(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
Locality and descent
The notion of -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.
Given an (∞,1)-site , let be the class of monomorphisms in the (∞,1)-category of (∞,1)-presheaves that correspond to covering (∞,1)-sieve?s
on objects , where is the (∞,1)-Yoneda embedding.
Then an (∞,1)-presheaf is an -sheaf if it is an -local object. That is, if for all such the morphism
is an equivalence.
This is the analog of the ordinary sheaf condition. The ∞-groupoid is also called the descent-∞-groupoid of relative to the covering encoded by .
An (,1)-sheaf is also called an ∞-stack with values in ∞-groupoids.
The practice of writing “-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 -stack: it can be an -sheaf or more specifically a hypercomplete -sheaf. This is a distinction that only appears in (∞,1)-topos theory, not in (n,1)-topos theory for finite .
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 | -truncated | homotopy n-type | n-groupoid | | h--groupoid | h-level | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h--groupoid
Section 6.2.2 in