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(infinity,2)-sheaf

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Higher topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

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

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

X:𝒞 opCat (,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 Func(𝒞 op,Cat (,1))Func(\mathcal{C}^{op}, Cat_{(\infty,1)}).

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

Sh (,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 (,2)(\infty,2)-sheaf

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

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

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

Propositon

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

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

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

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

Remark

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

References

Section 6.1.3 of

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