nLab Lie n-groupoid

Idea

The geometric infinity-stacks within all smooth infinity-groupoids are called Lie $\infty$-groupoids. These are equivalently those smooth $\infty$-groupoids which have a presentation by Kan-fibrant simplicial manifolds. See there for more.

In the language of groupoids, the Kan conditions (see below) correspond to composing and inverting various morphisms. For example, the existence of a composition for arrows is given by the condition $\Kan(2,1)$, whereas the composition of an arrow with the inverse of another is given by $\Kan(2,0)$ and $\Kan(2,2)$:

Here we recall that a simplicial set $X$ is Kan if any map from the horn $\Lambda[m,j]$ to $X$ ($m\ge 1$, $j=0,\dots,m$), extends to a map from $\Delta[m]$. Let us call $\Kan(m,j)$ the Kan condition for the horn $\Lambda[m,j]$. A Kan simplicial set is therefore a simplicial set satisfying $\Kan(m,j)$ for all $m\ge 1$ and $0\leq j\leq m$.

Note that the composition of two arrows is in general not unique, but any two of them can be joined by a $2$-morphism $h$ given by $\Kan(3,2)$.

The associativity can be given by $\Kan(2, 1)!$ and $Kan(3, 2)$ illustrated below:

In an $n$-groupoid, the only well-defined composition law is the one for $n$-morphisms. This motivates the following definition.

Definition

An $n$-groupoid ($n\in\N \cup \infty$) $X$ is a simplicial set that satisfies $\Kan(m,j)$ for all $0\leq j\leq m \geq 1$ and $\Kan!(m,j)$ for all $0 \leq j \leq m$ $\geq n+1$, where

$\Kan(m,j)$: Any map $\Lambda[m,j]\to X$ extends to a map $\Delta[m]\to X$.

$\Kan!(m,j)$: Any map $\Lambda[m,j]\to X$ extends to a unique map $\Delta[m]\to X$.

Related $n$Lab entries

• Lie differentiation