nLab geometric infinity-stack

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 inductively that a geometric $n$-stack is one which has an $n$-atlas and such that its diagonal is $(n-1)$-representable (Toën-Vezzosi 04, def. 1.3.3.1).

Then there is a result which says that such geometric $n$-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

• Every object in the image of $Spec : T Alg_\infty^{op} \to \mathbf{H}$ is a geometric $\infty$-stack.

• Over the étale site an algebraic stack that is a geometric stack is also a geometric $\infty$-stack.

• Every schematic homotopy type is given by a geometric $\infty$-stack.

• Kan-fibrant simplicial manifolds serve to present geometric $\infty$-stacks in higher differential geometry, the Lie infinity-groupoids

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 $T$ 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 lisse.

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).

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.

Related concepts

• geometric stack

• geometric $\infty$-stack, function algebras on ∞-stacks

• Arin-Luire representability theorem?

References

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

• Bertrand Toën, Affine stacks (Champs affines) (arXiv:math/0012219)

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

• Bertrand Toën, Gabriele Vezzosi Homotopical Algebraic Geometry II: geometric stacks and applications (arXiv:math/0404373)

and specifically in E-∞ geometry in

• Jacob Lurie, Representability Theorems

• Jonathan Pridham, Representability of derived stacks (arXiv:1011.2189)

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

• Jonathan Pridham, Presenting higher stacks as simplicial schemes, Advances in Mathematics, Volume 238, Pages 184-245 (arXiv:0905.4044)