nLab
path infinity-groupoid

Idea

The path ∞-groupoid $\Pi(X)$ of a generalized smooth space $X$ is a geometric version of the fundamental ∞-groupoid of $X$: its 1-morphisms form the geometric space of paths in $X$, and so forth. Its truncations to lower categorical degree yield

• path groupoids

• path n-groupoids.

For details see at shape via cohesive path ∞-groupoid.