Paths and cylinders
A thin homotopy between paths in a topological space (with the standard interval) is a homotopy which, roughly speaking, has zero area.
Thin homotopies in smooth spaces
A (smooth) homotopy between smooth paths in a smooth space is called thin if the rank of its differential is less than 2 for all .
Thin homotopies in topological spaces
The following is taken from
We define a finite tree to be a one-dimensional finite polyhedron.
A homotopy between paths and in the topological space is called thin if factors through a finite tree,
such that the paths , are piecewise-linear.
When is Hausdorff, points, paths and thin homotopies in form a bigroupoid.
The definition of path groupoids and path n-groupoids as strict or semi-strict n-groupoids typically involves taking morphisms to be thin homotopy classes of paths. See there for more details.
The parallel transport of a connection on a bundle is an assignment of fiber-homomorphisms to paths in a manifold that is invariant under thin homotopy.