nLab topological interval

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

$$\eta \colon f \,\Rightarrow_L\, g$$

is a continuous function

$$\eta \;\colon\; X \times I \longrightarrow Y$$

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.