

**Proof.**
Omitted.
$\square$

Thus we define the corresponding standard coverings of affines as follows.

Definition 34.4.5. Let $T$ be an affine scheme. A *standard étale covering* of $T$ is a family $\{ f_ j : U_ j \to T\} _{j = 1, \ldots , m}$ with each $U_ j$ is affine and étale over $T$ and $T = \bigcup f_ j(U_ j)$.

In the definition above we do **not** assume the morphisms $f_ j$ are standard étale. The reason is that if we did then the standard étale coverings would not define a site on $\textit{Aff}/S$, for example because of Algebra, Lemma 10.144.2 part (4). On the other hand, an étale morphism of affines is automatically standard smooth, see Algebra, Lemma 10.143.2. Hence a standard étale covering is a standard smooth covering and a standard syntomic covering.

Definition 34.4.6. A *big étale site* is any site $\mathit{Sch}_{\acute{e}tale}$ as in Sites, Definition 7.6.2 constructed as follows:

Choose any set of schemes $S_0$, and any set of étale coverings $\text{Cov}_0$ among these schemes.

As underlying category take any category $\mathit{Sch}_\alpha $ constructed as in Sets, Lemma 3.9.2 starting with the set $S_0$.

Choose any set of coverings as in Sets, Lemma 3.11.1 starting with the category $\mathit{Sch}_\alpha $ and the class of étale coverings, and the set $\text{Cov}_0$ chosen above.

See the remarks following Definition 34.3.5 for motivation and explanation regarding the definition of big sites.

Before we continue with the introduction of the big étale site of a scheme $S$, let us point out that the topology on a big étale site $\mathit{Sch}_{\acute{e}tale}$ is in some sense induced from the étale topology on the category of all schemes.

Lemma 34.4.7. Let $\mathit{Sch}_{\acute{e}tale}$ be a big étale site as in Definition 34.4.6. Let $T \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{\acute{e}tale})$. Let $\{ T_ i \to T\} _{i \in I}$ be an arbitrary étale covering of $T$.

There exists a covering $\{ U_ j \to T\} _{j \in J}$ of $T$ in the site $\mathit{Sch}_{\acute{e}tale}$ which refines $\{ T_ i \to T\} _{i \in I}$.

If $\{ T_ i \to T\} _{i \in I}$ is a standard étale covering, then it is tautologically equivalent to a covering in $\mathit{Sch}_{\acute{e}tale}$.

If $\{ T_ i \to T\} _{i \in I}$ is a Zariski covering, then it is tautologically equivalent to a covering in $\mathit{Sch}_{\acute{e}tale}$.

**Proof.**
For each $i$ choose an affine open covering $T_ i = \bigcup _{j \in J_ i} T_{ij}$ such that each $T_{ij}$ maps into an affine open subscheme of $T$. By Lemma 34.4.3 the refinement $\{ T_{ij} \to T\} _{i \in I, j \in J_ i}$ is an étale covering of $T$ as well. Hence we may assume each $T_ i$ is affine, and maps into an affine open $W_ i$ of $T$. Applying Sets, Lemma 3.9.9 we see that $W_ i$ is isomorphic to an object of $\mathit{Sch}_{\acute{e}tale}$. But then $T_ i$ as a finite type scheme over $W_ i$ is isomorphic to an object $V_ i$ of $\mathit{Sch}_{\acute{e}tale}$ by a second application of Sets, Lemma 3.9.9. The covering $\{ V_ i \to T\} _{i \in I}$ refines $\{ T_ i \to T\} _{i \in I}$ (because they are isomorphic). Moreover, $\{ V_ i \to T\} _{i \in I}$ is combinatorially equivalent to a covering $\{ U_ j \to T\} _{j \in J}$ of $T$ in the site $\mathit{Sch}_{\acute{e}tale}$ by Sets, Lemma 3.9.9. The covering $\{ U_ j \to T\} _{j \in J}$ is a refinement as in (1). In the situation of (2), (3) each of the schemes $T_ i$ is isomorphic to an object of $\mathit{Sch}_{\acute{e}tale}$ by Sets, Lemma 3.9.9, and another application of Sets, Lemma 3.11.1 gives what we want.
$\square$

Definition 34.4.8. Let $S$ be a scheme. Let $\mathit{Sch}_{\acute{e}tale}$ be a big étale site containing $S$.

The

*big étale site of $S$*, denoted $(\mathit{Sch}/S)_{\acute{e}tale}$, is the site $\mathit{Sch}_{\acute{e}tale}/S$ introduced in Sites, Section 7.25.The

*small étale site of $S$*, which we denote $S_{\acute{e}tale}$, is the full subcategory of $(\mathit{Sch}/S)_{\acute{e}tale}$ whose objects are those $U/S$ such that $U \to S$ is étale. A covering of $S_{\acute{e}tale}$ is any covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{\acute{e}tale}$ with $U \in \mathop{\mathrm{Ob}}\nolimits (S_{\acute{e}tale})$.The

*big affine étale site of $S$*, denoted $(\textit{Aff}/S)_{\acute{e}tale}$, is the full subcategory of $(\mathit{Sch}/S)_{\acute{e}tale}$ whose objects are affine $U/S$. A covering of $(\textit{Aff}/S)_{\acute{e}tale}$ is any covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{\acute{e}tale}$ which is a standard étale covering.

It is not completely clear that the big affine étale site or the small étale site are sites. We check this now.

Lemma 34.4.9. Let $S$ be a scheme. Let $\mathit{Sch}_{\acute{e}tale}$ be a big étale site containing $S$. Both $S_{\acute{e}tale}$ and $(\textit{Aff}/S)_{\acute{e}tale}$ are sites.

**Proof.**
Let us show that $S_{\acute{e}tale}$ is a site. It is a category with a given set of families of morphisms with fixed target. Thus we have to show properties (1), (2) and (3) of Sites, Definition 7.6.2. Since $(\mathit{Sch}/S)_{\acute{e}tale}$ is a site, it suffices to prove that given any covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{\acute{e}tale}$ with $U \in \mathop{\mathrm{Ob}}\nolimits (S_{\acute{e}tale})$ we also have $U_ i \in \mathop{\mathrm{Ob}}\nolimits (S_{\acute{e}tale})$. This follows from the definitions as the composition of étale morphisms is an étale morphism.

Let us show that $(\textit{Aff}/S)_{\acute{e}tale}$ is a site. Reasoning as above, it suffices to show that the collection of standard étale coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition 7.6.2. This is clear since for example, given a standard étale covering $\{ T_ i \to T\} _{i\in I}$ and for each $i$ we have a standard étale covering $\{ T_{ij} \to T_ i\} _{j\in J_ i}$, then $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is a standard étale covering because $\bigcup _{i\in I} J_ i$ is finite and each $T_{ij}$ is affine. $\square$

Lemma 34.4.10. Let $S$ be a scheme. Let $\mathit{Sch}_{\acute{e}tale}$ be a big étale site containing $S$. The underlying categories of the sites $\mathit{Sch}_{\acute{e}tale}$, $(\mathit{Sch}/S)_{\acute{e}tale}$, $S_{\acute{e}tale}$, and $(\textit{Aff}/S)_{\acute{e}tale}$ have fibre products. In each case the obvious functor into the category $\mathit{Sch}$ of all schemes commutes with taking fibre products. The categories $(\mathit{Sch}/S)_{\acute{e}tale}$, and $S_{\acute{e}tale}$ both have a final object, namely $S/S$.

**Proof.**
For $\mathit{Sch}_{\acute{e}tale}$ it is true by construction, see Sets, Lemma 3.9.9. Suppose we have $U \to S$, $V \to U$, $W \to U$ morphisms of schemes with $U, V, W \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{\acute{e}tale})$. The fibre product $V \times _ U W$ in $\mathit{Sch}_{\acute{e}tale}$ is a fibre product in $\mathit{Sch}$ and is the fibre product of $V/S$ with $W/S$ over $U/S$ in the category of all schemes over $S$, and hence also a fibre product in $(\mathit{Sch}/S)_{\acute{e}tale}$. This proves the result for $(\mathit{Sch}/S)_{\acute{e}tale}$. If $U \to S$, $V \to U$ and $W \to U$ are étale then so is $V \times _ U W \to S$ and hence we get the result for $S_{\acute{e}tale}$. If $U, V, W$ are affine, so is $V \times _ U W$ and hence the result for $(\textit{Aff}/S)_{\acute{e}tale}$.
$\square$

Next, we check that the big affine site defines the same topos as the big site.

Lemma 34.4.11. Let $S$ be a scheme. Let $\mathit{Sch}_{\acute{e}tale}$ be a big étale site containing $S$. The functor $(\textit{Aff}/S)_{\acute{e}tale}\to (\mathit{Sch}/S)_{\acute{e}tale}$ is special cocontinuous and induces an equivalence of topoi from $\mathop{\mathit{Sh}}\nolimits ((\textit{Aff}/S)_{\acute{e}tale})$ to $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale})$.

**Proof.**
The notion of a special cocontinuous functor is introduced in Sites, Definition 7.29.2. Thus we have to verify assumptions (1) – (5) of Sites, Lemma 7.29.1. Denote the inclusion functor $u : (\textit{Aff}/S)_{\acute{e}tale}\to (\mathit{Sch}/S)_{\acute{e}tale}$. Being cocontinuous just means that any étale covering of $T/S$, $T$ affine, can be refined by a standard étale covering of $T$. This is the content of Lemma 34.4.4. Hence (1) holds. We see $u$ is continuous simply because a standard étale covering is a étale covering. Hence (2) holds. Parts (3) and (4) follow immediately from the fact that $u$ is fully faithful. And finally condition (5) follows from the fact that every scheme has an affine open covering.
$\square$

Next, we establish some relationships between the topoi associated to these sites.

Lemma 34.4.12. Let $\mathit{Sch}_{\acute{e}tale}$ be a big étale site. Let $f : T \to S$ be a morphism in $\mathit{Sch}_{\acute{e}tale}$. The functor $T_{\acute{e}tale}\to (\mathit{Sch}/S)_{\acute{e}tale}$ is cocontinuous and induces a morphism of topoi

For a sheaf $\mathcal{G}$ on $(\mathit{Sch}/S)_{\acute{e}tale}$ we have the formula $(i_ f^{-1}\mathcal{G})(U/T) = \mathcal{G}(U/S)$. The functor $i_ f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes with fibre products and equalizers.

**Proof.**
Denote the functor $u : T_{\acute{e}tale}\to (\mathit{Sch}/S)_{\acute{e}tale}$. In other words, given an étale morphism $j : U \to T$ corresponding to an object of $T_{\acute{e}tale}$ we set $u(U \to T) = (f \circ j : U \to S)$. This functor commutes with fibre products, see Lemma 34.4.10. Let $a, b : U \to V$ be two morphisms in $T_{\acute{e}tale}$. In this case the equalizer of $a$ and $b$ (in the category of schemes) is

which is a fibre product of schemes étale over $T$, hence étale over $T$. Thus $T_{\acute{e}tale}$ has equalizers and $u$ commutes with them. It is clearly cocontinuous. It is also continuous as $u$ transforms coverings to coverings and commutes with fibre products. Hence the Lemma follows from Sites, Lemmas 7.21.5 and 7.21.6. $\square$

Lemma 34.4.13. Let $S$ be a scheme. Let $\mathit{Sch}_{\acute{e}tale}$ be a big étale site containing $S$. The inclusion functor $S_{\acute{e}tale}\to (\mathit{Sch}/S)_{\acute{e}tale}$ satisfies the hypotheses of Sites, Lemma 7.21.8 and hence induces a morphism of sites

and a morphism of topoi

such that $\pi _ S \circ i_ S = \text{id}$. Moreover, $i_ S = i_{\text{id}_ S}$ with $i_{\text{id}_ S}$ as in Lemma 34.4.12. In particular the functor $i_ S^{-1} = \pi _{S, *}$ is described by the rule $i_ S^{-1}(\mathcal{G})(U/S) = \mathcal{G}(U/S)$.

**Proof.**
In this case the functor $u : S_{\acute{e}tale}\to (\mathit{Sch}/S)_{\acute{e}tale}$, in addition to the properties seen in the proof of Lemma 34.4.12 above, also is fully faithful and transforms the final object into the final object. The lemma follows from Sites, Lemma 7.21.8.
$\square$

Definition 34.4.14. In the situation of Lemma 34.4.13 the functor $i_ S^{-1} = \pi _{S, *}$ is often called the *restriction to the small étale site*, and for a sheaf $\mathcal{F}$ on the big étale site we denote $\mathcal{F}|_{S_{\acute{e}tale}}$ this restriction.

With this notation in place we have for a sheaf $\mathcal{F}$ on the big site and a sheaf $\mathcal{G}$ on the small site that

Moreover, we have $(i_{S, *}\mathcal{G})|_{S_{\acute{e}tale}} = \mathcal{G}$ and we have $(\pi _ S^{-1}\mathcal{G})|_{S_{\acute{e}tale}} = \mathcal{G}$.

Lemma 34.4.15. Let $\mathit{Sch}_{\acute{e}tale}$ be a big étale site. Let $f : T \to S$ be a morphism in $\mathit{Sch}_{\acute{e}tale}$. The functor

is cocontinuous, and has a continuous right adjoint

They induce the same morphism of topoi

We have $f_{big}^{-1}(\mathcal{G})(U/T) = \mathcal{G}(U/S)$. We have $f_{big, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times _ S T/T)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers.

**Proof.**
The functor $u$ is cocontinuous, continuous and commutes with fibre products and equalizers (details omitted; compare with the proof of Lemma 34.4.12). Hence Sites, Lemmas 7.21.5 and 7.21.6 apply and we deduce the formula for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover, the functor $v$ is a right adjoint because given $U/T$ and $V/S$ we have $\mathop{Mor}\nolimits _ S(u(U), V) = \mathop{Mor}\nolimits _ T(U, V \times _ S T)$ as desired. Thus we may apply Sites, Lemmas 7.22.1 and 7.22.2 to get the formula for $f_{big, *}$.
$\square$

Lemma 34.4.16. Let $\mathit{Sch}_{\acute{e}tale}$ be a big étale site. Let $f : T \to S$ be a morphism in $\mathit{Sch}_{\acute{e}tale}$.

We have $i_ f = f_{big} \circ i_ T$ with $i_ f$ as in Lemma 34.4.12 and $i_ T$ as in Lemma 34.4.13.

The functor $S_{\acute{e}tale}\to T_{\acute{e}tale}$, $(U \to S) \mapsto (U \times _ S T \to T)$ is continuous and induces a morphism of sites

\[ f_{small} : T_{\acute{e}tale}\longrightarrow S_{\acute{e}tale} \]We have $f_{small, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times _ S T/T)$.

We have a commutative diagram of morphisms of sites

\[ \xymatrix{ T_{\acute{e}tale}\ar[d]_{f_{small}} & (\mathit{Sch}/T)_{\acute{e}tale}\ar[d]^{f_{big}} \ar[l]^{\pi _ T}\\ S_{\acute{e}tale}& (\mathit{Sch}/S)_{\acute{e}tale}\ar[l]_{\pi _ S} } \]so that $f_{small} \circ \pi _ T = \pi _ S \circ f_{big}$ as morphisms of topoi.

We have $f_{small} = \pi _ S \circ f_{big} \circ i_ T = \pi _ S \circ i_ f$.

**Proof.**
The equality $i_ f = f_{big} \circ i_ T$ follows from the equality $i_ f^{-1} = i_ T^{-1} \circ f_{big}^{-1}$ which is clear from the descriptions of these functors above. Thus we see (1).

The functor $u : S_{\acute{e}tale}\to T_{\acute{e}tale}$, $u(U \to S) = (U \times _ S T \to T)$ transforms coverings into coverings and commutes with fibre products, see Lemma 34.4.3 (3) and 34.4.10. Moreover, both $S_{\acute{e}tale}$, $T_{\acute{e}tale}$ have final objects, namely $S/S$ and $T/T$ and $u(S/S) = T/T$. Hence by Sites, Proposition 7.14.7 the functor $u$ corresponds to a morphism of sites $T_{\acute{e}tale}\to S_{\acute{e}tale}$. This in turn gives rise to the morphism of topoi, see Sites, Lemma 7.15.2. The description of the pushforward is clear from these references.

Part (3) follows because $\pi _ S$ and $\pi _ T$ are given by the inclusion functors and $f_{small}$ and $f_{big}$ by the base change functors $U \mapsto U \times _ S T$.

Statement (4) follows from (3) by precomposing with $i_ T$. $\square$

In the situation of the lemma, using the terminology of Definition 34.4.14 we have: for $\mathcal{F}$ a sheaf on the big étale site of $T$

This equality is clear from the commutativity of the diagram of sites of the lemma, since restriction to the small étale site of $T$, resp. $S$ is given by $\pi _{T, *}$, resp. $\pi _{S, *}$. A similar formula involving pullbacks and restrictions is false.

Lemma 34.4.17. Given schemes $X$, $Y$, $Y$ in $\mathit{Sch}_{\acute{e}tale}$ and morphisms $f : X \to Y$, $g : Y \to Z$ we have $g_{big} \circ f_{big} = (g \circ f)_{big}$ and $g_{small} \circ f_{small} = (g \circ f)_{small}$.

**Proof.**
This follows from the simple description of pushforward and pullback for the functors on the big sites from Lemma 34.4.15. For the functors on the small sites this follows from the description of the pushforward functors in Lemma 34.4.16.
$\square$

Lemma 34.4.18. Let $\mathit{Sch}_{\acute{e}tale}$ be a big étale site. Consider a cartesian diagram

in $\mathit{Sch}_{\acute{e}tale}$. Then $i_ g^{-1} \circ f_{big, *} = f'_{small, *} \circ (i_{g'})^{-1}$ and $g_{big}^{-1} \circ f_{big, *} = f'_{big, *} \circ (g'_{big})^{-1}$.

**Proof.**
Since the diagram is cartesian, we have for $U'/S'$ that $U' \times _{S'} T' = U' \times _ S T$. Hence both $i_ g^{-1} \circ f_{big, *}$ and $f'_{small, *} \circ (i_{g'})^{-1}$ send a sheaf $\mathcal{F}$ on $(\mathit{Sch}/T)_{\acute{e}tale}$ to the sheaf $U' \mapsto \mathcal{F}(U' \times _{S'} T')$ on $S'_{\acute{e}tale}$ (use Lemmas 34.4.12 and 34.4.15). The second equality can be proved in the same manner or can be deduced from the very general Sites, Lemma 7.28.1.
$\square$

We can think about a sheaf on the big étale site of $S$ as a collection of “usual” sheaves on all schemes over $S$.

Lemma 34.4.19. Let $S$ be a scheme contained in a big étale site $\mathit{Sch}_{\acute{e}tale}$. A sheaf $\mathcal{F}$ on the big étale site $(\mathit{Sch}/S)_{\acute{e}tale}$ is given by the following data:

for every $T/S \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale})$ a sheaf $\mathcal{F}_ T$ on $T_{\acute{e}tale}$,

for every $f : T' \to T$ in $(\mathit{Sch}/S)_{\acute{e}tale}$ a map $c_ f : f_{small}^{-1}\mathcal{F}_ T \to \mathcal{F}_{T'}$.

These data are subject to the following conditions:

given any $f : T' \to T$ and $g : T'' \to T'$ in $(\mathit{Sch}/S)_{\acute{e}tale}$ the composition $c_ g \circ g_{small}^{-1}c_ f$ is equal to $c_{f \circ g}$, and

if $f : T' \to T$ in $(\mathit{Sch}/S)_{\acute{e}tale}$ is étale then $c_ f$ is an isomorphism.

**Proof.**
Given a sheaf $\mathcal{F}$ on $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale})$ we set $\mathcal{F}_ T = i_ p^{-1}\mathcal{F}$ where $p : T \to S$ is the structure morphism. Note that $\mathcal{F}_ T(U) = \mathcal{F}(U/S)$ for any $U \to T$ in $T_{\acute{e}tale}$ see Lemma 34.4.12. Hence given $f : T' \to T$ over $S$ and $U \to T$ we get a canonical map $\mathcal{F}_ T(U) = \mathcal{F}(U/S) \to \mathcal{F}(U \times _ T T'/S) = \mathcal{F}_{T'}(U \times _ T T')$ where the middle is the restriction map of $\mathcal{F}$ with respect to the morphism $U \times _ T T' \to U$ over $S$. The collection of these maps are compatible with restrictions, and hence define a map $c'_ f : \mathcal{F}_ T \to f_{small, *}\mathcal{F}_{T'}$ where $u : T_{\acute{e}tale}\to T'_{\acute{e}tale}$ is the base change functor associated to $f$. By adjunction of $f_{small, *}$ (see Sites, Section 7.13) with $f_{small}^{-1}$ this is the same as a map $c_ f : f_{small}^{-1}\mathcal{F}_ T \to \mathcal{F}_{T'}$. It is clear that $c'_{f \circ g}$ is the composition of $c'_ f$ and $f_{small, *}c'_ g$, since composition of restriction maps of $\mathcal{F}$ gives restriction maps, and this gives the desired relationship among $c_ f$, $c_ g$ and $c_{f \circ g}$.

Conversely, given a system $(\mathcal{F}_ T, c_ f)$ as in the lemma we may define a presheaf $\mathcal{F}$ on $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale})$ by simply setting $\mathcal{F}(T/S) = \mathcal{F}_ T(T)$. As restriction mapping, given $f : T' \to T$ we set for $s \in \mathcal{F}(T)$ the pullback $f^*(s)$ equal to $c_ f(s)$ where we think of $c_ f$ as a map $\mathcal{F}_ T \to f_{small, *}\mathcal{F}_{T'}$ again. The condition on the $c_ f$ guarantees that pullbacks satisfy the required functoriality property. We omit the verification that this is a sheaf. It is clear that the constructions so defined are mutually inverse. $\square$

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