nLab path n-groupoid

Idea

A path $n$-groupoid $P_n(X)$ of a smooth space (or generalized smooth space) $X$ is a diffeological n-groupoid which is

• a generalization of the path groupoid to higher categorical dimension

• a truncation of sorts of an path ∞-groupoid of $X$.

Its j-morphisms are given by (possibly equivalence classes of) $j$-dimensional smooth paths in $X$, i.e. usually smooth maps $\gamma : D^j \to X$. Composition is by gluing of such maps.

Path 1-groupoid

See path groupoid.

Path 2-groupoid

Definitions of path 2-groupoids as strict 2-groupoids internal to diffeological spaces appear (at least) in

• John Baez, Urs Schreiber, Higher gauge theory (arXiv)

• Urs Schreiber, Konrad Waldorf, Smooth functors vs. differential forms (arXiv)

• João Faria Martins, Roger Picken, On 2-dimensional holonomy (arXiv)

For the underlying notion of fundamental 2-groupoid see there.

Path 3-groupoid

A realization of the path 3-groupoid as a Gray-groupoid internal to diffeological spaces appears in

• João Faria Martins, Roger Picken, The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module (arXiv)