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\section*{loop space}

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

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

[[!include topology - contents]]

\hypertarget{mapping_space}{}\paragraph*{{Mapping space}}\label{mapping_space}

[[!include mapping space - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{AInfinityStructure}{Homotopy-associative structure}\dotfill \pageref*{AInfinityStructure} \linebreak
\noindent\hyperlink{local_homotopy_properties}{Local homotopy properties}\dotfill \pageref*{local_homotopy_properties} \linebreak
\noindent\hyperlink{models}{Models}\dotfill \pageref*{models} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{ReferencesExamples}{Examples}\dotfill \pageref*{ReferencesExamples} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In the strict sense of the word a \emph{loop space} in [[topology]] for a given [[pointed topological space]] $X$ is the [[mapping space]] (with its [[compact-open topology]], see the example \href{compact-open+topology#LoopSpace}{there}) $Maps_\ast(S^1, X)$ of [[continuous functions]] from the [[circle]] to $X$, such that they take the given basepoint of the circle to the prescribed basepoint in $X$ (or if one drops this condition, then one speaks of the \emph{[[free loop space]]}). One such continuous function may be thought of as a continuous loop in $X$, and hence the mapping space is the space of all these loops.

If here $X$ is equipped with further structure, such as [[smooth structure]] (e.g. a [[smooth manifold]]), then one may in good cases find such extra structure also on the loop space, for instance to form a \emph{[[smooth loop space]]}, etc. See at \emph{[[manifolds of mapping spaces]]} for more on this.

If one regards this construction not in [[point-set topology]] but in [[classical model structure on topological spaces|classical homotopy theory of topological spaces]] ([[homotopy hypothesis|equivalently]] [[∞Grpd]]), then, up to [[weak homotopy equivalence]], the loop space is equivalent to the [[homotopy fiber product]] of the basepoint inclusion $\ast \to X$ along itself.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let [[Top]] be a [[nice category of spaces|nice category of topological spaces]], in particular one which is [[complete category|complete]], [[cocomplete category|cocomplete]], and [[cartesian closed category|cartesian closed]]. Let $(S^1, pt)$ be the [[circle]], i.e., 1-dimensional [[sphere]], with chosen basepoint, and let $(X, *)$ be a space with a chosen [[pointed set|basepoint]]. Then the \textbf{loop space} of $X$ (at $*$) is an [[internal hom]]

\begin{displaymath}
\Omega X = hom((S^1, pt), (X, *))
\end{displaymath}
in the category $Top_*$ of based spaces. Explicitly, it is given by the [[pullback]] in $Top$

\begin{displaymath}
\itexarray{
\Omega X & \to & 1\\ 
 \downarrow & & \downarrow *\\
X^{S^1} & \underset{X^{pt}}{\to} & X^1
}
\end{displaymath}
(using [[exponential object|exponentials]] to denote internal homs in $Top$), in other words the [[function set|function space]] of basepoint-preserving maps $S^1 \to X$, whose basepoint is the constant map $S^1 \to X$ at the basepoint of $X$.

The category $Top_*$ is [[symmetric monoidal category|symmetric]] monoidal [[monoidal closed category|closed]]; its monoidal product is called the \textbf{[[smash product]]}, often denoted $\wedge$. In particular, the loop space functor

\begin{displaymath}
\Omega = hom((S^1, pt), -): Top_* \to Top_*
\end{displaymath}
has a [[left adjoint]] obtained by taking smash product with $(S^1, pt)$. This left adjoint $S: Top_* \to Top_*$ is called the \textbf{[[suspension]] functor}. Explicitly, the suspension $S X$ is formed as the [[pushout]]

\begin{displaymath}
\itexarray{
 & 1 \times X + S^1 \times 1& \to & 1\\
(pt \times X, S^1 \times *) & \downarrow & & \downarrow \\
 & S^1 \times X & \to & S X
}
\end{displaymath}
with basepoint provided by the right vertical arrow.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{AInfinityStructure}{}\subsubsection*{{Homotopy-associative structure}}\label{AInfinityStructure}

A loop space is an example of a [[A-∞ space]], in particular it is an [[H-space]]. Loop spaces admit this rich algebraic structure which arises from the fact that the based space $S^1$ carries a correspondingly rich co-algebraic structure, starting from the fact that the based space $S^1$ is an H-[[cogroup]].

The description of this structure on loop spaces has been the very motivation for the introduction of the notion of [[operad]] and [[algebra over an operad]] in (\hyperlink{May}{May}).

An important theoretical consideration is when an H-space, and particularly one having the type of a [[CW-complex]], has the homotopy type of a loop space of another CW-complex: $X \simeq \Omega Y$. In this circumstance, one calls $Y$ a \textbf{[[delooping]]} of $X$; an important example is where $X$ carries a [[topological group]] structure $G$, and $Y$ is the [[classifying space]] of $G$.

The most basic fact about deloopings is the shift in [[homotopy group]]s:

\begin{itemize}%
\item $\pi_n(\Omega Y) \cong \pi_{n+1}(Y)$

\end{itemize}
which follows straight from the [[adjunction]] $S \dashv \Omega$ plus the fact that the suspension of $S^n$ is $S^{n+1}$. (This isomorphism needs to be developed at greater length.)

The modern study of the question ``when can an H-space be [[delooping|delooped]]?'' was inaugurated by [[Jim Stasheff]]. The basic theorem is as follows (all spaces assumed to be CW-complexes):

\begin{theorem}
\label{}\hypertarget{}{}
An [[H-space]] $X$ admits a delooping if and only if the [[monoid]] $\pi_0(X)$ induced from the H-space structure is a [[group]], and the H-space $X$ structure can be extended to a structure of [[algebra over an operad]] over [[Jim Stasheff|Stasheff]]`s [[A-∞ operad]] $K$.

\end{theorem}
This is due to (\hyperlink{Stasheff}{Stasheff}). The analogous statement holds true in every [[(∞,1)-topos]] other than [[Top]]. Details on this more general statement are at [[loop space object]] and at [[groupoid object in an (∞,1)-category]].

\hypertarget{local_homotopy_properties}{}\subsubsection*{{Local homotopy properties}}\label{local_homotopy_properties}

Let the space $X$ be [[locally connected space|locally 0-connected]] and [[semi-locally simply connected space|semi-locally 1-connected]] (i.e. it admits a [[universal covering space]]). The loop space $\Omega X$ for any basepoint is locally path connected, as is the free loop space $X^{S^1}$. If $X$ is locally 1-connected and admits a basis of open sets $U$ such that $\pi_2(U) \to \pi_2(X)$ is the zero map, then $\Omega X$ is locally 0-connected and semi-locally 1-connected, and so admits a universal covering space.

In general, if $X$ is locally $n$-[[n-connected space|connected]], $\Omega X$ is locally $(n-1)$-connected. This process can obviously be iterated up to $n$ times, so that $\Omega^n X$ is locally 0-connected. This can be weakened to locally $(n-1)$-connected and semi-locally $n$-connected: this is just like the $n=1$ case but replacing $\pi_1$ with $\pi_n$ (as was done in the previous paragraph with $\pi_2$). We will actually define a space to be semi-locally $n$-connected to include the condition that it is locally $(n-1)$-connected. This result was proved for more general mapping spaces $X^P$ and various subspaces when $X$ is Hausdorff and $P$ a finite [[polyhedron]] in (\hyperlink{Wada}{Wada}) but a much simpler and direct proof for general $X$ and $P = I$ or $P= S^1$ is possible.

\begin{utheorem}
The fundamental $n$-groupoid of a space $X$ ([[Trimble n-category|Trimblean]] for choice) can be topologised to be an internal $n$-groupoid in $\Top$ when $X$ is semi-locally $n$-connected. Furthermore, the homotopy groups of the $n$-groupoid, \emph{a priori} topological groups, are discrete.

\end{utheorem}
For $n=2$, this is in [[David Roberts]]'s [[Fundamental Bigroupoids and 2-Covering Spaces|thesis]]. For $n=1$, it has been known for ages and is in [[Ronnie Brown]]'s topology textbook.

\hypertarget{models}{}\subsection*{{Models}}\label{models}

There is a [[Quillen equivalence]]

\begin{displaymath}
(G \dashv \bar W) 
   \;\colon\; 
  sGrp
   \stackrel{\overset{\Omega}{\leftarrow}}{\underset{\bar W}{\to}}
  sSet_0
\end{displaymath}
between the [[model structure on simplicial groups]] and the [[model structure on reduced simplicial sets]], thus exhibiting both of these as models for [[infinity-groups]] (\hyperlink{Kan58}{Kan 58}). Its [[left adjoint]] $G$, the \emph{[[simplicial loop space]] construction}, is a concrete model for the loop space construction with values in [[simplicial groups]].

See also [[simplicial group]] and [[groupoid object in an (∞,1)-category]] for more details.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[path space]]


\item [[loop space object]], [[free loop space object]],

\begin{itemize}%
\item [[Sullivan model of loop space]]


\item [[delooping]], [[looping and delooping]]


\item \textbf{loop space}, [[free loop space]],


\item [[infinite loop space]]


\item [[formal loop space]]


\item [[derived loop space]]


\item [[smooth loop space]]



\end{itemize}

\item [[caloron correspondence]]


\item [[suspension object]]

\begin{itemize}%
\item [[suspension]]

\end{itemize}


\end{itemize}
[[!include k-monoidal table]]

\begin{itemize}%
\item [[A-infinity space]], [[model structure for dendroidal Cartesian fibrations]]

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item [[Daniel Kan]], \emph{On homotopy theory and c.s.s. groups}, Ann. of Math. 68 (1958), 38-53


\item [[Jim Stasheff]], \emph{Homotopy associative $H$-spaces I, II}, Trans. Amer. Math. Soc. 108, 1963, 275-312


\item [[Peter May]], \emph{The geometry of iterated loop spaces} Lecture Notes in Mathematics 271 (1970) (\href{http://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf}{pdf})


\item H. Wada, \emph{Local connectivity of mapping spaces}, Duke Mathematical Journal, vol ? (1955) pp 419-425



\end{itemize}
The simplicial loop group functor is discussed in chapter V, section 5 of

\begin{itemize}%
\item [[Paul Goerss]], [[Rick Jardine]], \emph{Simplicial homotopy theory} (\href{http://www.maths.abdn.ac.uk/~bensondj/html/archive/goerss-jardine.html}{web})

\end{itemize}
See also the references at [[looping and delooping]].

\hypertarget{ReferencesExamples}{}\subsubsection*{{Examples}}\label{ReferencesExamples}

On loop spaces of [[configuration spaces of points]]:

\begin{itemize}%
\item [[Edward Fadell]], [[Sufian Husseini]], \emph{The space of loops on configuration spaces and the Majer-Terracini index}, Topol. Methods Nonlinear Anal. Volume 11, Number 2 (1998), 249-271 (\href{https://projecteuclid.org/euclid.tmna/1476842829}{euclid:tmna/1476842829})


\item [[Fred Cohen]], [[Samuel Gitler]], \emph{Loop spaces of configuration spaces, braid-like groups, and knots}, In: Aguadé J., Broto C., [[Carles Casacuberta]] (eds.) \emph{Cohomological Methods in Homotopy Theory}. Progress in Mathematics, vol 196. Birkhäuser, Basel (\href{https://doi.org/10.1007/978-3-0348-8312-2_7}{doi:10.1007/978-3-0348-8312-2\_7})


\item [[Toshitake Kohno]], \emph{Loop spaces of configuration spaces and finite type invariants}, Geom. Topol. Monogr. 4 (2002) 143-160 (\href{https://arxiv.org/abs/math/0211056}{arXiv:math/0211056})


\item [[Fred Cohen]], [[Samuel Gitler]], \emph{On loop spaces of configuration spaces}, Trans. Amer. Math. Soc. \textbf{354} (2002), no. 5, 1705--1748, (\href{https://www.jstor.org/stable/2693715}{jstor:2693715}, \href{http://www.ams.org/mathscinet-getitem?mr=1881013}{MR2002m:55020})

(on [[ordinary homology]] of [[loop spaces]] of configuration spaces)



\end{itemize}
[[!redirects loop space]] [[!redirects loop spaces]]

[[!redirects based loop space]] [[!redirects based loop spaces]]



\end{document}