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(infinity,2)-sheaf

Idea

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

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

$$X:{𝒞}^{\mathrm{op}}\to {\mathrm{Cat}}_{(\mathrm{\infty },1)}$$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 $\mathrm{Func}({𝒞}^{\mathrm{op}},{\mathrm{Cat}}_{(\mathrm{\infty },1)})$.

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

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

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

Examples

Codomain fibration / canonical $(\mathrm{\infty },2)$-sheaf

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

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

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

Propositon

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

$$\mathrm{Cod}:{𝒳}^{\mathrm{op}}\to {\mathrm{CAT}}_{(\mathrm{\infty },1)}$$Cod : \mathcal{X}^{op} \to CAT_{(\infty,1)}

$$\mathrm{Cod}:A\mapsto {𝒳}_{/A}$$Cod : A \mapsto \mathcal{X}_{/A}

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

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

Remark

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

Related concepts

• sheaf

• 2-sheaf, (2,1)-sheaf

• (∞,1)-sheaf / ∞-stack

• $(\mathrm{\infty },2)$-sheaf

• (∞,n)-sheaf

References

Section 6.1.3 of

• Jacob Lurie, Higher Topos Theory