A path -groupoid of a smooth space (or generalized smooth space) is
a generalization of the path groupoid to higher categorical dimension
a truncation of sorts of an path infinity-groupoid of .
Its j-morphisms are given by (possibly equivalence classes of) -dimensional smooth paths in , i.e. usually smooth maps . Composition is by gluing of such maps.
Definitions of path 2-groupoids as strict 2-groupoids internal to diffeological spaces appear (at least) in
Baez, Schreiber, Higher gauge theory (arXiv)
Schreiber, Waldorf, Smooth functors vs. differential forms (arXiv)
J. F. Martins, R. Picken, On 2-dimensional holonomy (arXiv)
A realization of the path 3-groupoid as a Gray-groupoid internal to diffeological spaces appears in