2-natural transformation?
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
A $(2,1)$-sheaf is a sheaf with values in groupoids. This is traditionally called a stack.
Let $C$ be a (2,1)-site. Write Grpd for the (2,1)-category of groupoids, functors and natural isomorphisms.
A $(2,1)$-sheaf on $C$ is equivalently
a 1-truncated (∞,1)-sheaf on $C$.
The (2,1)-category of a $(2,1)$-sheaves on a (2,1)-site forms a (2,1)-topos.
There are model category presentations of this $(2,1)$-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 | (-1)-groupoid/truth value | (0,1)-sheaf | 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 | h-2-groupoid |
h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | (4,1)-sheaf | h-3-groupoid |
h-level $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | (n+1,1)-sheaf | h-$n$-groupoid |
h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid |