Spahn sheaf on a sheaf (changes)

Showing changes from revision #22 to #23: Added | ~~Removed~~ | ~~Chan~~ged

Motivation

Let $X\in H$ be a space (an object of a category $H$ of spaces), let $Sh(X)$ be the category of sheaves on the frame of opens on $X$ , let $(H/X)^{et}$ denote the wide subcategory of $H/X$ with only étale morphisms. Then there is an adjoint equivalence

(L\dashv \Gamma):(H/X)^{et}\stackrel{\Gamma}{\to}Sh(X)

where

$\Gamma$ sends an étale morphism $f:U\to X$ to the sheaf of local sections of $f$ .
$L$ sends a sheaf on $X$ to its espace étale.

Très petit topos

We wish to clarify in which sense also the $(\infty,1)$ - topos $(H/X)^{fet}$ can be regarded as an $(\infty,1)$ -sheaftopos on $X$ . One formulation of this is to show that $((H/X)^{fet},O_{(H/X)^{fet}})$ is a locally representable $H$ -structured $(\infty,1)$ -topos - and that the representation is exhibited by formally étale morphisms.

We assume that $fet$ is the admissible class defined by an infinitesimal modality $\Box$ on $H$ .

Definition (universal $G$ -structure, classifying topos)

(1) A $G$ -structure $O$ on an $(\infty,1)$ -topos is called universal if for every $(\infty,1)$ -topos $X$ composition with $O$ induces an equivalence of $(\infty,1)$ -categories

Fun^*(K,T)\to Str_G(T)

where $Fun^*(K,T)$ denotes the geometric morphisms $f$ with inverse image $f^*:K\to T$ .

(2) In this case we say $O$ exhibits $K$ as classifying $(\infty,1)$ -topos for $G$ -structures on $X$ .

Remark

$H$ , $H/X$ , and $(H/X)^fet$ are $(H/X)^{fet}$ -structured $(\infty,1)$ -toposes.

Proof

The classifying topos for $(G/X)^{fet}$ -structures is $H$ and the $(\infty,1)$ -toposes in question are linked with $H$ by geometric morphisms. We obtain the required structures as the image of

Fun^*(H,H/X)\to Str_{(H/X)^{fet}}(H/X)

respectively for $H$ and $(H/X)^{fet}$ in place of “H/X”.

Local representability of the très petit topos

Relative-, and absolute spectra

Definition (pro objects)

Let $C$ be an $(\infty,1)$ -category. We have $Ind(C^{op})\simeq Pro(C)^{op}$ . A pro object in $C$ is a formal limit of a cofiltered diagram in $C$ . A cofiltered diagram is defined to be a finite diagram $F$ having a cone (i.e. a family of natural transformation $\kappa_c\to F$ for all $c\in C$ , where $\kappa_c$ denotes the constant functor having value $c$ ). So we have

Pro(C)=\{F:D\to C | D\,is\,finite,\,cofiltered\}

and the hom sets are

Pro(C)(F,G)=lim_{e\in E}colim_{d\in D}C(F(d),G(e))

We have (more or less) synonyms:

pro object, cofiltered, having a cone
ind object, filtered, having a cocone

Definition 2.2.2 (admissible morphism in $Pro(H)$ )

Let $H$ be a geometry. A morphism $U\to X$ in $Pro(H)$ is called $Pro(H)$ - admissible if it arises as a pullback

\array{ U&\to&j(U^\prime) \\ \downarrow&&\downarrow^{j(f)} \\ X&\to&j(X^\prime) }

in $Pro(H)$ where $f:U^\prime \to X^\prime$ is admissible in $H$ .

Remark 2.2.7

~~Let~~ $Pro(H)^{ad}/X$ ~~$H$~~ is essentially small, hence the presheaf- and sheaf category on it can be defined in the usual way. ~~be a geometry, then every admissible morphism~~ ~~$U\to X$~~ in ~~$Pro(G)$~~ ~~arises from a pullback diagram~~

~~$\array{ U&\to&j(U^\prime) \\ \downarrow&&\downarrow^{j(f)} \\ X&\to&j(X^\prime) }$~~

Definition (2.2.9 $Spec_X$ as a $H$ -structured topos)

Let $H$ be a geometry, let $X\in H$ be an object. Then we define the $H$ -stuctured topos $(Spec X, O_{Spec X})$ by:

Spec X:=Sh(Pro(H)^{ad}/ X)

O_{Spec X}:=(H\stackrel{\tilde{O}_{Spec X}}{\to} Psh(Pro(H)^{ad} /X)\stackrel{L}{\to} Spec X:=Sh(Pro(H)^{ad}/X))

where $L$ is sheafification (left adjoint to the inclusion and

\tilde{O}_{Spec X}:=(H\times (Pro(H)^{ad}/X)^{op}\to H\times Pro(H)^{op}=Lex(H,\infty Grpd)\stackrel{ev}{\to} \infty Grpd

Proof

We have to show that $O_{Spec X}$ is a $H$ -structure on $Spec X$ .

~~Restriction of $H$ -structured $(\infty,1)$ -toposes, $H$ -schemes~~

Lemma 2.2.14

Let $H$ be a geometry, let $X\in Proj(H)$ a pro object of $H$ . Let $j:Pro(H)^{ad}/X \to Spec X$ denote the composite of the top arrows, let $proj$ denote the projection in

\array{ &&Pro(H)^{ad}/X&\stackrel{Y}{\hookrightarrow} Psh(Pro(H)^{ad}/X)\stackrel{Sh(-)}{\to} Sh(Pro(H)^{ad}/X)=Spec X \\ &&\downarrow^{proj}&&\nearrow^{Lan_{proj} j} \\ H&\stackrel{y}{\hookrightarrow}& Pro(H) }

and let $Y$ and $y$ denote Yoneda embeddings. Then we have the equivalence

((Lan_{proj} j)\circ y)\simeq O_{Spec X}

Digression (DAG V, Prop.2.3.7)

(1) A morphisms $f:(X,O_{G,X})\to (Y,O_{G,Y})$ is called étale if (1a) the underlying geometric morphism of $(\infty,1)$ -toposes is étale and (1b) the induced map $f^*:X\to Y$ is an equivalence in $Str_G(\mathfrak{U})$

(2) Condition (1b) is equivalent to the requirement that $f$ is $p$ -cocartesian for $p:LTop(G)\to LTop$ the projection.

(3) Being an étale geometric morphism of structured $(\infty,1)$ -toposes is a local property:

If there is an effective epimorphism $\coprod_i U_i\to *_X$ to the terminal object of $X$ , and $f:(X,O_{G,X})\to (Y,O_{G,Y})$ in $LTop(G)^{op}$ a morphism such that

$f_{|U_i}:((X/U_i,(O_{G,X})_{|U_i})\to (Y,O_{G,Y})$

is étale, then $f$ is étale.

Restriction of $H$ -structured $(\infty,1)$ -toposes, $H$ -schemes

Definition Digression (Restriction (DAG 2.3.3, V, DAG Prop.2.3.7) chapter 2.3)

~~Let~~ (1) A morphisms $f : (X, O_{G, X}) \to (Y, O_{G, Y})$ ~~(X,O_{G,X})~~ f:(X,O_{G,X})\to (Y,O_{G,Y}) be is a called ~~structured~~ étale if (1a) the underlying geometric morphism of $(\infty,1)$ ~~-topos,~~ -toposes ~~let~~ is ~~$U\in X$~~ étale be and an (1b) ~~object.~~ the induced map $f^*:X\to Y$ is an equivalence in $Str_G(\mathfrak{U})$

~~(1)~~ (2) ~~The~~ Condition ~~restriction~~ (1b) of is equivalent to the requirement that $X f$ X f to is $U p$ U p -cocartesian is for ~~defined~~ to be ~~the~~ ~~slice~~ $X p / : U LTop (G) \to LTop$ ~~X/U~~ p:LTop(G)\to LTop . the projection.

~~(2)~~ (3) ~~The~~ Being ~~restriction~~ an étale geometric morphism of structured $(O_{G, X} ∞)_{| U}, 1)$ ~~(O_{G,X})_{|~~ (\infty,1) U} -toposes of is a local property: ~~$O_{G,X}$~~ to ~~$U$~~ ~~is defined to be composite~~

~~$G\stackrel{O_{G,X}}{\to}X\stackrel{p^*}{\to}X/U$~~

If there is an effective epimorphism $\coprod_i U_i\to *_X$ to the terminal object of $X$ , and $f:(X,O_{G,X})\to (Y,O_{G,Y})$ in $LTop(G)^{op}$ a morphism such that

~~where $p^*$ is base change along $p:U\to *$ .~~

f_{|U_i}:((X/U_i,(O_{G,X})_{|U_i})\to (Y,O_{G,Y})

is étale, then $f$ is étale.

Definition ( (Restriction 2.3.3, DAG chapter 2.3) $LTop$ , $LTop(G)$ )

~~(1)~~ Let $LTop (X, O_{G, X})$ ~~LTop~~ (X,O_{G,X}) is ~~defined~~ to be ~~the~~ a structured $(\infty,1)$ ~~-category~~ -topos, ~~which~~ let ~~objects~~ ~~are~~ $(U ∞ \in, X 1)$ ~~(\infty,1)~~ U\in X ~~-toposes~~ ~~and~~ be ~~which~~ an ~~morphisms~~ object. ~~are~~ ~~geometric~~ ~~morphisms~~ of ~~$(\infty,1)$~~ ~~-toposes~~ ~~$f$~~ ~~such that the inverse image~~ ~~$f^*$~~ ~~preserves small colimits and finite limits.~~

~~(2)~~ (1) ~~For~~ The a restriction ~~geometry~~ of $G X$ G X we to ~~identify~~ ~~with~~ $LTop U (G)$ ~~LTop(G)~~ U is defined to be the slice $(X ∞ /, U 1)$ ~~(\infty,1)~~ X/U ~~-category~~ . ~~which~~ ~~has~~ as ~~objects~~ ~~$G$~~ ~~-structures~~ ~~$O:G\to H$~~ ~~on some~~ ~~$(\infty,1)$~~ ~~-topos~~ ~~$H$~~ ~~and morphisms are those natural transformations~~ ~~$f^*$~~ ~~which (1.) are inverse images of geometric morphisms and (2.) whose naturality square is a pullback square in every admissible morphism.~~

(2) The restriction $(O_{G,X})_{| U}$ of $O_{G,X}$ to $U$ is defined to be composite

G\stackrel{O_{G,X}}{\to}X\stackrel{p^*}{\to}X/U

where $p^*$ is base change along $p:U\to *$ .

Definition (relative- ( and absolute spectrum) $LTop$ , $LTop(G)$ )

~~Let~~ (1) $p LTop : G \to G_{0}$ ~~p:G\to~~ LTop ~~G_0~~ is defined to be a the ~~morphism~~ of ~~geometries.~~ ~~Let~~ $p^{*} : = (- ∞), \circ 1 p : LTop (G_{0}) \to L Top (G)$ ~~p^*:=(-)\circ~~ (\infty,1) ~~p:LTop(G_0)\to~~ L ~~Top(G)~~ -category ~~the~~ which ~~restriction~~ objects ~~functor.~~ are $(\infty,1)$ -toposes and which morphisms are geometric morphisms of $(\infty,1)$ -toposes $f$ such that the inverse image $f^*$ preserves small colimits and finite limits.

~~(1)~~ (2) ~~Then~~ For ~~there~~ a is geometry an ~~adjunction~~ $G$ we identify with $LTop(G)$ the $(\infty,1)$ -category which has as objects $G$ -structures $O:G\to H$ on some $(\infty,1)$ -topos $H$ and morphisms are those natural transformations $f^*$ which (1.) are inverse images of geometric morphisms and (2.) whose naturality square is a pullback square in every admissible morphism.

~~$(Spec_{G,G_0}\dashv p^*):L Top(G_0)\stackrel{p^*}{\to}LTop(G)$~~
~~where the left adjoint is called a relative spectrum functor.~~
~~(2) Let now $G_0$ be the discrete geometry underlying $G$ . Then~~
~~$Spec_G:= Spec_{G,G_0}\circ \iota$~~
~~is called absolute spectrum functor; here $\iota:Ind(G^{op})\hookrightarrow LTop(G_0)$ denotes the inclusion of the ind objects of $G$ .~~

Definition (relative- and absolute spectrum)

Let $p:G\to G_0$ be a morphism of geometries. Let $p^*:=(-)\circ p:LTop(G_0)\to L Top(G)$ the restriction functor.

(1) Then there is an adjunction

(Spec_{G,G_0}\dashv p^*):L Top(G_0)\stackrel{p^*}{\to}LTop(G)

where the left adjoint is called a relative spectrum functor.

(2) Let now $G_0$ be the discrete geometry underlying $G$ . Then

Spec_G:= Spec_{G,G_0}\circ \iota

is called absolute spectrum functor; here $\iota:Ind(G^{op})\hookrightarrow LTop(G_0)$ denotes the inclusion of the ind objects of $G$ .

A construction of the absolute spectrum (DAGV section 2.2)

Remark (The $H$ -structured global section functor $\Gamma_H$ )

Let $p:\overline{LTop}\to LTop$ be the universal topos fibration. We interpret objects of $\overline{LTop}$ as pairs $(X,x)$ where $X$ is a topos and $x\in X$ is an object of $X$ .

(…)

The global section functor $\Gamma:\overline{LTop}\to\infty Grpd$ factors through $Lex(H,\infty Grpd)\simeq Pro(H)^{op}$ . (…) The resulting map $\Gamma_H:LTop(H)\to Pro(H)^{op}$ is called $H$ -structured global section functor.

Proposition (Theorem 2.2.12, $(\Spec_H\dashv \Gamma_H)$ )

Let $H$ be a geometry, let $X$ be an object of $Pro(G)$ .

(1) Then

(Spec_H\dashv \Gamma_H):LTop(H)\stackrel{\Gamma_H}{\to}Pro_H

is an adjunction.

(2) $O_Y$ factors as

H\stackrel{j}{\to}Pro (H)\stackrel{\overline{O_Y}}{\to} Y

where $j$ denotes the Yoneda embedding and $\overline{O_Y}$ preserves small limits. And we have the identification of mapping spaces

Pro(H)^{op}(X,\Gamma_H(Y,O_Y))\simeq Y(*, \overline{O_Y}(X))

Digression (Example 2.3.8)

Let $H$ be a geometry, let $f:U\to X$ be an admissible morphism in $Pro(H)$ . Then

Spec_H U\to Spec_H X

is an étale morphism of absolute spectra.

Proof

This follows from the previous Proposition (Theorem 2.2.12).

A direct proof goes as follows: Let $f:U\to X$ be admissible in $Pro (H)$ . Then $Spec_H$ preserves finite limits, hence we can assume that $f$ arises from an admissible morphism $f_0:U_U\to X_0$ in $H$ .

Let $Spec_H X=:(\chi, O_\chi)$ such that $O_\chi(X_0)$ has a canonical global section $\eta:1_\chi\to O_\chi(X_0)$ . Let $Y:=hfiber(\eta, O_\chi(f_0))$ and let

(\Upsilon,O_\Upsilon):=(\chi/Y, O_\chi |Y)

Then there is a canonical global section $\eta^\prime$ of $O_\Upsilon(U_0)$ .

This means that $\eta^\prime$ exhibits $(\Upsilon,O_\Upsilon)$ as an absolute spectrum $U$ such that $Spec_H f$ can be identified with the étale map $(\Upsilon, O_\Upsilon)\to (\chi,O_\chi)$ .

Definition

A $G$ -structured (∞,1)-topos $(X,O_{G,X})$ is called locally representable (aka a $G$ -scheme) if

there exists a collection $\{U_i \in X\}$

such that

the $\{U_i\}$ cover $X$ in that the canonical morphism $\coprod_i U_i \to {*}$ (with ${*}$ the terminal object of $X$ ) is an effective epimorphism;
for every $U_i$ there exists an equivalence

$(X/{U_i}, O_{G,X}|_{U_i}) \simeq Spec_{G} A_i$

of structured $(\infty,1)$ -toposes for some $A_i \in Pro(G)$ (in the (∞,1)-category of pro-objects in $G$ ). In other words $(X,O_{G,X})$ is assumed to be locally equivalent to an absolute spectrum (aka affine scheme) of a pro object in $G$ .

Remark

If $(C,O_C)$ is a structured topos and $(C/U,(O_C)_{|U})$ is an restriction thereof, then $(C,O_C)\to (C/U,(O_C)_{|U})$ is an étale morphism of structured toposes.

Remark Lemma

~~There~~ If ~~exists~~ (cond) a admissible morphisms in the essential image of the Yoneda embedding $(H / X)^{fet} ↪ Pro ((H / X)^{fet})$ H (H/X)^{fet}\hookrightarrow Pro((H/X)^{fet}) ~~-structure~~ are preserved (pulled back to formally étale morphisms) by pullback we have: ~~$O$~~ on ~~$(H/X)^{fet}$~~ ~~such that~~ ~~$((H/X)^{fet},O)$~~ ~~is a locally representable~~ ~~$H$~~ ~~-structured~~ ~~$(\infty,1)$~~ ~~-topos.~~

\array{ (H/X)^{fet}/U&\simeq &Sh((H/X)^{fet})/U\\ &\simeq&Sh(Pro(H/X)^{ad}/U)&:=& Spec U}

where the first equivalence holds since $(H/X)^{fet}$ is a topos, the second one is implied by DAGV Notation 2.2.2 and Remark 2.2.7 and (cond), the $:=$ is definition 2.2.9. wrt. the geometry $(H/X)$

Proof Remark

~~$O:H\to E$~~ There exists a ~~has to satisfy~~ $H$ -structure $O$ on $(H/X)^{fet}$ such that $((H/X)^{fet},O)$ is a locally representable $H$ -structured $(\infty,1)$ -topos.

$O$ is left exact

$O$ satisfies codescent: For every collection of admissible (i.e. formally étale) morphisms $\{U_i\to X\}$ in $H$ which generate a covering sieve on $X$ , the induced map $\coprod_i O(U_i)\to O(X)$ is an effective epimorphism in $E$ .

The terminal object in $E:=(H/X)^{fet}$ is $id_X$ the identity on $X$ . The collection of all formally étale effective epimorphisms (in $H$ ) with codomain $X$ covers $X$ . By HTT Remark 6.2.3.6. they cover $id_X$ in the slice.
Now we choose $O:=r\circ p^*$ to be the composit of base change (this functor is exact) $b^*:H\to H/X$ along $b:X\to *$ followed by the coreflector (that we have a coreflector is shown (reference)) $r:H/X\to E$ (this functor is right adjoint and hence left exact). In total $O$ is left exact and since our cover consists only of formally étale morphisms $r$ and hence $O$ preserve the cover.
~~Now we describe the restriction of $(E,O)$ to an element $U$ of the cover:~~
~~Let $U\to id_X$ be an element of the cover; i.e. a formally étale effective epimorphism $U\to X$ .~~
~~The restriction $(E/U,O_{| U})$ of $(E,O)$ to $U$ is given by:~~

Objects of $E/U$ are cocones $\array{A&\to &X\\\searrow &&\swarrow\\&U&}$ where $A\to X$ is formally étale. Morphisms are pyramids with four faces and tip $U$ .

The restriction of the $H$ -structure $O$ is given as follows:

~~$H\stackrel{O}{\to}E\stackrel{p^*}{\to}E/U$~~
~~where $p^*$ is base change along $p:U\to *$ .~~
~~Now we show that $(E, O)$ is locally equivalent to an absolute spectrum:~~
Let $H_0$ denote the discrete geometry (admissible morphisms are precisely all equivalences) with underlying category $H$ . Let $h:H_0\to H$ be a morphism of geometries (i.e. $h$ preserves finite limits, maps admissible morphism to such, the image of an admissible cover is an admissible cover). Then there is an adjunction
~~$(Spec_{H_0,H}\dashv p^*):LTop(H_0)\stackrel{Spec_{H_0,H}}{\to} LTop(H)$~~
~~and the absolute spectrum $Spec_H$ is defined to be the composit~~
~~$Ind(H^{op})\simeq Pro(H)^{op}\simeq Lex(H, \infty Grpd)\hookrightarrow LTop(H_0)\stackrel{Spec_{H_0,H}}{\to}LTop(H)$~~
~~(Pro objects in $H$ are cofiltered diagrams in $H$ or -equivalently - filtered diagrams in $H^{op}$ )~~

Proof

$O:H\to E$ has to satisfy

$O$ is left exact
$O$ satisfies codescent: For every collection of admissible (i.e. formally étale) morphisms $\{U_i\to X\}$ in $H$ which generate a covering sieve on $X$ , the induced map $\coprod_i O(U_i)\to O(X)$ is an effective epimorphism in $E$ .

The terminal object in $E:=(H/X)^{fet}$ is $id_X$ the identity on $X$ . The collection of all formally étale effective epimorphisms (in $H$ ) with codomain $X$ covers $X$ . By HTT Remark 6.2.3.6. they cover $id_X$ in the slice.

Now we choose $O:=r\circ p^*$ to be the composit of base change (this functor is exact) $b^*:H\to H/X$ along $b:X\to *$ followed by the coreflector (that we have a coreflector is shown (reference)) $r:H/X\to E$ (this functor is right adjoint and hence left exact). In total $O$ is left exact and since our cover consists only of formally étale morphisms $r$ and hence $O$ preserve the cover.

Now we describe the restriction of $(E,O)$ to an element $U$ of the cover:

Let $U\to id_X$ be an element of the cover; i.e. a formally étale effective epimorphism $U\to X$ .

The restriction $(E/U,O_{| U})$ of $(E,O)$ to $U$ is given by:

Objects of $E/U$ are cocones $\array{A&\to &X\\\searrow &&\swarrow\\&U&}$ where $A\to X$ is formally étale. Morphisms are pyramids with four faces and tip $U$ .
The restriction of the $H$ -structure $O$ is given as follows:

H\stackrel{O}{\to}E\stackrel{p^*}{\to}E/U

where $p^*$ is base change along $p:U\to *$ .

Now we show that $(E, O)$ is locally equivalent to an absolute spectrum:

Let $H_0$ denote the discrete geometry (admissible morphisms are precisely all equivalences) with underlying category $H$ . Let $h:H_0\to H$ be a morphism of geometries (i.e. $h$ preserves finite limits, maps admissible morphism to such, the image of an admissible cover is an admissible cover). Then there is an adjunction

(Spec_{H_0,H}\dashv p^*):LTop(H_0)\stackrel{Spec_{H_0,H}}{\to} LTop(H)

and the absolute spectrum $Spec_H$ is defined to be the composit

Ind(H^{op})\simeq Pro(H)^{op}\simeq Lex(H, \infty Grpd)\hookrightarrow LTop(H_0)\stackrel{Spec_{H_0,H}}{\to}LTop(H)

(Pro objects in $H$ are cofiltered diagrams in $H$ or -equivalently - filtered diagrams in $H^{op}$ )

Last revised on December 18, 2012 at 00:33:11. See the history of this page for a list of all contributions to it.

Spahn sheaf on a sheaf (changes)

Motivation

Très petit topos

Definition (universal GG-structure, classifying topos)

Remark

Proof

Local representability of the très petit topos

Relative-, and absolute spectra

Definition (pro objects)

Definition 2.2.2 (admissible morphism in Pro(H)Pro(H))

Remark 2.2.7

Definition (2.2.9 Spec XSpec_X as a HH-structured topos)

Proof

Restriction of HH-structured (∞,1)(\infty,1)-toposes, HH-schemes

Lemma 2.2.14

Digression (DAG V, Prop.2.3.7)

Restriction of HH-structured (∞,1)(\infty,1)-toposes, HH-schemes

Definition Digression (Restriction (DAG 2.3.3, V, DAG Prop.2.3.7) chapter 2.3)

Definition ( (Restriction 2.3.3, DAG chapter 2.3)LTopLTop, LTop(G)LTop(G))

Definition (relative- ( and absolute spectrum)LTopLTop, LTop(G)LTop(G))

Definition (relative- and absolute spectrum)

A construction of the absolute spectrum (DAGV section 2.2)

Remark (The HH-structured global section functor Γ H\Gamma_H)

Proposition (Theorem 2.2.12, (Spec H⊣Γ H)(\Spec_H\dashv \Gamma_H))

Digression (Example 2.3.8)

Proof

Definition

Remark

Remark Lemma

Proof Remark

Proof

Definition (universal $G$ -structure, classifying topos)

Definition 2.2.2 (admissible morphism in $Pro(H)$ )

Definition (2.2.9 $Spec_X$ as a $H$ -structured topos)

Restriction of $H$ -structured $(\infty,1)$ -toposes, $H$ -schemes

Restriction of $H$ -structured $(\infty,1)$ -toposes, $H$ -schemes

Definition ( (Restriction 2.3.3, DAG chapter 2.3) $LTop$ , $LTop(G)$ )

Definition (relative- ( and absolute spectrum) $LTop$ , $LTop(G)$ )

Remark (The $H$ -structured global section functor $\Gamma_H$ )

Proposition (Theorem 2.2.12, $(\Spec_H\dashv \Gamma_H)$ )