Contents

# Contents

## Idea

Generally, a topological interval is a (bounded) interval in the real line (an open interval $(a,b)$ or a closed interval $[a,b]$ or a half-open interval $(a,b]$ or $[a,b)$) equipped with the subspace topology of the Euclidean metric topology.

Specifically in the context of topological homotopy theory, the standard topological interval object is the closed interval $[0,1]$ equipped with the continuous functions

1. $const_0 \;\colon\; \ast \to [0,1]$

2. $const_1 \;\colon\; \ast \to [0,1]$

which include the point space as the two endpoints, respectively.

Together with the unique function $[0,1] \to \ast$ this yields the factorization of the codiagonal on the point space

$\nabla_{\ast} \;\colon\; \ast \sqcup \ast \overset{(const_0,const_1)}{\longrightarrow} [0,1] \overset{\exists!}{\longrightarrow} \ast$

which exhibits an example of an interval object in the general sense of model category theory with respect to the classical model structure on topological spaces.

## Induced constructions

### Topological cylinder

For $X$ a topological space, then the product topological space $X \times [0,1]$ with the topological interval is the standard topological cylinder over $X$. Via the above inclusion functions, this inherits a factorization of the codiagonal of $X$ (the canonical continuous function out of the disjoint union space of $X$ with itself to $X$):

$\nabla_X \;\colon\; X \sqcup X \overset{(id_X \times const_0, id_X \times const_1)}{\longrightarrow} X \times [0,1] \overet{pr_1}{\longrightarrow} X \,.$

Accordingly, with respect to the classical model structure on topological spaces this is an example a of cylinder object.

### Left homotopy

Given $X,Y$ two topological spaces and $f,g \;\colon\; X \longrightarrow Y$ two continuous functions,then a left homotopy

$\eta \colon f \,\Rightarrow_L\, g$
$\eta \;\colon\; X \times I \longrightarrow Y$

out of the product topological space of $X$ with the topological interval, such that this fits into a commuting diagram of the form $\array{ X \\ {}^{\mathllap{(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{f}} \\ X \times I &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{g}} \\ X } \,.$

(graphics grabbed from J. Tauber here)

### Path space

For $X$ a topological space, then its path space is the mapping space $X^{[0,1]}$, out of the topological interval into $X$, i.e. the set of continuous function $\gamma \;\colon\; [0,1] \to X$ equipped with the compact-open topology.

The two endpoint inclusions $\ast \colon [0,1]$ and the unique projection $[0,1] \to \ast$ induce continuous functions

$X \overset{}{\longrightarrow} X^{[0,1]} \overset{X^{(const_0,const_1)}}{\longrightarrow} X \times X$

(inclusion of constant paths and endpoint evaluation of paths).

## Properties

### Freyd’s characterization

The topological interval $[0, 1]$ may be characterized by a coalgebraic definition first identified by Freyd:

Let $Top_{\ast, \ast}$ be the category of topological spaces $X$ equipped with a pair $x_0, x_1$ of distinct points, for example $I = ([0, 1]; 0, 1)$. Let $F \colon Top_{\ast, \ast} \to Top_{\ast, \ast}$ be the functor defined on objects by

$F(X; x_0, x_1) = (X \vee X, y_0, y_1) \,,$

where $X \vee X$ denotes the quotient space of the disjoint union space $X \sqcup X$ formed by identifying the element $x_1$ of the first copy of $X$ with $x_0$ of the second copy of $X$, where $y_0$ is identified with $x_0$ of the first copy, and $y_1$ is identified with $x_1$ of the second copy.

For $I = ([0, 1]; 0, 1)$ there is an evident identification $F(I) = ([0, 2]; 0, 2)$, and moreover there is an $F$-coalgebra structure $I \to F(I)$ given by “multiplication by $2$”.

###### Theorem

(Freyd)

The topological interval $I = ([0, 1], 0, 1)$ is the terminal F-coalgebra.

For more information, see coalgebra of the real interval, which shows in particular how the interval structure of $[0, 1]$ may be defined by coinduction.

Last revised on July 2, 2017 at 11:54:08. See the history of this page for a list of all contributions to it.