nLab
(infinity,1)-sheaf

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. For a presheaf $A : C^{\op} \to E$ with values in an arbitrary ∞-category, we say it is a sheaf iff $E(e, A(-))$ is a sheaf for every object $e$ of $E$.

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

As in the 1-categorial case, the sheaf condition for a covering sieve can be translated into a condition on a covering family that generates it:

Proposition

Let $\{ u_i \to c \}$ be a family of morphisms of $C$ that generate the sieve corresponding to $\eta : U \hookrightarrow j(c)$, and let $r_\bullet : \mathbf{\Delta}^{\op} \to PSh_C$ be the Čech nerve of $\amalg_i j(u_i) \to j(c)$. Then a presheaf $A$ is local with respect to $\eta$ iff the induced map $A(c) \to \lim A(r_\bullet)$ is an equivalence.

Thus, a presheaf $A$ is a sheaf iff every covering sieve contains a generating family satisfying this condition. Spelling out the description of the Čech nerve, the condition is that we have

$$A(c) \simeq \lim\left( \prod_i A(u_i) \stackrel{\to}{\to} \prod_{i,j} PSh_C(j(u_i) \times_{j(c)} j(u_j), A) \stackrel{\to}{\stackrel{\to}{\to}} \cdots \right)$$

If $C$ has pullbacks, this simplifies to

$$A(c) \simeq \lim\left( \prod_i A(u_i) \stackrel{\to}{\to} \prod_{i,j} A(u_i \times_c u_j) \stackrel{\to}{\stackrel{\to}{\to}} \cdots \right)$$

and furthermore this formulation applies to presheaves with values in an arbitrary ∞-category.

Proof

Taking colimits of Čech nerve computes $(-1)$-truncations in $(PSh_C)/j(X)$, so $\colim(r_\bullet)$ is the subobject of $j(c)$ corresponding to the sieve $\eta$. We have

$$PSh_C(\colim(r_\bullet), A) \simeq \lim PSh_C(r_\bullet, A)$$

and so the theorem follows.

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 $n$.

Related concepts

• sheaf

• 2-sheaf / stack

• $(\infty,1)$-sheaf / ∞-stack

  • sheaf of spectra

  • sheaf of L-∞ algebras

• (∞,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	contractible-if-inhabited	(-1)-groupoid/​truth value	(0,1)-sheaf/​ideal	mere proposition/​h-proposition
h-level 2	0-truncated	homotopy 0-type	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	(3,1)-sheaf/​2-stack	h-2-groupoid
h-level 5	3-truncated	homotopy 3-type	3-groupoid	(4,1)-sheaf/​3-stack	h-3-groupoid
h-level $n+2$	$n$-truncated	homotopy n-type	n-groupoid	(n+1,1)-sheaf/​n-stack	h-$n$-groupoid
h-level $\infty$	untruncated	homotopy type	∞-groupoid	(∞,1)-sheaf/​∞-stack	h-$\infty$-groupoid

References

Section 6.2.2 in

• Jacob Lurie, Higher Topos Theory