nLab (infinity,2)-sheaf

Idea

An $(\infty,2)$-sheaf or $(\infty,2)$-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 $\mathcal{X}$.

Related concepts

• sheaf

• 2-sheaf, (2,1)-sheaf

• (∞,1)-sheaf / ∞-stack

• $(\infty,2)$-sheaf

• (∞,n)-sheaf

References

• Jacob Lurie, section 6.1.3 of Higher Topos Theory

Discussion of a local model structure on simplicial presheaves $[S^op, sSet_{Joyal}]_{loc}$ with respect to the Joyal model structure $sSet_{Joyal}$ for quasicategories is in

• Nicholas Meadows, The Local Joyal Model Structure (arXiv:1507.08723)

and with respect to the model structure for complete Segal spaces in

• Nicholas Meadows, Local Complete Segal Spaces (arXiv:1607.05794)