nLab
geometric infinity-stack

Idea

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

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

Definition

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

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

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

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

Definition

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

$$X\simeq \underrightarrow{\lim }{K}_{•}$$X \simeq {\lim_\to} K_\bullet

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

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

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, definition 4.1.4.

Properties

Proposition

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

Proof

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

Examples

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

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

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

Related concepts

• geometric stack

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

References

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

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