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(infinity,1)-sheaf

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

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

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

$$A(c)\simeq {\mathrm{Psh}}_C(j(c),A)\stackrel{{\mathrm{PSh}}_C(\eta ,,A)}{\to }\mathrm{PSh}(U,A)$$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 ${\mathrm{PSh}}_C(U,A)$ 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

homotopy level	n-truncation	homotopy theory	higher category theory	higher topos theory	homotopy type theory
h-level 0	(-2)-truncated	contractible space	(-2)-groupoid		true/unit type/contractible type
h-level 1	(-1)-truncated		(-1)-groupoid/truth value		h-proposition
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 $\mathrm{\infty }$	untruncated	homotopy type	∞-groupoid	(∞,1)-sheaf/∞-stack	h-$\mathrm{\infty }$-groupoid

References

Section 6.2.2 in

• Jacob Lurie, Higher Topos Theory