homotopy theory, (∞,1)-category theory, homotopy type theory
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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.
The $n$-groupoids form an (n+1,1)-category, nGrpd.
A general model for ∞-groupoids is provided by Kan complexes (the fibrant objects in the classical model structure on simplicial sets which presents ∞Grpd).
In this context an $n$-groupoid in the general sense is modeled by a Kan complex all of whose homotopy groups vanish in degree $k \gt n$. In this generality one also speaks of a homotopy $n$-type or an $n$-truncated object of ∞Grpd.
Every such $n$-type is equivalent to a “small” model, an $(n+1)$-coskeletal Kan complex (see there): one in which every $k$-sphere $\partial \Delta^{k+1}$ for $k \geq n+1$ has a unique filler.
Yet a bit smaller than this are Kan complexes that is an $n$-hypergroupoid, where in addition to these sphere-fillers also the horn fillers in degree $n+1$ are unique.
(For 1-groupoids/1-hypergroupoids this situation is further spelled out here).
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 $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | (n+1,1)-sheaf/n-stack | h-$n$-groupoid |
h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid |
Discussion of homotopy $n$-types/$n$-truncated objects in homotopy type theory:
Univalent Foundations Project, §7 of: Homotopy Type Theory – Univalent Foundations of Mathematics (2013) [web, pdf]
Nicolai Kraus, Truncation levels in Homotopy Type Theory, Nottingham (2015) [pdf, eprints:28986]
Felix Cherubini, Egbert Rijke, Thm. 3.10 of: Modal Descent, Mathematical Structures in Computer Science , 31 4 (2021) 363-391 [doi:10.1017/S0960129520000201, arXiv:2003.09713]
Last revised on February 14, 2023 at 14:03:55. See the history of this page for a list of all contributions to it.