nLab
(infinity,1)-sheaf

Context

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Locality and descent

Contents

Idea

The notion of (,1)(\infty,1)-sheaf (or ∞-stack or geometric homotopy type) is the analog in (∞,1)-category theory of the notion of sheaf (geometric type?) in ordinary category theory.

See (∞,1)-category of (∞,1)-sheaves for more.

Definition

Given an (∞,1)-site CC, let SS be the class of monomorphisms in the (∞,1)-category of (∞,1)-presheaves PSh (,1)(C)PSh_{(\infty,1)}(C) that correspond to covering (∞,1)-sieve?s

η:Uj(c) \eta : U \hookrightarrow j(c)

on objects cCc \in C, where jj is the (∞,1)-Yoneda embedding.

Then an (∞,1)-presheaf APSh (,1)(C)A \in PSh_{(\infty,1)}(C) is an (,1)(\infty,1)-sheaf if it is an SS-local object. That is, if for all such η\eta the morphism

A(c)Psh C(j(c),A)PSh C(η,,A)PSh(U,A) A(c) \simeq Psh_C(j(c),A) \stackrel{PSh_C(\eta,,A)}{\to} PSh(U,A)

is an equivalence.

This is the analog of the ordinary sheaf condition. The ∞-groupoid PSh C(U,A)PSh_C(U,A) is also called the descent-∞-groupoid of AA relative to the covering encoded by UU.

Terminology

An (\infty,1)-sheaf is also called an ∞-stack with values in ∞-groupoids.

The practice of writing “\infty-sheaf” instead of ∞-stack is a rather reasonable one, since a stack is nothing but a 2-sheaf.

Notice however that there is ambiguity in what precisely one may mean by an \infty-stack: it can be an (,1)(\infty,1)-sheaf or more specifically a hypercomplete (,1)(\infty,1)-sheaf. This is a distinction that only appears in (∞,1)-topos theory, not in (n,1)-topos theory for finite nn.

homotopy leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/unit type/contractible type
h-level 1(-1)-truncated(-1)-groupoid/truth valuemere proposition, h-proposition
h-level 20-truncateddiscrete space0-groupoid/setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/groupoid(2,1)-sheaf/stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoidh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoidh-3-groupoid
h-level n+2n+2nn-truncatedhomotopy n-typen-groupoidh-nn-groupoid
h-level \inftyuntruncatedhomotopy type∞-groupoid(∞,1)-sheaf/∞-stackh-\infty-groupoid

References

Section 6.2.2 in

Revised on January 3, 2014 17:05:26 by Urs Schreiber (82.113.98.138)