Spahn locally representable structured (infinity,1)-topos

Admissible structure

(…)

Geometry

G=(G̲,Ad G)G=(\underline G, Ad_G) is a geometry, if G̲\underline G is an essentially small (,1)(\infty,1)-category with finite limits, which is idempotent complete, Ad GAd_G is an admissible structure in G̲\underline G

Structured (,1)(\infty,1)-topos

Structured (,1)(\infty,1)-topos (X,O G,X)(X,O_{G,X}) where

  • (1) XX is an (,1)(\infty,1)-topos

  • (2) G=(G̲,Ad G)G=(\underline G, Ad_G) is a geometry, i.e. G̲\underline G is an essentially small (,1)(\infty,1)-category with finite limits, which is idempotent complete, Ad GAd_G is an admissible structure in G̲\underline G.

  • (3) O G,X:GXO_{G,X}:G\to X is a GG-structure (aka “structure sheaf”) on XX, i.e. a functor which is

    • (3a) left exact

    • (3b) induces a jointly epimorphic family in the image of O G,XO_{G,X} (i.e. every covering sieve consisting of admissible morphisms on an object of XX 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 GG is called discrete geometry if

  • precisely equivalences in G̲\underline G are admissible

  • the Grothendieck topology on G̲\underline G is trivial.

Spectra (relative (G,G 0)(G,G_0)-spectrum, absolute GG-spectrum)

LTopL Top denotes the (,1)(\infty,1)-category of (,1)(\infty,1)-toposes with morphisms being geometric morphisms ff such that the inverse image functor f *f^* preserves small colimits and finite limits.

LTop(G)L Top (G) is called the opposite (,1)(\infty,1)-category of that of GG-structured (,1)(\infty,1)-topoi.

Definition (Relative Spectrum functor, absolute spectrum functor (=“affine scheme”))

Let p:GG 0p:G\to G_0 be a morphism of geometries. Let p *:=()p:LTop(G 0)LTop(G)p^*:=(-)\circ p:LTop(G_0)\to L Top(G) the restriction functor.

(1) Then there is an adjunction

(Spec G,G 0p *):LTop(G 0)p *LTop(G)(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 0G_0 be the discrete geometry underlying GG. Then

Spec G:=Spec G,G 0ιSpec_G:= Spec_{G,G_0}\circ \iota

is called absolute spectrum functor; here ι:Ind(G op)LTop(G 0)\iota:Ind(G^{op})\hookrightarrow LTop(G_0) denotes the inclusion of the ind objects of GG.

GG-schemes

Let GG be a geometry (for structured (∞,1)-toposes).

A GG-structured (∞,1)-topos (X,O G,X)(X,O_{G,X}) is a GG-scheme if

  • there exists a collection {U iX}\{U_i \in X\}

such that

  • the {U i}\{U_i\} cover XX in that the canonical morphism iU i*\coprod_i U_i \to {*} (with *{*} the terminal object of XX) is an effective epimorphism;

  • for every U iU_i there exists an equivalence

    (X/U i,O G,X| U i)Spec G,XA i (X/{U_i}, O_{G,X}|_{U_i}) \simeq Spec_{G,X} A_i

    of structured (,1)(\infty,1)-toposes for some A iPro(G)A_i \in Pro(G) (in the (∞,1)-category of pro-objects of GG).

Last revised on December 16, 2012 at 04:23:10. See the history of this page for a list of all contributions to it.