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 $G$ ).