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\begin{document}

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\section*{loop space object}

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

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory}

[[!include quasi-category theory contents]]

\hypertarget{stabe_homotopy_theory}{}\paragraph*{{Stabe homotopy theory}}\label{stabe_homotopy_theory}

[[!include stable homotopy theory - contents]]

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

[[!include mapping space - contents]]

\hypertarget{loop_space_objects}{}\section*{{Loop space objects}}\label{loop_space_objects}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{explicit_constructions}{Explicit constructions}\dotfill \pageref*{explicit_constructions} \linebreak
\noindent\hyperlink{free_loop_space_objects}{Free loop space objects}\dotfill \pageref*{free_loop_space_objects} \linebreak
\noindent\hyperlink{based_loop_space_objects}{Based loop space objects}\dotfill \pageref*{based_loop_space_objects} \linebreak
\noindent\hyperlink{remarks_2}{Remarks}\dotfill \pageref*{remarks_2} \linebreak
\noindent\hyperlink{remarks_3}{Remarks}\dotfill \pageref*{remarks_3} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

In the [[(∞,1)-topos]] [[Top]] the construction of a [[loop space]] of a given [[topological space]] is familiar.

This construction may be generalized to any other [[(∞,1)-topos]] and in fact to any other [[(∞,1)-category]] with [[homotopy pullbacks]].

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

Loop space objects are defined in any [[(∞,1)-category]] $\mathbf{C}$ with [[homotopy pullbacks]]: for $X$ any [[pointed object]] of $\mathbf{C}$ with point ${*} \to X$, its [[loop space object]] is [[generalized the|the]] [[homotopy pullback]] $\Omega X$ of this point along itself:

\begin{displaymath}
\itexarray{
    \Omega X &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    {*} &\to& X
  }
  \,.
\end{displaymath}
A ([[generalised element|generalised]]) [[global point|element]] of $\Omega X$ may be thought of as a [[loop]] in $X$ at the [[base point]] $*$.

\hypertarget{remarks}{}\subsubsection*{{Remarks}}\label{remarks}

Since $\mathbf{C}(X,-)$ commutes with homotopy limits, one has a natural homotopy equivalence $\Omega\mathbf{C}(X,Y)\simeq \mathbf{C}(X,\Omega Y)$, for any two objects $X$ and $Y$ in $\mathbf{C}$.

See also

\begin{itemize}%
\item [[fibration sequence]]


\item [[delooping]]


\item [[stable (∞,1)-category]]



\end{itemize}
\hypertarget{explicit_constructions}{}\subsection*{{Explicit constructions}}\label{explicit_constructions}

Usually the [[(∞,1)-category]] in question is [[presentable (infinity,1)-category|presented]] by concrete 1-categorical data, such as that of a [[model category]]. In that case the above [[homotopy pullback]] has various realizations as an ordinary [[pullback]].

Notably it may be expressed using [[path objects]] which may come from [[interval objects]]. Even if the context is not (or not manifestly) that of a [[homotopical category]], an [[interval object]] may still exist and may be used as indicated in the following to construct loop space objects.

\hypertarget{free_loop_space_objects}{}\subsubsection*{{Free loop space objects}}\label{free_loop_space_objects}

In a category with [[interval object]] the \textbf{free loop space object} is the part of the [[path object]] $B^I = [I,B]$ which consists of closed paths, namely the [[pullback]]

\begin{displaymath}
\itexarray{
    \Lambda B &\to& [I,B]
    \\
    \downarrow && \downarrow^{\mathrlap{d_0 \times d_1}}
    \\
    B &\stackrel{Id \times Id}{\to}& B \times B
  }
  \,.
\end{displaymath}
This is the same as the image of the [[co-span co-trace]] $cotr(I)$ of the interval object (which is the interval object closed to a loop!, see the examples at [[co-span co-trace]]) in $B$:

\begin{displaymath}
\left[
  \itexarray{
     && cotr(I)
     \\
     & \nearrow && \nwarrow
     \\
     pt &&&& I
     \\
     & {}_{Id \sqcup Id}\nwarrow && \nearrow_{in \sqcup out}
     \\
     && pt \sqcup pt
  }
  \;\;\;\;,\;\;\;\;
  B
  \right]
  \;,\;\;\;\;
  \simeq
  \;,\;\;\;\;
  \itexarray{
     && \Lambda B
     \\
     & \swarrow && \searrow
     \\
     B &&&& [I,B]
     \\
     & {}_{Id \times Id}\searrow && \swarrow_{d_0 \times d_1}
     \\
     && B \times B
  }
\end{displaymath}
\hypertarget{based_loop_space_objects}{}\subsubsection*{{Based loop space objects}}\label{based_loop_space_objects}

If $B$ is a [[pointed object]] with point $pt \stackrel{pt_B}{\to} B$ then the \textbf{based loop space object} of $B$ is the pullback $\Omega_{pt} B$ in

\begin{displaymath}
\itexarray{
    \Omega_{pt}B &\to& [I,B]
    \\
    \downarrow && \downarrow^{d_0 \times d_1}
    \\
    pt
    &\stackrel{pt_B \times pt_B}{\to}&
    B \times B
  }
  \,.
\end{displaymath}
\hypertarget{remarks_2}{}\paragraph*{{Remarks}}\label{remarks_2}

\begin{itemize}%
\item $\Omega_{pt}B$ is the fiber of the [[generalized universal bundle]] $\mathbf{E}_{pt}B \to B$.


\item the based loop space object $\Omega_{pt} B$ is the pullback of the free loop space object $\Lambda B$ to the point

\begin{displaymath}
\itexarray{
  \Omega_{pt} B &\to& \Lambda B
  \\
  \downarrow && \downarrow
  \\
  pt &\stackrel{pt_B}{\to}& B 
}
\,.
\end{displaymath}


\end{itemize}
\hypertarget{remarks_3}{}\subsection*{{Remarks}}\label{remarks_3}

\begin{itemize}%
\item The loop space object $B$ can be regarded as the homotopy trace on the identity span on $B$, as described at [[span trace]].


\item The free loop space object inherits the structure of an $A_\infty$-[[A-infinity-category|category]] from that of the [[path object]] $[I,B]$.


\item In a [[generalized smooth space|suitable extension]] of $\operatorname{Diff}$, this construction does \textbf{not} give the usual \emph{smooth} loop space (free or based). It gives the space of paths with coincident endpoints rather than the space of smooth maps from the circle. Thus the [[smooth loop space]] is not a loop space object.



\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item Let $C =$ [[Top]] with the standard [[interval object]]. Then for $B= X$ a topological space $\Lambda B = \Lambda X$ is the ordinary free [[loop space]] of $X$.

The generalization of this to \emph{smooth} spaces is discussed at [[smooth loop space]].


\item Let $C =$ [[Grpd]] with the standard interval object $I = \{a \stackrel{\simeq}{\to} b\}$ and let $\mathbf{B}G$ be the one-object groupoid corresponding to a group $G$, then

\begin{displaymath}
\Lambda \mathbf{B}G = G//_{Ad}G
\end{displaymath}
is the [[action groupoid]] of $G$ acting on itself by its adjoint action. Notice the example at [[co-span co-trace]] which says that the cotrace on $I$ is $cotr(I) = \mathbf{B}\mathbb{Z}$, and indeed

\begin{displaymath}
\Lambda \mathbf{B}G =  [\mathbf{B}\mathbb{Z}, \mathbf{B}G]  
\,.
\end{displaymath}
The role of this $\Lambda \mathbf{B}G$ as a loop object is amplified in particular in

\begin{itemize}%
\item Simon Willerton, \emph{The twisted Drinfeld double of a finite group via gerbes and finite groupoids} (\href{http://arxiv.org/abs/math.QA/0503266}{arXiv})


\item Bruce Bartlett, \emph{On unitary 2-representations of finite groups and topological quantum field theory} (\href{http://arxiv.org/abs/0901.3975}{arXiv})



\end{itemize}

\item On the other hand, the \emph{based} loop object of $\mathbf{B}G$ is just $G$:

\begin{displaymath}
\Omega \mathbf{B}G = G
\,.
\end{displaymath}


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

\begin{itemize}%
\item \textbf{loop space object}, [[free loop space object]],

\begin{itemize}%
\item [[delooping]], [[looping and delooping]]


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



\end{itemize}

\item [[suspension object]]

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

\end{itemize}


\end{itemize}
[[!redirects loop space object]] [[!redirects loop space objects]] [[!redirects loop object]] [[!redirects loop objects]] [[!redirects loop space functor]] [[!redirects loop space functors]]



\end{document}