(infinity,1)-sheaf (changes) in nLab

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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:U↪j(c) \eta : U \hookrightarrow j(c)

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

Then an (∞,1)-presheaf A∈PSh (∞,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. For a presheaf A:C op→EA : C^{\op} \to E with values in an arbitrary ∞-category, we say it is a sheaf iff E(e,A(−))E(e, A(-)) is a sheaf for every object ee of EE.

This is the analog of the ordinary sheaf condition for covering sieves. 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.

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:

Thus, a presheaf AA 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)≃lim(∏ iA(u i)→→∏ i,jPSh C(j(u i)× j(c)j(u j),A)→→→⋯) 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 CC has pullbacks, this simplifies to

A(c)≃lim(∏ iA(u i)→→∏ i,jA(u i× cu j)→→→⋯) 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.

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.

• sheaf

• 2-sheaf / stack

• (∞,1)(\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+2n+2	nn-truncated	homotopy n-type	n-groupoid	(n+1,1)-sheaf/​n-stack	h-nn-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