nLab geometric infinity-stack

Theorems

$(\infty,1)$-Topos Theory

(∞,1)-topos theory

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

A geometric $\infty$-stack is an ∞-stack over a geometry with function theory $(\mathcal{O} \dashv Spec)$ which is an ∞-groupoid that is degreewise an $\infty$-stack in the image of $Spec$.

This generalizes the notion of geometric stack from topos theory to (∞,1)-topos theory.

Definition

The text below follows (Toën 00). Needs to be expanded and (Toën-Vezzosi 04, Lurie) needs to be brought into the picture

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

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

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

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

Definition

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

$X \simeq {\lim_\to} K_\bullet$

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

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

2. the target map $d_1 : K_1 \to K_0$ is (…sufficiently well behaved…)

For $T$ the theory of commutative associative algebras over a commutative ring $k$ and $C$ the fpqc topology this appears as Toën 00, definition 4.1.4.

Properties

General

Proposition

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

Proof

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

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.

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 April 8, 2014 10:05:04 by Urs Schreiber (82.113.106.176)