An $n$-groupoid is an
n-category in which all k-morphisms are equivalences;
$n$-truncated ∞-groupoid (a homotopy n-type).
In terms of a known notion of (n,r)-category, we can define an $n$-groupoid explicitly as an $\infty$-category such that:
Or we define an $n$-groupoid abstractly as an n-truncated object in the (∞,1)-category ∞Grpd.
A general model for ∞-groupoids are Kan complexes. In this context an $n$-groupoid in the general sense is modeled by a Kan complex all whose homotopy groups vanish in degree $k \gt n$. In this generality one also speaks of a homotopy n-type.
Every such $n$-type is equivalent to a “small” model, an $(n+1)$-coskeletal Kan complex: one in which every $k$-sphere $\partial \Delta^{k+1}$ for $k \geq n+1$ has a unique filler.
Even a bit smaller than this is a Kan complex that is an $n$-hypergroupoid, where in addition to these spheres also the horn fillers in degree $n+1$ are unique.
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 | mere proposition, h-proposition | ||
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 $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | h-$n$-groupoid | |
h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid |