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\begin{document}

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\section*{delooping}

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

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

[[!include quasi-category theory contents]]

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - 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{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{characterization_of_deloopable_objects}{Characterization of deloopable objects}\dotfill \pageref*{characterization_of_deloopable_objects} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{topological_loop_spaces}{Topological loop spaces}\dotfill \pageref*{topological_loop_spaces} \linebreak
\noindent\hyperlink{delooping_of_a_group_to_a_groupoid}{Delooping of a group to a groupoid}\dotfill \pageref*{delooping_of_a_group_to_a_groupoid} \linebreak
\noindent\hyperlink{deloopings_of_higher_categorical_structures}{Deloopings of higher categorical structures}\dotfill \pageref*{deloopings_of_higher_categorical_structures} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

The \emph{delooping} of an object $A$ is, if it exists, a uniquely [[pointed object]] $\mathbf{B} A$ such that $A$ is the [[loop space object]] of $\mathbf{B} A$:

\begin{displaymath}
A \simeq \Omega(\mathbf{B} A)
\end{displaymath}
In particular, if $A = G$ is a [[group]] then its delooping

\begin{itemize}%
\item in the context [[Top]] is the [[classifying space]] $\mathcal{B}G$


\item in the context [[∞-Grpd]] is the one-object groupoid $\mathbf{B}G$.



\end{itemize}
Under the [[homotopy hypothesis]] these two objects are identified: the [[geometric realization]] of the groupoid $\mathbf{B}G$ is the classifying space $\mathcal{B}G$:

\begin{displaymath}
|\mathbf{B}G| \simeq \mathcal{B}G
  \,.
\end{displaymath}
See [[looping and delooping]].

\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}
Conversely, if $A$ is given and a [[homotopy pullback]] diagram

\begin{displaymath}
\itexarray{
    A &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    {*} &\to& \mathbf{B}A
  }
\end{displaymath}
exists, with the point ${*} \to \mathbf{B} A$ being essentially unique, by the above $A$ has been realized as the [[loop space object]] of $\mathbf{B} A$

\begin{displaymath}
A = \Omega \mathbf{B} A
\end{displaymath}
and we say that $\mathbf{B} A$ is the \textbf{delooping} of $A$.

See the section  at [[groupoid object in an (∞,1)-category]] for more.

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

If $\mathbf{C}$ is even a [[stable (∞,1)-category]] then all deloopings exist and are then also denoted $\Sigma A$ and called the \textbf{[[suspension]]} of $A$.

\hypertarget{characterization_of_deloopable_objects}{}\subsection*{{Characterization of deloopable objects}}\label{characterization_of_deloopable_objects}

In section 6.1.3 of

\begin{itemize}%
\item [[Jacob Lurie]], [[Higher Topos Theory]]

\end{itemize}
a definition of [[groupoid object in an (infinity,1)-category]] $\mathbf{C}$ is given as a homotopy simplicial object, i.e. a [[(infinity,1)-functor]]

\begin{displaymath}
C  : \Delta^{op} \to \mathbf{C}
\end{displaymath}
\begin{displaymath}
\cdots
  C_2 \stackrel{\to}\rightrightarrows
  C_1 
  \rightrightarrows 
 C_0
\end{displaymath}
satisfying certain conditions (prop. 6.1.2.6) which are such that if $C_0 = {*}$ is the [[point]] we have an internal \emph{group} in a homotopical sense, given by an object $C_1$ equipped with a coherently associative multiplication operation $C_1 \times C_1 \to C_1$ generalizing that of [[Jim Stasheff|Stasheff]] [[H-space]] from the $(\infty,1)$-category [[Top]] to arbitrary $(\infty,1)$-categories.

Lurie calls the groupoid object $C$ an \emph{effective} [[groupoid object in an (infinity,1)-category]] precisely if it arises as the delooping, in the above sense, of some object $\mathbf{B}C$.

One of the characterizing properties of an [[(infinity,1)-topos]] is that \emph{every} groupoid object in it is effective.

This is the analog of [[Jim Stasheff|Stasheff]]`s classical result about [[H-spaces]].

See the remark at the very end of section 6.1.2 in [[Higher Topos Theory|HTT]].

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

\hypertarget{topological_loop_spaces}{}\subsubsection*{{Topological loop spaces}}\label{topological_loop_spaces}

For $C =$ [[Top]] the [[(infinity,1)-category]] of [[topological spaces]], a space is deloopable if it is an [[A-infinity-space]] and hence homotopy equivalent to a [[loop space]].

\hypertarget{delooping_of_a_group_to_a_groupoid}{}\subsubsection*{{Delooping of a group to a groupoid}}\label{delooping_of_a_group_to_a_groupoid}

Let $G$ be a [[group]] regarded as a [[discrete category|discrete groupoid]] in the [[(∞,1)-topos]] [[∞Grpd]] of [[∞-groupoids]].

Then $\mathbf{B} G$ exists and is, up to equivalence, the [[groupoid]]

\begin{itemize}%
\item with a single object $\bullet$,


\item with $Hom_{\mathbf{B} G}(\bullet, \bullet) = G$, or equivalently $Aut_{\mathbf{B}G}(\bullet) = G$,


\item and with composition of morphisms in $\mathbf{B} G$ being given by the product operation in the group.



\end{itemize}
More informally but more suggestively we may write

\begin{displaymath}
\mathbf{B} G = \{ \bullet \stackrel{g}{\to} \bullet | g \in G\}
\end{displaymath}
or

\begin{displaymath}
\mathbf{B}G = \{ 
     \bullet \righttoleftarrow g \;|\; g \in G 
  \}
\end{displaymath}
to emphasize that there is really only a single object.

Notice how the [[homotopy pullback]] works in this simple case:

the universal 2-cell $\eta$

\begin{displaymath}
\itexarray{
    G &\to& {*}
    \\
    \downarrow &\Downarrow^{\eta}& \downarrow
    \\
    {*} &\to& \mathbf{B}G
  }
\end{displaymath}
filling this [[2-limit]] diagram is the [[natural transformation]] from the constant [[functor]]

\begin{displaymath}
G \to {*} \to \mathbf{B}G
\end{displaymath}
to itself, whose component map

\begin{displaymath}
\eta : Obj(G) \to Mor(\mathbf{B}G)
\end{displaymath}
is just the identity map, using that $Obj(G) = G$ and $Mor(\mathbf{B}G) = G$.

\hypertarget{deloopings_of_higher_categorical_structures}{}\subsection*{{Deloopings of higher categorical structures}}\label{deloopings_of_higher_categorical_structures}

There is also a notion of delooping which takes a pointed $(n, k+1)$-category $C$ to a pointed $(n+1, k)$-category $\mathbf{B} C$ in which $\mathbf{B} C$ has a single $0$-cell $\bullet$, and where $\hom(\bullet, \bullet) = C$. This is a tautological construction if one accepts the [[delooping hypothesis]], which views a $(n, k+1)$-category $C$ as a special type of $(n+k+1)$-category, namely a pointed $k$-connected $(n+k+1)$-category: by viewing such as \emph{a fortiori} a pointed $(k-1)$-connected $(n+k+1)$-category, we get the delooping $\mathbf{B} C$.

This is just a generalization of the fact that a [[monoid]] $M$ gives rise to a one-object category (which we are denoting $\mathbf{B} M$). For an important example: a [[monoidal category]] $M$ has an associated \textbf{delooping [[bicategory]]} $\mathbf{B} M$, where

\begin{itemize}%
\item $\mathbf{B} M$ has a single $0$-cell $\bullet$,


\item the $1$-cells $\bullet \to \bullet$ of $\mathbf{B} M$ are named by objects of $M$, and the composite of $\bullet \stackrel{a}{\to} \bullet \stackrel{b}{\to} \bullet$ is $\bullet \stackrel{a \otimes b}{\to} \bullet$ (using the monoidal product $\otimes$ of $M$),


\item the $2$-cells of $\mathbf{B} M$ are similarly named by morphisms of $M$; the vertical composition of $2$-cells in $\mathbf{B} M$ is given by composition of morphisms of $M$, and the horizontal composition of $2$-cells in $\mathbf{B} M$ is given by taking the monoidal product of the morphisms that name them in $M$.



\end{itemize}
Along similar lines, the delooping of a [[braided monoidal category]] produces a [[monoidal bicategory]], and delooping of \emph{that} is a [[tricategory]] or (weak) $3$-category. See [[delooping hypothesis]] for more.

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

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

\begin{itemize}%
\item \textbf{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 deloopings]] [[!redirects delooped group]]



\end{document}