Spahn locally representable structured (infinity,1)-topos

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.

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 $G$).