# nLab (infinity,1)-sheaf

### Context

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

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Idea

The notion of $(\infty,1)$-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 $C$, let $S$ be the class of monomorphisms in the (∞,1)-category of (∞,1)-presheaves $PSh_{(\infty,1)}(C)$ that correspond to covering (∞,1)-sieve?s

$\eta : U \hookrightarrow j(c)$

on objects $c \in C$, where $j$ is the (∞,1)-Yoneda embedding.

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

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

is an equivalence.

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

## 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 $(\infty,1)$-sheaf or more specifically a hypercomplete $(\infty,1)$-sheaf. This is a distinction that only appears in (∞,1)-topos theory, not in (n,1)-topos theory for finite $n$.

h-level 2 | 0-truncated | discrete space | 0-groupoid/set | sheaf | h-set h-level 3 | 1-truncated | homotopy 1-type | 1-groupoid/groupoid | (2,1)-sheaf/stack | h-groupoid h-level 4 | 2-truncated | homotopy 2-type | 2-groupoid | | h-2-groupoid h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | | h-3-groupoid h-level $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | | h-$n$-groupoid | h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid

## References

Section 6.2.2 in

Revised on September 26, 2013 20:49:29 by Urs Schreiber (158.109.1.23)