path in nLab

Paths

Paths

Definitions

In topology, a (parametrised, oriented) path in a space XX is a map (a morphism in an appropriate category of spaces, such as a continuous function between topological space) to XX from the topological interval 𝕀=[0,1]\mathbb{I} = [0,1].

A path from aa to bb is a path ff such that f(0)=af(0) = a and f(1)=bf(1) = b.
An unparametrised path is an equivalence class of paths, such that ff and gg are equivalent if there is an increasing automorphism ϕ\phi of 𝕀\mathbb{I} such that g=f∘ϕg = f \circ \phi. An unoriented path is an equivalence class of paths such that ff is equivalent to (x↦f(1−x))(x \mapsto f(1 - x)).

If PP is a path, then its reverse path¹, denoted P¯\overline{P}, is defined to be the composite P∘(t↦1−t)P \circ ( t\mapsto 1-t ). The operation P↦P¯P\mapsto\overline{P} is called path reversal.

A Moore path is defined like a path, except for having another domain: replace [0,1][0,1] with the interval [0,n][0,n] for some natural number (or, more commonly, any non-negative real number) nn. All of these variations can be combined, of course. (For unoriented paths, one usually says ‘between aa and bb’ instead of ‘from aa to bb’. Also, a Moore path from aa to bb has f(n)=bf(n) = b instead of f(1)=bf(1) = b. Finally, there is not much difference between unparametrised paths and unparametrised Moore paths, since we may interpret (t↦nt)(t \mapsto n t) as a reparametrisation ϕ\phi.)

In graph theory, a path is a list of edges, each of which ends where the next begins. Actually, this is a special case of the above, if we use Moore paths and interpret [0,n][0,n] as the linear graph with n+1n + 1 vertices and nn edges; in this way, the other variations become meaningful. (However, as the only directed graph automorphism of [0,n][0,n] is the identity, parametrisation is trivial for directed graphs and equivalent to orientation for undirected graphs. Note that a non-Moore path is simply an edge, one of the fundamental ingredients of a graph.)

Concatenation

Given a Moore path ff from aa to bb and a Moore path gg from bb to cc, the concatenation of ff and gg is a Moore path f;gf ; g or g∘fg \circ f from aa to cc. If the domain of ff is [0,m][0,m] and the domain of gg is [0,n][0,n], then the domain of f;gf ; g is [0,m+n][0,m+n], and

(f;g)(x)≔{f(x) x≤m g(x−m) x≥m. (f ; g)(x) \coloneqq \left \{ \array { f(x) & \quad x \leq m \\ g(x-m) & \quad x \geq m .} \right .

In this way, we get a (strict) category whose objects are points in XX and whose morphisms are Moore paths in XX, with concatenation as composition. This category is called the Moore path category.

Often we are more interested in a quotient category of the Moore path category. If we use unparametrised paths (in which case we may use paths with domain 𝕀\mathbb{I} if we wish), then we get the unparametrised path category. If XX is a smooth space, then we may additionally identify paths related through a thin homotopy to get the path groupoid. Finally, if XX is a continuous space and we identify paths related through any (endpoint-preserving) homotopy, then we get the fundamental groupoid of XX.

In graph theory, the Moore path category is known as the free category on the graph.

• path-connected topological space

• locally path-connected topological space

• loop

• path space,

• fundamental group, fundamental groupoid

• fundamental theorem of covering spaces

• homotopy of smooth paths relative to their endpoints

References

E.g.

• Tammo tom Dieck, Algebraic Topology, European Mathematical Society, 2008

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