(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
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 .
-sheaf / ∞-stack
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 | -truncated | homotopy n-type | n-groupoid | (n+1,1)-sheaf/n-stack | h--groupoid |
h-level | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h--groupoid |
Section 6.2.2 in
Last revised on February 20, 2015 at 02:32:50. See the history of this page for a list of all contributions to it.