Spahn sheaf on a sheaf (Rev #9, changes)

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

(2) where In this case we say$O{\mathrm{Fun}}^{*}(K,T)$ O Fun^*(K,T) exhibits denotes the geometric morphisms$\mathrm{<span><del\; class="diffmod">\; K</del><ins\; class="diffmod">\; f</ins></span>}$ K f as with classifying inverse image$(f^{*}\mathrm{\infty }:,K1\to )T$ (\infty,1) f^*:K\to T-topos for $G$-structures on $X$.

Definition (pro objects in $(H/X)^{fet}$)

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

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 $G$.