# nLab n-groupoid

### Context

#### Higher category theory

higher category theory

# Contents

## Idea

An $n$-groupoid is an

## Definitions

In terms of a known notion of (n,r)-category, we can define an $n$-groupoid explicitly as an $\infty$-category such that:

• every $j$-morphism (at any level) is an equivalence;
• every parallel pair of $j$-morphisms is equivalent, for $j > n$.

Or we define an $n$-groupoid abstractly as an n-truncated object in the (∞,1)-category ∞Grpd.

## Models

### As Kan complexes

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 leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/unit type/contractible type
h-level 1(-1)-truncated(-1)-groupoid/truth valuemere proposition, h-proposition
h-level 20-truncateddiscrete space0-groupoid/setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/groupoid(2,1)-sheaf/stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoidh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoidh-3-groupoid
h-level $n+2$$n$-truncatedhomotopy n-typen-groupoidh-$n$-groupoid
h-level $\infty$untruncatedhomotopy type∞-groupoid(∞,1)-sheaf/∞-stackh-$\infty$-groupoid

Revised on April 29, 2013 21:44:32 by Urs Schreiber (89.204.138.79)