# nLab geometric infinity-stack

## Theorems

#### $\left(\infty ,1\right)$-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 $\left(𝒪⊣\mathrm{Spec}\right)$ which is an ∞-groupoid that is degreewise an $\infty$-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

$\left(𝒪⊣\mathrm{Spec}\right):\left(T{\mathrm{Alg}}^{\Delta }{\right)}^{\mathrm{op}}\stackrel{\stackrel{𝒪}{←}}{\underset{\mathrm{Spec}}{\to }}{\mathrm{Sh}}_{\infty }\left(C\right)=:H\phantom{\rule{thinmathspace}{0ex}},$(\mathcal{O} \dashv Spec) : (T Alg^{\Delta})^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}} Sh_\infty(C) =: \mathbf{H} \,,

where $H:={\mathrm{Sh}}_{\infty }\left(C\right)$ 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 $\infty$-stack over $C$ if there is it is the (∞,1)-colimit

$X\simeq \underset{\to }{\mathrm{lim}}{K}_{•}$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 $\mathrm{Spec}:\left(T{\mathrm{Alg}}^{\Delta }{\right)}^{\mathrm{op}}\to H$;

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 $\infty$-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 $\mathrm{Spec}:T{\mathrm{Alg}}_{\infty }^{\mathrm{op}}\to H$ 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.

## References

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

Revised on November 4, 2011 13:53:33 by Stephan (79.227.148.132)