## Admissible structure (...) ## Geometry $G=(\underline G, Ad_G)$ is a *geometry*, if $\underline G$ is an essentially small $(\infty,1)$-category with finite limits, which is idempotent complete, $Ad_G$ is an admissible structure in $\underline G$ ## Structured $(\infty,1)$-topos Structured $(\infty,1)$-topos $(X,O_{G,X})$ where * (1) $X$ is an $(\infty,1)$-topos * (2) $G=(\underline G, Ad_G)$ is a *geometry*, i.e. $\underline G$ is an essentially small $(\infty,1)$-category with finite limits, which is idempotent complete, $Ad_G$ is an admissible structure in $\underline G$. * (3) $O_{G,X}:G\to X$ is a *$G$-structure (aka "structure sheaf") on $X$*, i.e. a functor which is * (3a) left exact * (3b) induces a jointly epimorphic family in the image of $O_{G,X}$ (i.e. every covering sieve consisting of admissible morphisms on an object of $X$ induces an effective epimorphism out of the product of the images...) A *morphism of geometries* (called *transformation* by Lurie) is defined to be a functor satisfying (3a),(3b), and takes admissible morphisms to such. A morphism of geometries is called *local morphism of geometries* if all its naturality squares are pullbacks. A geometry $G$ is called *discrete geometry* if * precisely equivalences in $\underline G$ are admissible * the Grothendieck topology on $\underline G$ is trivial. ## Spectra (relative $(G,G_0)$-spectrum, absolute $G$-spectrum) $L Top$ denotes the $(\infty,1)$-category of $(\infty,1)$-toposes with morphisms being geometric morphisms $f$ such that the inverse image functor $f^*$ preserves small colimits and finite limits. $L Top (G)$ is called *the opposite $(\infty,1)$-category of that of $G$-structured $(\infty,1)$-topoi*. +-- {: .num_defn} ###### Definition (Relative Spectrum functor, absolute spectrum functor (="affine scheme")) 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 $G$. =-- ## $G$-schemes Let $G$ be a geometry (for structured (∞,1)-toposes). A $G$-structured (∞,1)-topos $(X,O_{G,X})$ is 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 $$ (X/{U_i}, O_{G,X}|_{U_i}) \simeq Spec_{G,X} A_i $$ of structured $(\infty,1)$-toposes for some $A_i \in Pro(G)$ (in the (∞,1)-category of pro-objects of $G$).