nLab thin homotopy

Idea

A thin homotopy between paths $f,g: I \to X$ in a topological space $X$ (with $I = [0,1]$ the standard interval) is a homotopy $I\times I \to X$ which, roughly speaking, has zero area.

Definition

Thin homotopies in smooth spaces

A (smooth) homotopy $F: I \times I \to X$ between smooth paths in a smooth space $X$ is called thin if the rank of its differential $d F(s,t): T_{s,t} I \times I \to T_{F(s),F(t)} X$ is less than 2 for all $s,t \in I \times I$.

(More to go here…)

Thin homotopies in topological spaces

The following is taken from

• K. A. Hardie, K. H. Kamps, R.W. Kieboom, A homotopy 2-groupoid of a Hausdorff space, Appl. Cat. Str.8 (2000).

We define a finite tree to be a one-dimensional finite polyhedron.

A homotopy $F:I\times I \to X$ between paths $F(-,0)$ and $F(-,1)$ in the topological space $X$ is called thin if $F$ factors through a finite tree,

$$I\times I \stackrel{F_0}{\to} T \stackrel{F_1}{\to} X$$

such that the paths $F_0(-,0):I\to T$, $F_0(-,1):I\to T$ are piecewise-linear.

When $X$ is Hausdorff, points, paths and thin homotopies in $X$ form a bigroupoid.

Applications

Path n-groupoids

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.

Parallel transport

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.