Spahn locally representable structured (infinity,1)-topos (Rev #2)

Admissible structure

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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 $(G_0,G)$-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.

$G$-schemes