## 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 $H$. +-- {: .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 $$Pro(C)(F,G)=lim_{e\in E}colim_{d\in D}C(F(d),G(e))$$ =-- +-- {: .num_defn} ###### Definition (restriction) =-- +-- {: .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 the underlying geometric morphism of $(\infty,1)$-toposes is [[nLab:etale geometric morphism|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. =-- +-- {: .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 $(H/X)^{fet}$. =--