Spahn sheaf on a sheaf (Rev #6, changes)

Showing changes from revision #5 to #6: 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)

(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

~~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 $G$ .

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

Revision on December 16, 2012 at 02:54:54 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

Spahn sheaf on a sheaf (Rev #6, changes)

Motivation

Très petit topos

Definition (universal GG-structure, classifying topos)

Remark

Proof

Local representability of the très petit topos

Definition

Definition

Remark

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