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\begin{document}

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\section*{loop}

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

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

[[!include topology - contents]]

\hypertarget{loops}{}\section*{{Loops}}\label{loops}

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


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

In [[topology]], a (parametrised, oriented) \textbf{loop} in a [[space]] $X$ is a map (a [[morphism]] in an appropriate [[category]] of spaces, such as a [[continuous function]] between [[topological spaces]]) to $X$ from the [[circle]] $S^1 = \mathbb{R}/\mathbb{Z}$. Hence a continuous [[path]] whose endpoints coincide.

A \textbf{loop at $a$} is a loop $f$ such that $f(k) = a$ for any (hence every) [[integer]] $k$. An \textbf{unparametrised loop} is an [[equivalence class]] of loops, such that $f$ and $g$ are equivalent if there is an [[monotone function|increasing]] [[automorphism]] $\phi$ of $S^1$ such that $g = f \circ \phi$. An \textbf{unoriented loop} is an equivalence class of loops such that $f$ is equivalent to $(x \mapsto f(-x))$. A \textbf{Moore loop} has domain $\mathbb{R}/n \mathbb{Z}$ for some [[natural number]] (or possibly any [[real number]]) $n$. All of these variations can be combined, of course. (A Moore loop at $a$ has $f(k n) = a$ instead of $f(k) = a$. Also, a Moore loop for $n = 0$ is simply a [[point]], so possibly there is a better way to define this to avoid making this exception. Finally, there is not much difference between unparametrised loops and unparametrised Moore loops, since we may interpret $(t \mapsto n t)$ as a reparametrisation $\phi$.)

In [[graph theory]], a \textbf{loop} is an edge whose endpoints are the same vertex. Actually, this is a special case of the above, if we interpret $S^1$ as the graph with $1$ vertex and $1$ edge; in this way, the other variations become meaningful. In this context, a Moore loop is called a \textbf{cycle}. (However, as the only directed graph automorphism of $S^1$ is the [[identity morphism|identity]], parametrisation is trivial for directed graphs and equivalent to orientation for undirected graphs.)

Every loop may be interpreted as a [[path]]. Sometimes a loop, say at $a$, is \emph{defined} to be a path from $a$ to $a$. However, this is correct only in certain contexts. In graph theory, it's incorrect, but only because of terminological conventions; the idea is sound. In [[continuous spaces]], it is also correct. However, in [[smooth spaces]], it is not correct, since the [[derivatives]] at the endpoints should also agree; the same holds in many other more structured contexts.

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

Given two Moore loops $f$ and $g$ at $a$, the \textbf{concatenation} of $f$ and $g$ is a Moore loop $f ; g$ or $g \circ f$ at $a$. If the domain of $f$ is $\mathbb{R}/m \mathbb{Z}$ and the domain of $g$ is $\mathbb{R}/n \mathbb{Z}$, then the domain of $f ; g$ is $\mathbb{R}/(m+n) \mathbb{Z}$, 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 [[monoid]] of Moore loops in $X$ at $a$, with concatenation as multiplication. This monoid may called the \textbf{[[Moore loop monoid]]}.

Often we are more interested in a [[quotient]] monoid of the Moore loop monoid. If we use unparametrised loops (in which case we may use loops with domain $S^1$ if we wish), then we get the \textbf{unparametrised [[loop monoid]]}. If $X$ is a [[smooth space]], then we may additionally identify loops related through a [[thin homotopy]] to get the \textbf{[[loop group]]}. Finally, if $X$ is a [[continuous space]] and we identify loops related through any (basepoint-preserving) [[homotopy]], then we get the \textbf{[[fundamental group]]} of $X$.

See [[looping]] and [[delooping]] for more.

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

\begin{itemize}%
\item [[path]],
\item [[loop space]],
\item [[fundamental group]].

\end{itemize}
[[!redirects loop]] [[!redirects loops]]

[[!redirects Moore loop]] [[!redirects Moore loops]]



\end{document}