nLab path space

Idea

In topology the path space of a topological space $X$ is a topological space of all continuous paths in $X$.

In topological homotopy theory the path space construction serves to exhibit homotopies in the guise of right homotopies. This situation generalizes to many other model categories and one speaks more generally of path space objects in this case.

For exposition in the context of point-set topology see at Introduction to Topology – 1 around this example.

For exposition in the context of topological homotopy theory see at Introduction to Homotopy Theory around this definition.

Definition

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 functions $\gamma \;\colon\; [0,1] \to X$ 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

Relation to loop space

The fiber product of the projection with the diagonal on $X$ is the free loop space $\mathcal{L}X$ of $X$:

$$\array{ \mathcal{L}X &\longrightarrow& X^{[0,1]} \\ \downarrow &(pb)& \downarrow^{\mathrlap{X^{(const_0,const_1)}}} \\ X &\underset{\Delta_X}{\longrightarrow}& X \times X }$$

If $X$ is equipped with a choice of basepoint $x \colon \ast \to X$ (making it a pointed topological space), then the further fiber product with this basepoint inclusion is the based loop space $\Omega_x X$:

$$\array{ \Omega_x X &\longrightarrow& \mathcal{L}X &\longrightarrow& X^{[0,1]} \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow^{\mathrlap{X^{(const_0,const_1)}}} \\ \ast &\underset{x}{\longrightarrow}& X &\underset{\Delta_X}{\longrightarrow}& X \times X }$$

Related concepts

• loop space

• path groupoid;

• fundamental groupoid;

• topological complexity

• path space object

• identity type