nLab path space




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory

Mapping space

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



Paths and cylinders

Homotopy groups

Basic facts




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

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.


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

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

XX [0,1]X (const 0,const 1)X×X 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).


Relation to loop space

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

X X [0,1] (pb) X (const 0,const 1) X Δ X X×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 XX is equipped with a choice of basepoint x:*Xx \colon \ast \to X (making it a pointed topological space), then the further fiber product with this basepoint inclusion is the based loop space Ω xX\Omega_x X:

Ω xX X X [0,1] (pb) (pb) X (const 0,const 1) * x X Δ X 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 }

Last revised on February 14, 2024 at 07:51:09. See the history of this page for a list of all contributions to it.