## 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 $$Fun^*(K,T)\to Str_G(T)$$ (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 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 of $G$). 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 =--