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\begin{document}

%-------------------------------------------------------------------


\section*{looping}

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

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher category theory - contents]]

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{ForPlainGroupsDeloopingToGroupoids}{For plain groups delooping to groupoids}\dotfill \pageref*{ForPlainGroupsDeloopingToGroupoids} \linebreak
\noindent\hyperlink{for_topological_spaces_and_groupoids}{For topological spaces and $\infty$-groupoids}\dotfill \pageref*{for_topological_spaces_and_groupoids} \linebreak
\noindent\hyperlink{for_parameterized_groupoids_stacks__sheaves}{For parameterized $\infty$-groupoids ($\infty$-stacks / $(\infty,1)$-sheaves)}\dotfill \pageref*{for_parameterized_groupoids_stacks__sheaves} \linebreak
\noindent\hyperlink{for_cohesive_groupoids}{For cohesive $\infty$-groupoids}\dotfill \pageref*{for_cohesive_groupoids} \linebreak
\noindent\hyperlink{for_categories}{For $(\infty,n)$-categories}\dotfill \pageref*{for_categories} \linebreak
\noindent\hyperlink{relation_to_looping_and_suspension}{Relation to looping and suspension}\dotfill \pageref*{relation_to_looping_and_suspension} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

For $X$ any kind of [[space]] (or possibly a [[directed space]], viewed as some sort of [[category]] or higher dimensional analogue of one), its [[loop space object]]s $\Omega_x X$ canonically inherit a [[monoidal category|monoidal structure]], coming from concatenation of [[loops]].

If $x \in X$ is essentially unique, then $\Omega_x X$ equipped with this monoidal structure remembers all of the structure of $X$: we say $X \simeq B \Omega_x X$ call $B A$ the \emph{[[delooping]]} of the monoidal object $A$.

What all these terms (``loops'' $\Omega$, ``delooping'' $B$ etc.) mean in detail and how they are \emph{presented} concretely depends on the given setup. We discuss some of these below in the section \hyperlink{Examples}{Examples}.

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

\hypertarget{ForPlainGroupsDeloopingToGroupoids}{}\subsubsection*{{For plain groups delooping to groupoids}}\label{ForPlainGroupsDeloopingToGroupoids}

Write

\begin{itemize}%
\item [[Grpd]] for the [[(2,1)-category]] of [[groupoids]] ([[objects]] are [[groupoids]], [[1-morphisms]] are [[functors]] between these and [[2-morphisms]] are [[natural transformations]] between those, which are necessarily [[natural isomorphisms]]),


\item [[Grp]] for the [[1-category]] of [[groups]] ([[discrete groups]]), also regarded as a [[(2,1)-category]];


\item $Grpd^{\ast/}$ for the $(2,1)$-category of [[pointed objects]] in [[Grpd]],


\item $Grpd_{\geq 1} \hookrightarrow Grpd$ for the [[full sub-(infinity,1)-category|full sub-(2,1)-category]] of [[connected object in an (infinity,1)-topos|connected]] groupoids, those for which $\pi_0 \simeq \ast$;


\item $Grp^{\ast/}_{\geq 1}$ for the [[pointed objects]] in connected groupoids.


\item $\pi_1(X,x) \in Grp$ for the [[fundamental group]] of a pointed groupoid $(\ast \stackrel{x}{\to} X) \in Grpd^{\ast/}$ at the given basepoint.


\item $\mathbf{B}G \in Grpd$, given a group $G$, for the groupoid $(G\stackrel{\longrightarrow}{\longrightarrow} \ast)$, with composition given by the product in the group. There are two possible choices of conventions, we agree that

\begin{displaymath}
\itexarray{
    && \ast
    \\
    & {}^{\mathllap{g_1}}\nearrow && \searrow^{\mathrlap{g_2}}
    \\
    \ast && \underset{g_1 \cdot g_2}{\longrightarrow} && \ast
  }
  \,.
\end{displaymath}


\end{itemize}
\begin{prop}
\label{SkeletalRepresentativesForConnectedGroupoids}\hypertarget{SkeletalRepresentativesForConnectedGroupoids}{}
The [[(2,1)-category]] $Grp_{\geq 1}$ of connected groupoids is equivalent to its [[full sub-(infinity,1)-category|full sub-(2,1)-category]] on those objects of the form $\mathbf{B}G$, for $G$ a group.

\end{prop}
\begin{proof}
Given a connected groupoid $X$, pick any basepoint $x\in X$ and consider the canonical inclusion $\mathbf{B}\pi_1(X,x) \longrightarrow X$. By construction this is [[fully faithful functor|fully faithful]] and by assumption of connectedness it is [[essentially surjective functor|essentially surjective]], hence it is an [[equivalence of groupoids]].

\end{proof}
\begin{prop}
\label{HomsBetweenBGs}\hypertarget{HomsBetweenBGs}{}
The [[hom-groupoids]] between connected groupoids with fundamental groups $G$ and $H$, respectively, are equivalent to the [[action groupoids]] of the set of group [[homomorphisms]] $G \to H$ [[action|acted]] on by [[conjugation]] with elements of $H$:

\begin{displaymath}
Grpd(\mathbf{B}G, \mathbf{B}H)
  \simeq
   Grp(G,H)//_{ad}H
\end{displaymath}
Given two group homomorphisms $\phi_1, \phi_2 \colon G \longrightarrow H$ then an [[isomorphism]] between them in this hom-groupoid is an element $h \in H$ such that

\begin{displaymath}
\phi_2 = Ad_h \circ \phi_1 \coloneqq h^{-1}\cdot \phi_1(-) \cdot h
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By direct inspection of the naturality square for the [[natural transformations]] which are the morphisms in $Grpd(\mathbf{B}G, \mathbf{B}H)$:

\begin{displaymath}
\itexarray{
    \ast && && \ast &\stackrel{h}{\longrightarrow}& \ast
    \\
    \downarrow^{\mathrlap{g_1}} && && \downarrow^{\mathrlap{\phi_1(g_1)}} && 
    \downarrow^{\mathrlap{\phi_2(g_1)}}
    \\
    \ast && && \ast &\stackrel{h}{\longrightarrow}& \ast
    \\
    \downarrow^{\mathrlap{g_2}}
    && &&
    \downarrow^{\mathrlap{\phi_1(g_2)}}
    &&
    \downarrow^{\mathrlap{\phi_2(g_2)}}
    \\
    \ast && && \ast &\stackrel{h}{\longrightarrow}& \ast
  }
  \,.
\end{displaymath}
\end{proof}
\begin{remark}
\label{piAs2Functor}\hypertarget{piAs2Functor}{}
The operation of forming $\pi_1$ is equivalently the operation of forming the [[homotopy fiber product]] of the point inclusion with itself, and hence extends to a [[(2,1)-functor]]

\begin{displaymath}
\pi_1 \colon Grpd^{\ast/} \longrightarrow Grp
  \,.
\end{displaymath}
\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
Restricted to [[connected object in an (infinity,1)-topos|connected groupoids]] among the pointed groupoids, the functor $\pi_1 \colon Grpd^{\ast/}_{\geq 1} \longrightarrow Grp$ of remark \ref{piAs2Functor} is an [[equivalence of (2,1)-categories]].

\end{prop}
\begin{proof}
It is clear that the functor is essentially surjective: for $G$ any [[group]] then $\pi_1(\mathbf{B}G,\ast) \simeq G$.

The more interesting point to notice is that $\pi_1$ is indeed a fully faithful [[(2,1)-functor]], in that for any $(X,x), (Y,y) \in Grpd^{\ast/}_{\geq 1}$ then the functor

\begin{displaymath}
(\pi_1)_{X,Y}
  \colon
  Grpd^{\ast/}((X,y),(Y,y))
  \longrightarrow
  Grp(\pi_1(X,x), \pi_1(Y,y))
\end{displaymath}
is an [[equivalence of groupoids|equivalence]] of [[hom-groupoids]]. By prop. \ref{SkeletalRepresentativesForConnectedGroupoids} it is sufficient to check this for $X = \mathbf{B}G$ and $Y = \mathbf{B}H$ with their canonical basepoints, hence to check that for any two groups $G,H$ the functor

\begin{displaymath}
(\pi_1)_{X,Y}
  \;\colon\;
  Grpd^{\ast/}((\mathbf{B}G,\ast),(\mathbf{B}H,\ast))
  \longrightarrow
  Grp(G,H)
\end{displaymath}
is an equivalence.

To see this, observe that, by definition of [[pointed objects]] via the [[undercategory]] under the point, a morphism in $Grpd^{\ast/}$ between groupoids of this form $\mathbf{B}(-)$ is a diagram in $Grp$ (unpointed) of the form

\begin{displaymath}
\itexarray{
    && \ast
    \\
    & \swarrow &\swArrow_{h}& \searrow
    \\
    \mathbf{B}G
    &&
    \underset{\mathbf{B}\phi}{\longrightarrow}
    &&
    \mathbf{B}H
  }
\end{displaymath}
where the [[natural isomorphism]] is equivalently just the choice of an element $h \in H$. Hence these morphisms are pairs $(\phi,h)$ of a group homomorphism and an element of the domain.

We claim that the [[(2,1)-functor]] $\pi_1$ takes such $(\phi,h)$ to the homomorphism $Ad_{h^{-1}} \circ \phi \;\colon\; G \longrightarrow H$. To see this, consider via remark \ref{piAs2Functor} this functor as forming loops:

\begin{displaymath}
\pi_1(\mathbf{B}G,\ast)
  =
  \left\{
    \itexarray{
      && \ast
      \\
      & \swarrow && \searrow
      \\
      \ast && \swArrow_{\mathrlap{g}} && \ast
      \\
      & \searrow && \swarrow
      \\
      && \mathbf{B}G
    }
  \right\}_{g\in G}
  \,.
\end{displaymath}
This shows that on a morphism as above this acts by forming the [[pasting]]

\begin{displaymath}
\itexarray{
    && \ast
    \\
    & \swarrow && \searrow
    \\
    \ast && \swArrow_{\mathrlap{g}} && \ast
    \\
    & \searrow && \swarrow &\swArrow_{\mathrlap{h}}& \searrow
    \\
    && \mathbf{B}G && \underset{\phi}{\longrightarrow} && \mathbf{B}H
  }
  \;\;\;\;
  =
  \:\;\;\;
  \itexarray{
    && && \ast
    \\
    && & \swarrow && \searrow
    \\
    && \ast && \swArrow_{\mathrlap{h\phi(g)h^{-1}}} && \ast
    \\
    & \swarrow & \swArrow_{\mathrlap{h}} & \searrow && \swarrow 
    \\
    \mathbf{B}G && \underset{\phi}{\longrightarrow} && \mathbf{B}H
  }
  \,.
\end{displaymath}
Unwinding the [[whiskering]] of [[natural transformations]] here, the claim follows, as indicated by the label of the upper 2-morphisms on the right.

One observes now that these extra labels $h$ are precisely the information that ``trivializes'' the conjugation action which in prop. \ref{HomsBetweenBGs} prevents the bare set of group homomorphism: a [[2-morphism]] $(\phi_1, h_1) \Rightarrow (\phi_2,h_2)$ in $Grp^{\ast/}$ is a natural isomorphism of groupoids

\begin{displaymath}
\itexarray{
    \mathbf{B}G
    &\stackrel{\phi_1}{\longrightarrow}& 
    \mathbf{B}H
    \\
    {}^{\mathllap{id}}\downarrow &\Downarrow^{\mathrlap{h}}& \downarrow^{\mathrlap{id}}
    \\
    \mathbf{B}G
    &\underset{\phi_2}{\longrightarrow}& 
    \mathbf{B}H    
  }
\end{displaymath}
(encoding a conjugation relation $\phi_2 = Ad_{h} \circ \phi_1$ as above) such that we have the [[pasting]] relation

\begin{displaymath}
\itexarray{
    && \ast
    \\
    & \swarrow &\swArrow_{h_1}& \searrow
    \\
    \mathbf{B}G
    &&
    \stackrel{\phi_1}{\longrightarrow}
    &&
    \mathbf{B}H
    \\
    {}^{\mathllap{id}}\downarrow && \Downarrow^{\mathrlap{h}} && \downarrow^{\mathrlap{id}}
    \\
    \mathbf{B}G
    &&\underset{\phi_2}{\longrightarrow} && 
    \mathbf{B}H    
  }
  \;\;\;\;\;
  =
  \;\;\;\;\;  
  \itexarray{
    && \ast
    \\
    & \swarrow &\swArrow_{h_2}& \searrow
    \\
    \mathbf{B}G
    &&
    \underset{\phi_2}{\longrightarrow}
    &&
    \mathbf{B}H
  }
  \,.
\end{displaymath}
But this says in components that $h_2 = h_1\cdot h$. Hence there is a \emph{at most one} morphism in $Grpd^{\ast/}((\mathbf{B}G,\ast),(\mathbf{B}H,\ast))$ from $(\phi_1,h_1)$ to $(\phi_2,h_2)$: it exists if $\phi_2 = Ad_h \circ \phi_1$ and $h_2 = h_1\cdot h$.

But since, by the previous argument, the functor $\pi_1$ takes $(\phi_1,h_1)$ to $Ad_{h_1^{-1}} \circ \phi_1$, this means that such a morphism exists precisely if both $(\phi_1,h_1)$ and $(\phi_2,h_2)$ are taken to the same group homomorphism by $\pi_1$

\begin{displaymath}
Ad_{h_2^{-1}} \circ \phi_2
  =
  Ad_{h^{-1}\cdot h_1^{-1}}\circ \phi_2
  = 
  Ad_{h_1^{-1}} \circ \phi_1
  \,.
\end{displaymath}
This establishes that $\pi_1$ is alspo an equivalence on all [[hom-groupoids]].

\end{proof}
This proof also shows that $\mathbf{B}(-)$ is in fact the inverse equivalence:

\begin{cor}
\label{}\hypertarget{}{}
There is an [[equivalence of (2,1)-categories]] between pointed connected groupoids and plain groups

\begin{displaymath}
Grp
  \stackrel{\underoverset{\simeq}{\pi_1 = \Omega_\ast}{\longleftarrow}}{\underset{\mathbf{B}}{\longrightarrow}}
  Grpd^{\ast/}_{\geq 1}
\end{displaymath}
given by forming [[loop space objects]] and by forming deloopings.

\end{cor}
\hypertarget{for_topological_spaces_and_groupoids}{}\subsubsection*{{For topological spaces and $\infty$-groupoids}}\label{for_topological_spaces_and_groupoids}

There is an [[equivalence of (∞,1)-categories]]

\begin{displaymath}
\infty Grpd^{\ast/}_{\geq 1} 
    \stackrel{\overset{B}{\leftarrow}}{\underset{\Omega}{\to}}
  \infty Group
\end{displaymath}
between [[pointed object|pointed]] [[connected]] [[∞-groupoid]]s and [[∞-group]]s, where $\Omega$ forms [[loop space object]]s.

This is [[presentable (∞,1)-category|presented]] by a [[Quillen equivalence]] of [[model categories]]

\begin{displaymath}
sSet_* 
    \stackrel{\overset{\bar W}{\leftarrow}}{\underset{G}{\to}}
  sGrp
\end{displaymath}
between the [[model structure on reduced simplicial sets]] and the [[transferred model structure]] on [[simplicial group]]s along the [[forgetful functor]] to the [[model structure on simplicial sets]].

(See [[groupoid object in an (infinity,1)-category]] for more details on this Quillen equivalence.)

\hypertarget{for_parameterized_groupoids_stacks__sheaves}{}\subsubsection*{{For parameterized $\infty$-groupoids ($\infty$-stacks / $(\infty,1)$-sheaves)}}\label{for_parameterized_groupoids_stacks__sheaves}

The following result makes precise for \emph{parameterized [[∞-groupoid]]s} -- for [[∞-stack]]s -- the general statement that $k$-fold [[delooping]] provides a correspondence between [[n-category|n-categories]] that have trivial [[k-morphism|r-morphism]]s for $r \lt k$ and [[k-tuply monoidal n-category|k-tuply monoidal n-categories]].

\begin{defn}
\label{}\hypertarget{}{}
An [[Ek-algebra]] object $A$ in an [[(∞,1)-topos]] $\mathbf{H}$ is called \textbf{groupal} if its [[homotopy groups in an (∞,1)-topos|connected components]] $\pi_0(A) \in \mathbf{H}_{\leq 0}$ is a [[group object]].

Write $Mon^{gp}_{\mathbb{E}[k]}(\mathbf{H})$ for the [[(∞,1)-category]] of groupal $E_k$-algebra objects in $\mathbf{H}$.

A groupal $E_1$-algebra -- hence an groupal [[A-∞ algebra]] object in $\mathbf{H}$ -- we call an \emph{[[∞-group]]} in $\mathbf{H}$. Write $\infty Grp(\mathbf{H})$ for the [[(∞,1)-category]] of [[∞-group]]s in $\mathbf{H}$.

\end{defn}
\begin{theorem}
\label{LoopingDeloopingForHigherStacks}\hypertarget{LoopingDeloopingForHigherStacks}{}
Let $k \gt 0$, let $\mathbf{H}$ be an [[(∞,1)-category of (∞,1)-sheaves]] and let $\mathbf{H}_*^{\geq k}$ denote the [[full subcategory]] of the category $\mathbf{H}_{*}$ of [[pointed objects]], spanned by those pointed objects thar are $k-1$-[[connected]] (i.e. their first $k$ [[homotopy groups in an (∞,1)-topos|homotopy sheaves]]) vanish. Then there is a canonical equivalence of [[(∞,1)-category|(∞,1)-categories]]

\begin{displaymath}
\mathbf{H}^{\ast/}_{\geq k} \simeq Mon^{gp}_{\mathbb{E}[k]}(\mathbf{H}) \,.
\end{displaymath}
between the pointed $(k-1)$-[[connected]] objects and the groupal [[Ek-algebra]] objects in $\mathbf{H}$.

\end{theorem}
This is (\hyperlink{LurieAlgebra}{Lurie, Higher Algebra, theorem 5.1.3.6}).

Specifically for $\mathbf{H} =$ [[Top]], this reduces to the classical theorem by [[Peter May]]

\begin{theorem}
\label{}\hypertarget{}{}
Let $Y$ be a [[topological space]] equipped with an action of the [[little cubes operad]] $\mathcal{C}_k$ and suppose that $Y$ is grouplike. Then $Y$ is homotopy equivalent to a $k$-fold loop space $\Omega^k X$ for some pointed topological space $X$.

\end{theorem}
This is [[Ek-Algebras|EkAlg, theorem 1.3.16.]]

\begin{remark}
\label{LoopingDeloopingDegree1InTopos}\hypertarget{LoopingDeloopingDegree1InTopos}{}
For $k = 1$ we have a looping/delooping equivalence

\begin{displaymath}
Grp(\mathbf{H})
  \stackrel{\overset{\Omega}{\longleftarrow}}{\underset{\mathbf{B}}{\longrightarrow}}
  \mathbf{H}_{\geq 1}^{\ast /}
\end{displaymath}
between pointed connected objects in $\mathbf{H}$ and grouplike [[A-∞ algebra]] objects in $\mathbf{H}$: [[∞-group]] objects in $\mathbf{H}$.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
If the ambient [[(∞,1)-topos]] has [[homotopy dimension]] 0 then every connected object $E$ admits a point $* \to E$. Still, the [[(∞,1)-category]] of pointed connected objects differs from that of unpointed connected objects (because in the latter the [[natural transformation]]s may have nontrivial components on the point, while in the former case they may not).

The connected objects $E$ which fail to be [[∞-group]]s by failing to admit a point are of interest: these are the \emph{[[∞-gerbes]]} in the [[(∞,1)-topos]].

\end{remark}
\hypertarget{for_cohesive_groupoids}{}\subsubsection*{{For cohesive $\infty$-groupoids}}\label{for_cohesive_groupoids}

A special case of the parameterized $\infty$-groupoids above are [[cohesive ∞-groupoid]]s. Looping and delooping for these is discussed at [[cohesive (∞,1)-topos -- structures]] in the section .

\hypertarget{for_categories}{}\subsubsection*{{For $(\infty,n)$-categories}}\label{for_categories}

See [[delooping hypothesis]].

\hypertarget{relation_to_looping_and_suspension}{}\subsection*{{Relation to looping and suspension}}\label{relation_to_looping_and_suspension}

For $A$ any monoidal [[space]], we may [[forgetful functor|forget]] its monoidal structure and just remember the underlying [[space]]. The formation of [[loop space object]]s composed with this [[forgetful functor]] has a [[left adjoint]] $\Sigma$ which forms \emph{[[suspension objects]]}.

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

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

\begin{itemize}%
\item [[delooping]], \textbf{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}
[[!include k-monoidal table]]

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

Section 6.1.2 of

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

\end{itemize}
Section 5.1.3 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Algebra]]}

\end{itemize}
[[!redirects looping]] [[!redirects looping and delooping]]



\end{document}