thin homotopy

**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**

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.

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 here…)

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.

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.

Revised on October 27, 2016 13:10:05
by Anonymous Coward
(24.130.57.254)