nLab Lie n-groupoid




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)\Kan(2,1), whereas the composition of an arrow with the inverse of another is given by Kan(2,0)\Kan(2,0) and Kan(2,2)\Kan(2,2):

Here we recall that a simplicial set XX is Kan if any map from the horn Λ[m,j]\Lambda[m,j] to XX (m1m\ge 1, j=0,,mj=0,\dots,m), extends to a map from Δ[m]\Delta[m]. Let us call Kan(m,j)\Kan(m,j) the Kan condition for the horn Λ[m,j]\Lambda[m,j]. A Kan simplicial set is therefore a simplicial set satisfying Kan(m,j)\Kan(m,j) for all m1m\ge 1 and 0jm0\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 22-morphism hh given by Kan(3,2)\Kan(3,2).

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

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


An nn-groupoid (nNn\in\N \cup \infty) XX is a simplicial set that satisfies Kan(m,j)\Kan(m,j) for all 0jm10\leq j\leq m \geq 1 and Kan!(m,j)\Kan!(m,j) for all 0jm0 \leq j \leq m n+1\geq n+1, where

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

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

Last revised on April 21, 2024 at 13:16:03. See the history of this page for a list of all contributions to it.