# nLab (infinity,2)-sheaf

### Context

#### Higher category theory

higher category theory

## 1-categorical presentations

#### Higher topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Idea

An $(\infty,1)$-sheaf or $(\infty,1)$-stack is the higher analog of an (∞,1)-sheaf / ∞-stack.

For $\mathcal{C}$ an (∞,1)-category equipped with the structure of an (∞,1)-site, an $(\infty,2)$-sheaf on $\mathcal{C}$ is an (∞,1)-functor

$X : \mathcal{C}^{op} \to Cat_{(\infty,1)}$

to (∞,1)Cat, that satisfies descent: hence which is a local object with respect to the covering sieve inclusions in $Func(\mathcal{C}^{op}, Cat_{(\infty,1)})$.

The (∞,2)-category of $(\infty,2)$-sheaves

$Sh_{(\infty,2)}(\mathcal{C})$

is an (∞,2)-topos, the homotopy theory-generalization of a 2-topos of 2-sheaves.

## Examples

### Codomain fibration / canonical $(\infty,2)$-sheaf

Let $\mathcal{X}$ be an (∞,1)-topos, regarded as a (large) (∞,1)-site equipped with the canonical topology. Then an (∞,1)-functor

$A : \mathcal{X}^{op} \to CAT_{(\infty,1)}$

is an $(\infty,2)$-sheaf precisely if it preserves (∞,1)-limits (takes (∞,1)-colimits in $\mathcal{X}$ to (∞,1)-limits in (∞,1)Cat).

###### Propositon

For $\mathcal{X}$ an $(\infty,1)$-topos, the functor

$Cod : \mathcal{X}^{op} \to CAT_{(\infty,1)}$
$Cod : A \mapsto \mathcal{X}_{/A}$

is a (large) $(\infty,2)$-sheaf on $\mathcal{X}$ , regarded as a (∞,1)-site equipped with the canonical topology. Here $\mathcal{X}_{/A}$ is the slice (∞,1)-topos over $A$.

This is a special case of (Lurie, lemma 6.1.3.7).

###### Remark

The functor $Cod$ classifies the codomain fibration. Its fiberwise stabilization to the tangent (∞,1)-category is the $(\infty,2)$-sheaf of quasicoherent sheaves on $\mathcal{X}$.

## References

Section 6.1.3 of

Revised on November 27, 2012 10:47:40 by Urs Schreiber (82.169.65.155)