sheaf on a sheaf in Spahn

## 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 $((H/X)^{fet},{id_{(H/X)^{fet}}})$ 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 $H$.

+-- {: .num_defn}
###### 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 $X$.

=--


+-- {: .num_remark}
###### Remark

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

=--

+-- {: .proof}
###### 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

+-- {: .num_defn}
###### 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
=--

+-- {: .num_remark}
###### 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 [[nLab:etale geometric morphism|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.
=--

+-- {: .num_defn}
###### Definition (Restriction, DAG  chapter 2.3)
Let $(X,O_{G,X})$ be a structured $(\infty,1)-topos, let $U\in X$ 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 $p:U\to *$.
=--

+--{: .num_defn}
###### 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 $G$.

=--

+-- {: .num_defn}
###### 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 $G$.
=--

+-- {: .num_remark}
###### Remark
$((H/X)^{fet},id_{(H/X)^{fet}})$ is locally representable.
=--

+--{: .proof}
###### 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 $(H/X)^{fet}$ or -equivalently filtered diagrams in $((H/X)^{fet})^{op}$
=--