Contents

Idea

The notion of $(\mathrm{\infty },1)$-sheaf is the analog in (∞,1)-category theory of the notion of sheaf 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 ${\mathrm{PSh}}_{(\mathrm{\infty },1)}(C)$ that correspond to covering (∞,1)-sieve?s

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

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

is an equivalence.

This is the analog of the ordinary sheaf condition. The ∞-groupoid ${\mathrm{PSh}}_C(U,A)$ is also called the descent-∞-groupoid of $A$ relative to the covering encoded by $U$.

Terminology

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

The practice of writing “$\mathrm{\infty }$-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 $\mathrm{\infty }$-stack: it can be an $(\mathrm{\infty },1)$-sheaf or more specifically a hypercomplete $(\mathrm{\infty },1)$-sheaf. This is a distinction that only appears in (∞,1)-topos theory, not in (n,1)-topos theory for finite $n$.

Related concepts

• sheaf

• 2-sheaf / stack

• $(\mathrm{\infty },1)$-sheaf / ∞-stack

• (∞,2)-sheaf

• (∞,n)-sheaf

References

Section 6.2.2 in

• Jacob Lurie, Higher Topos Theory