Spahn sheaf on a sheaf (Rev #10, changes)

Showing changes from revision #9 to #10: 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 $X$ .

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 objects) in $(H/X)^{fet}$ )

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))

Definition (restriction)

~~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 Digression (DAG V, Prop.2.3.7)

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

(2) 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.

Proof Definition

~~The~~ A ~~terminal~~ ~~object~~ in $(G H / X)^{fet}$ ~~(H/X)^{fet}~~ G -structured is (∞,1)-topos ${id}_{X} (X, O_{G, X})$ ~~id_X~~ (X,O_{G,X}) ~~the~~ is ~~identity~~ called on ~~$X$~~ locally representable . ~~The~~ (aka ~~collection~~ a of ~~all~~ ~~formally~~ ~~étale~~ ~~effective~~ ~~epimorphisms~~ ~~(in~~ ~~$H$~~ $G$ -scheme ) ~~with~~ if ~~codomain~~ ~~$X$~~ ~~covers~~ ~~$X$~~ ~~and hence the cover~~ ~~$id_X$~~ ~~in the slice. Since our chosen~~ ~~$(H/X)^{fet}$~~ ~~-structure is the identity the cover is preserved by it.~~

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

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

~~Now~~ such we that ~~consider~~ ~~$((H/X)^{fet}/U,id_{(H/X)^{fet}/U})$~~ :

~~Objects of $(H/X)^{fet}/U$ are cocones $\array{A&\to &X\\\searrow &&\swarrow\\&U&}$ where $A\to X$ is formally étale. Morphisms are pyramids with three faces and tip $U$ .~~

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

$((H/X)^{fet},id_{(H/X)^{fet}})$ is locally representable.

Proof

The terminal object in $(H/X)^{fet}$ 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. Since our chosen $(H/X)^{fet}$ -structure is the identity the cover is preserved by it.

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

Now we consider $((H/X)^{fet}/U,id_{(H/X)^{fet}/U})$ :

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

Pro objects in $(H/X)^{fet}$ are cofiltered diagrams in $(H/X)^{fet}$ .

Revision on December 16, 2012 at 19:09:19 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

Spahn sheaf on a sheaf (Rev #10, changes)

Motivation

Très petit topos

Definition (universal GG-structure, classifying topos)

Remark

Proof

Local representability of the très petit topos

Definition (pro objects objects) in(H/X) fet(H/X)^{fet})

Definition (restriction)

Remark Digression (DAG V, Prop.2.3.7)

Proof Definition

Remark

Proof

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

Definition (pro objects objects) in $(H/X)^{fet}$ )