Paths

# Paths

## Definitions

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

A path from $a$ to $b$ is a path $f$ such that $f(0) = a$ and $f(1) = b$.
An unparametrised path is an equivalence class of paths, such that $f$ and $g$ are equivalent if there is an increasing automorphism $\phi$ of $\mathbb{I}$ such that $g = f \circ \phi$. An unoriented path is an equivalence class of paths such that $f$ is equivalent to $(x \mapsto f(1 - x))$.

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

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

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 $[0,n]$ as the linear graph with $n + 1$ vertices and $n$ edges; in this way, the other variations become meaningful. (However, as the only directed graph automorphism of $[0,n]$ 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 $f$ from $a$ to $b$ and a Moore path $g$ from $b$ to $c$, the concatenation of $f$ and $g$ is a Moore path $f ; g$ or $g \circ f$ from $a$ to $c$. If the domain of $f$ is $[0,m]$ and the domain of $g$ is $[0,n]$, then the domain of $f ; g$ is $[0,m+n]$, and

$(f ; g)(x) \coloneqq \left \{ \array { f(x) & \quad x \leq m \\ g(m+x) & \quad x \geq m .} \right .$

In this way, we get a (strict) category whose objects are points in $X$ and whose morphisms are Moore paths in $X$, 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 $\mathbb{I}$ if we wish), then we get the unparametrised path category. If $X$ is a smooth space, then we may additionally identify paths related through a thin homotopy to get the path groupoid. Finally, if $X$ is a continuous space and we identify paths related through any (endpoint-preserving) homotopy, then we get the fundamental groupoid of $X$.

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

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

1. Cf. e.g. Introduction to Topology – 2, or also Section 2.1; beware that that reference, (0) like many others, uses the term “inverse path”, even though the operation of concatenation of paths does not in and of itself yield a strict groupoid, in which $\overline{P}$ would be an inverse, and (1) that it uses $a$ and $b$ for the endpoints of the interval, not the endpoints of the paths in the space $X$, and (2) that it uses $P^-$ instead of $\overline{P}$, which however is less suited for notational iterating (compare $\overline{\overline{P}}=P$ with $(P^-)^-=P$), and that (3) the 2008 edition has a typo: “ $w(1-t)$ ” in loc. cit., when inverse path gets defined, should be $u(1-t)$.