Spahn sheaf on a sheaf (Rev #14, changes)

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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 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 if

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

Definition (pro objects)

Let $C$ be an $(\infty,1)$ -category. We have $Ind(C^{op})\simeq Pro(C)^{op}$ . A pro object in 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

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.

Definition (Restriction 2.3.3, DAG chapter 2.3)

Let $(X,O_{G,X})$ be a structured $(∞, 1) - topos, let$ ~~(\infty,1)-topos,~~ (\infty,1) ~~let~~ ~~U\in~~ -topos, X$ let be an ~~object.~~ $U\in X$ be an object.

(1) The restriction of $X$ to $U$ is defined to be the slice $X/U$ .

(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)

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 $G$ .

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

~~$((H/X)^{fet},O)$~~ If ~~is locally representable~~ $(C,O_C)$ ~~$H$~~ is a structured topos and ~~-structured~~ $(C/U,(O_C)_{|U})$ ~~$(\infty,1)$~~ is an restriction thereof, then ~~-topos.~~ $(C,O_C)\to (C/U,(O_C)_{|U})$ is an étale morphism of structured toposes.

Proof Remark

~~The terminal object in~~ $((H/X)^{fet},O)$ ~~$E:=(H/X)^{fet}$~~ is locally representable is $H$ ~~$id_X$~~ -structured ~~the identity on~~ $(\infty,1)$ ~~$X$~~ -topos.~~. The collection of all formally étale effective epimorphisms (in~~ ~~$H$~~ ~~) with codomain~~ ~~$X$~~ ~~covers~~ ~~$X$~~ ~~and hence the cover~~ ~~$id_X$~~ ~~in the slice.~~

We know that $E\hookrightarrow H$ is a reflective- and coreflective subcategory of $H$ (if not argue $O$ is leftexact, then an adjoint functor theorem implies that $E$ is coreflective subcat.). We think of $O=r$ being the reflector
~~Let $U\to id_X$ be an element of the cover; i.e. a formally étale effective epimorphism $U\to X$ . Since $E$ is a coreflective subcategory $O(U)\to O(X)$ is a cover of $O(X)$ .~~
~~The restriction $(E/U,(O_E)_{| U})$ of $(E,O_E)$ 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 $O:=O_E$ is given as follows:

~~$H\stackrel{O}{\to}E\stackrel{p^*}{\to}H/U$~~
~~where $p^*$ is base change along $p:U\to *$ . Pro objects in $E$ are cofiltered diagrams in $E$ or -equivalently filtered diagrams in $E^{op}$~~

Proof

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 $X$ and hence the cover $id_X$ in the slice. This follows from HTT Remark 6.2.3.6.

We know that $E\hookrightarrow H/X$ is a reflective- and coreflective subcategory of $H$ . $H/X\to H$ is a geometric morphism. Hence we can choose $O:H\to E$ to be a left adjoint which preserves effective epimorphisms.

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

The restriction $(E/U,(O_E)_{| U})$ of $(E,O_E)$ 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 $O:=O_E$ is given as follows:

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

where $p^*$ is base change along $p:U\to *$ . Pro objects in $E$ are cofiltered diagrams in $E$ or -equivalently filtered diagrams in $E^{op}$

Revision on December 17, 2012 at 04:39:16 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

Spahn sheaf on a sheaf (Rev #14, changes)

Motivation

Très petit topos

Definition (universal GG-structure, classifying topos)

Remark

Proof

Local representability of the très petit topos

Definition (pro objects)

Digression (DAG V, Prop.2.3.7)

Definition (Restriction 2.3.3, DAG chapter 2.3)

Definition (relative- and absolute spectrum)

Definition

Remark

Proof Remark

Proof

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