nLab
geometric infinity-stack

Context

Higher geometry

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

The concept of geometric \infty-stack is the refinement to ∞-stack of that of geometric stack.

There is an intrinsic definition which iterates that of geometric stacks and says inducively that a geometric nn-stack is one which has an nn-atlas and such that its diagonal is (n1)(n-1)-representable (Toën-Vezzosi 04, def. 1.3.3.1).

Then there is a result which says that such geometric nn-stacks are equivalently those represented by suitable Kan complex-objects in the given site (“internal infinity-groupoids” in the site) (Pridham 09).

(There is also a definition of “geometric \infty-stack” in (Toën 00, definition 4.1.4), which is however different.)

Definition

Properties

Presentation by Kan-fibrant simplicial objects

A presentation of geometric \infty-stacks, in some generality, by suitably Kan-fibrant simplicial objects is in (Pridham 09). See also at Kan-fibrant simplicial manifold.

Examples

Toen 00

The text below follows (Toën 00). Needs to be connected to the rest of the entry.

We consider the higher geometry encoded by a Lawvere theory TT via Isbell duality. Write TAlgT Alg for the category of algebras over a Lawvere theory and write TAlg ΔT Alg^{\Delta} for the (∞,1)-category of cosimplicial TT-algebras .

Consider a site TCTAlg opT \subset C \subset T Alg^{op} that satisfies the assumptions described at function algebras on ∞-stacks. Then, by the discussion given there, we have a pair of adjoint (∞,1)-functors

(𝒪Spec):(TAlg Δ) opSpec𝒪Sh (C)=:H, (\mathcal{O} \dashv Spec) : (T Alg^{\Delta})^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}} Sh_\infty(C) =: \mathbf{H} \,,

where H:=Sh (C)\mathbf{H} := Sh_\infty(C) is the (∞,1)-category of (∞,1)-sheaves over CC, the big topos for the higher geometry over CC.

Definition

An object XHX \in \mathbf{H} is called a geometric \infty-stack over CC if there is it is the (∞,1)-colimit

Xlim K X \simeq {\lim_\to} K_\bullet

over a groupoid object K :ΔHK_\bullet : \Delta \to \mathbf{H} in H\mathbf{H} such that

  1. K 0K_0 and K 1K_1 are in the image of Spec:(TAlg Δ) opHSpec : (T Alg^{\Delta})^{op} \to \mathbf{H};

  2. the target map d 1:K 1K 0d_1 : K_1 \to K_0 is lisse.

For TT the theory of commutative associative algebras over a commutative ring kk and CC the fpqc topology this appears as (Toën 00, definition 4.1.4).

Proposition

Geometric \infty-stacks are stable under (∞,1)-pullbacks along morphism in the image of SpecSpec.

Proof

Use that in the (∞,1)-topos H\mathbf{H} we have universal colimits and that SpecSpec is right adjoint.

References

The notion of geometric \infty-stack as a weak quotient of affine \infty-stacks is considered in section 4 of

More general theory in the context of derived algebraic geometry is in

and specifically in E-∞ geometry in

Discussion of presentation of geometric \infty-stacks by Kan-fibrant simplicial objects in the site is in

Revised on September 11, 2014 10:29:33 by Urs Schreiber (141.0.9.62)