## 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 $X$. ### Local representability of the très petit topos We assume that $fet$ is the admissible class defined by an infinitesimal modality $\Box$ on $H$. +-- {: .num_remark} ###### Remark $H$, $H/X$, and $(H/X)^fet$ are $(H/X)^fet$-structured $(\infty,1)$-toposes. =-- +-- {: .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 $$Fun^*(K,X)\to Str_G(X)$$ (2) In this case we say $O$ exhibits $K$ as classifying $(\infty,1)$-topos for $G$-structures on $X$. =--