Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
A -sheaf is a sheaf with values in groupoids. This is traditionally called a stack.
Let be a (2,1)-site. Write Grpd for the (2,1)-category of groupoids, functors and natural isomorphisms.
A -sheaf on is equivalently
a 1-truncated (∞,1)-sheaf on .
The (2,1)-category of a -sheaves on a (2,1)-site forms a (2,1)-topos.
There are model category presentations of this -topos. See model structure for (2,1)-sheaves.
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 |
Last revised on April 25, 2013 at 22:00:22. See the history of this page for a list of all contributions to it.