nLab
thin homotopy
Contents
Context
Homotopy theory
homotopy theory , (∞,1)-category theory , homotopy type theory

flavors: stable , equivariant , rational , p-adic , proper , geometric , cohesive , directed …

models: topological , simplicial , localic , …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents
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

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 groupoid s and path n-groupoid s 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.

Last revised on September 9, 2018 at 10:22:12.
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