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\begin{document}

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\section*{path}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{paths}{}\section*{{Paths}}\label{paths}

\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{concatenation}{Concatenation}\dotfill \pageref*{concatenation} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak


\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

In [[topology]], a (parametrised, oriented) \textbf{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 \textbf{path from $a$ to $b$} is a path $f$ such that $f(0) = a$ and $f(1) = b$.\newline An \textbf{unparametrised path} is an [[equivalence class]] of paths, such that $f$ and $g$ are equivalent if there is an [[monotone function|increasing]] [[automorphism]] $\phi$ of $\mathbb{I}$ such that $g = f \circ \phi$. An \textbf{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 \textbf{reverse path}\footnote{Cf. e.g. [[Introduction to Topology -- 2]], or also \hyperlink{tomDieck2008}{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 \emph{in and of itself} yield a [[strict category|strict groupoid]], in which $\overline{P}$ would be an inverse, and (1) that it uses $a$ and $b$ for the endpoints of the \emph{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 \emph{inverse path} gets defined, should be $u(1-t)$.} , denoted $\overline{P}$, is defined to be the composite $P \circ ( t\mapsto 1-t )$. The operation $P\mapsto\overline{P}$ is called \emph{path reversal}.

A \textbf{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 \textbf{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 morphism|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.)

\hypertarget{concatenation}{}\subsection*{{Concatenation}}\label{concatenation}

Given a Moore path $f$ from $a$ to $b$ and a Moore path $g$ from $b$ to $c$, the \textbf{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

\begin{displaymath}
(f ; g)(x) \coloneqq \left \{ \array { f(x) & \quad x \leq m \\ g(m+x) & \quad x \geq m .} \right .
\end{displaymath}
In this way, we get a ([[strict category|strict]]) [[category]] whose [[objects]] are [[global element|points]] in $X$ and whose [[morphisms]] are Moore paths in $X$, with concatenation as [[composition]]. This category is called the \textbf{[[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 \textbf{unparametrised [[path category]]}. If $X$ is a [[smooth space]], then we may additionally identify paths related through a [[thin homotopy]] to get the \textbf{[[path groupoid]]}. Finally, if $X$ is a [[continuous space]] and we identify paths related through any (endpoint-preserving) [[homotopy]], then we get the \textbf{[[fundamental groupoid]]} of $X$.

In graph theory, the Moore path category is known as the \textbf{[[free category]]} on the graph.

\hypertarget{references}{}\subsection*{{References}}\label{references}

 [[Tammo tom Dieck]], \emph{Algebraic Topology}, European Mathematical Society, 2008

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[path-connected topological space]]


\item [[locally path-connected topological space]]


\item [[loop]]


\item [[path space]],


\item [[fundamental group]], [[fundamental groupoid]]


\item [[fundamental theorem of covering spaces]]



\end{itemize}
[[!redirects path]] [[!redirects paths]]

[[!redirects Moore path]] [[!redirects Moore paths]]

[[!redirects path concatenation]] [[!redirects path concatenations]]

[[!redirects concatenation of paths]] [[!redirects concatenrations of paths]]

[[!redirects reverse path]] [[!redirects reverse paths]]

[[!redirects path reversal]] [[!redirects path reversion]] [[!redirects path reversals]] [[!redirects path reversions]]

[[!redirects constant path]] [[!redirects constant paths]]



\end{document}