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\begin{document}

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\section*{groupoid object in an (infinity,1)-category}

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

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{internal_categories}{}\paragraph*{{Internal $(\infty,1)$-Categories}}\label{internal_categories}

[[!include internal infinity-categories contents]]

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

[[!include quasi-category theory contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition_complete_segalspace_style}{Definition (complete Segal-space style)}\dotfill \pageref*{definition_complete_segalspace_style} \linebreak
\noindent\hyperlink{groupoid_object}{Groupoid object}\dotfill \pageref*{groupoid_object} \linebreak
\noindent\hyperlink{group_object}{Group object}\dotfill \pageref*{group_object} \linebreak
\noindent\hyperlink{RelationToQuotients}{Relation to $(\infty,1)$-quotients}\dotfill \pageref*{RelationToQuotients} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{EquivalentCharacterizations}{Equivalent characterizations}\dotfill \pageref*{EquivalentCharacterizations} \linebreak
\noindent\hyperlink{the_category_of_groupoid_objects}{The $(\infty,1)$-category of groupoid objects}\dotfill \pageref*{the_category_of_groupoid_objects} \linebreak
\noindent\hyperlink{cech_nerves}{Cech nerves}\dotfill \pageref*{cech_nerves} \linebreak
\noindent\hyperlink{Effective}{Effective quotients}\dotfill \pageref*{Effective} \linebreak
\noindent\hyperlink{Delooping}{Delooping}\dotfill \pageref*{Delooping} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{group_objects_in_an_topos}{Group objects in an $(\infty,1)$-topos}\dotfill \pageref*{group_objects_in_an_topos} \linebreak
\noindent\hyperlink{ModelsInInfGrpd}{Models for group objects in $\infty Grpd$}\dotfill \pageref*{ModelsInInfGrpd} \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}

A [[group object]] in an ordinary [[category]] $C$ with [[pullback]]s is an [[internalization|internal]] [[group]]. More generally, there is the notion of an [[internal groupoid]] in a category $C$.

By the logic of [[vertical categorification]], an \textbf{internal $\infty$-group} or \textbf{[[internal ∞-groupoid]]} may be defined as a group(oid) object internal to an [[(∞,1)-category]] $C$ with [[(∞,1)-pullbacks]]. As described there, in full generality this involves not only a weakening of the usual [[associativity]] and [[unit laws]] up to [[homotopy]], but requires specification of [[coherence]] laws of these homotopies up to higher homotopy, and so on.

A \emph{group object} in an [[(∞,1)-category]] generalizes and unifies two familiar concepts:

\begin{itemize}%
\item it is the generalization of the notion of groupal [[Jim Stasheff|Stasheff]] $A_\infty$-[[A-infinity-space|space]] from [[Top]] to more general [[(∞,1)-sheaf (∞,1)-toposes]]: an object that comes equipped with an associative and invertible [[monoid]] structure, up to [[coherence|coherent]] [[homotopy]], and possibly only [[horizontal categorification|partially defined]] (see also [[looping and delooping]] for more on this) ;


\item it generalizes the notion of \emph{[[equivalence relation]]} -- or rather the [[internalization|internal]] notion of \emph{[[congruence]]} -- from [[category theory]] to [[(∞,1)-category theory]].



\end{itemize}
Of particular relevance are such group objects that define [[quotient object|effective quotients]]

\begin{itemize}%
\item these are [[delooping|deloopable]];


\item these generalize the notion of [[regular epimorphism]];


\item these serve to characterize [[regular category|regular (∞,1)-categories]] -- such as [[∞-stack]] [[(∞,1)-topoi]] -- as those where every such object is an [[quotient object|effective quotient]].



\end{itemize}
A \emph{groupoid object} is then accordingly the [[horizontal categorification|many-object]] version of a [[group object]].

But notice the following. Since this is defined [[internalization|internal]] to an [[(∞,1)-category]], externally these look like genuine [[∞-groupoid]] and [[∞-group]] objects. For instance a [[group object]] in a [[(2,1)-category]] such as [[Grpd]] is, externally, a [[2-group]].

Also notice that if the ambient $(\infty,1)$-category is in fact an [[(∞,1)-topos]], then every object in there may already be thought of as an ``∞-groupoid with geometric structure'' (see for instance the discussion at [[cohesive (∞,1)-topos]], but this is true more generally). The relation between the \emph{internal groupoid objects} then and the objects themselves is (an [[horizontal categorification|oid-ification]]) of that of [[looping and delooping]]. Notably for $G$ any internal group object (externally an [[∞-group]]) the corresponging ordinary object is its [[delooping]] object $\mathbf{B}G$, and every [[pointed object|pointed]] [[n-connected object in an (infinity,1)-topos|connected]] object in the $(\infty,1)$-topos arises this way from an internal group object.

A groupoid object

\begin{displaymath}
\cdots C_2 \stackrel{\to}{\stackrel{\to}{\to}} C_1 \stackrel{\to}{\to} C_0
\end{displaymath}
being \textbf{effective} means that it is the [[?ech nerve]]

\begin{displaymath}
\cdots 
  C_0 \times_{C_0//C_1} C_0 \times_{C_0//C_1} C_0 
  \stackrel{\to}{\stackrel{\to}{\to}} 
  C_0 \times_{C_0//C_1} C_0  
 \stackrel{\to}{\to} C_0
\end{displaymath}
of its [[action groupoid]] $C_0//C_1$ (the [[(∞,1)-colimit]] over its diagram)

\begin{displaymath}
\cdots C_2 \stackrel{\to}{\stackrel{\to}{\to}} C_1 \stackrel{\to}{\to} C_0
  \to
  C_0//C_1 := colim_i C_i
  \,.
\end{displaymath}
Accordingly, groupoid objects in an $(\infty,1)$-category play a central role in the theory of [[principal ∞-bundles]].

Notice that one of the four characterizing properties of an [[(∞,1)-topos]] by the higher analog of the [[Giraud theorem]] is that every groupoid object is effective.

\hypertarget{definition_complete_segalspace_style}{}\subsection*{{Definition (complete Segal-space style)}}\label{definition_complete_segalspace_style}

\hypertarget{groupoid_object}{}\subsubsection*{{Groupoid object}}\label{groupoid_object}

The following definition follows in style the definition of a [[complete Segal space]] object.

\begin{defn}
\label{}\hypertarget{}{}
For $C$ an [[(∞,1)-category]], a \textbf{groupoid object} in $C$ is a [[simplicial object in an (∞,1)-category]]

\begin{displaymath}
A : \Delta^{op} \to C
\end{displaymath}
such that for all partitions $S \cup S'$ of $[n]$ that share precisely one vertex $s$, we have that

\begin{displaymath}
\itexarray{
    A([n]) &\to & A(S)
    \\
    \downarrow && \downarrow
    \\
    A(S') &\to& A(\{s\})
  }
\end{displaymath}
is a [[(∞,1)-pullback]] diagram in $C$. Here, by a partition $S \cup S'$ of $[n]$ that share precisely one vertex $s$, we mean two subsets $S$ and $S'$ of $\{0,1,\ldots,n\}$ whose union is $\{0,1,\ldots,n\}$ and whose intersection is the singleton $\{s\}$. The linear order on $[n]$ then restricts to the linear order on $S$ and $S'$.

The $(\infty,1)$-category of groupoid objects in $C$ is the full [[sub-(∞,1)-category]]

\begin{displaymath}
Grpd(C) \hookrightarrow Func(\Delta^{op}, C)
\end{displaymath}
of the [[(∞,1)-category of (∞,1)-functors]] on those objects that are groupoid objects.

\end{defn}
This is [[Higher Topos Theory|HTT, prop. 6.1.2.6, item 4'']] with [[Higher Topos Theory|HTT, def. 6.1.2.7]].

\begin{remark}
\label{}\hypertarget{}{}
If one requires the above condition only for those partitions that are order-preserving, then this yields the definition of a (pre-)[[category object in an (∞,1)-category]].

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
A groupoid object $A : \Delta^{op} \to C$ is the \textbf{[[Cech nerve]]} of a morphism $A_0 \to B$ if $A$ is the restriction of an augmented simplicial object $A^+ : \Delta^{op}_a \to C$ with $A^+_0 \to A^+_{-1}$ as the morphism $A_0 \to B$, such that the sub-[[diagram]]

\begin{displaymath}
\itexarray{
      A^+_1 &\to& A^+_0
      \\
      \downarrow && \downarrow
      \\
      A^+_0 &\to& A^+_{-1}
    }
\end{displaymath}
of $A^+$ is a [[(∞,1)-pullback]] diagram in $C$.

\end{defn}
This is [[Higher Topos Theory|HTT, below prop. 6.1.2.11]].

If $A$ is the [[?ech nerve]] of a morphism $A_0 \to A_{-1}$\newline then the groupoid object is [[delooping|deloopable]] in the groupoid sense.

\begin{defn}
\label{EffectiveGroupoid}\hypertarget{EffectiveGroupoid}{}
A groupoid object $A : \Delta^{op} \to C$ is an \textbf{effective [[quotient object]]} if the [[(∞,1)-colimit]] diagram $A^+ : \Delta_a^{op} \to C$ exists, such that $A$ is the [[Cech nerve]] of $A^+_0 \to A^+_{-1}$, i.e. of $A_0 \to \lim_\to A_\bullet$.

\end{defn}
\hypertarget{group_object}{}\subsubsection*{{Group object}}\label{group_object}

A \textbf{group object} is a groupoid object $U : \Delta^{op} \to C$ for which $U_0 \simeq *$ is a [[terminal object in a quasi-category|terminal object]].

([[Higher Topos Theory|HTT, def. 7.2.2.1]]).

It follows ([[Higher Topos Theory|HTT, prop. 7.2.2.4]]) that a group object is of the form

\begin{displaymath}
U = \left(
      \cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} *
  \right)
  \,.
\end{displaymath}
\hypertarget{RelationToQuotients}{}\subsubsection*{{Relation to $(\infty,1)$-quotients}}\label{RelationToQuotients}

\begin{remark}
\label{}\hypertarget{}{}
A groupoid object in an $(\infty,1)$-category

\begin{displaymath}
\left(  \cdots  A_2  \stackrel{\to}{\stackrel{\to}{\to}}
   A_1 \stackrel{\to}{\to} A_0
  \right)
\end{displaymath}
is the [[(∞,1)-category]] analog of an internal [[equivalence relation]] on $A_0$, which is just a pair of morphisms

\begin{displaymath}
R \stackrel{\to}{\to} A_0
   \,.
\end{displaymath}
The [[colimit]] ([[coequalizer]]) of the latter diagram is the [[quotient]] of $A_0$ by the relation $R$.

Analogously, the [[(∞,1)-colimit]]

\begin{displaymath}
\lim_\to (\Delta^{op} \stackrel{A}{\to} C)
\end{displaymath}
over the simplicial diagram $A : \Delta^{op} \to C$ is the corresponding \textbf{$(\infty,1)$-quotient}.

If we are given a [[model category]] [[presentable (∞,1)-category|presentation]] of the [[(∞,1)-category]] $C$, then this [[(∞,1)-colimit]] is presented by a [[homotopy colimit]] over the corresponding simplicial diagraM a \textbf{homotopy quotient} .

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

\hypertarget{EquivalentCharacterizations}{}\subsubsection*{{Equivalent characterizations}}\label{EquivalentCharacterizations}

We state in prop. \ref{EquivalentCharacterizationsViaSlicesAndPowering} below a list of equivalent conditions that characterize a [[simplicial object in an (∞,1)-category]] as a groupoid object. This uses the following basic notions, which we review here for convenience.

\begin{defn}
\label{}\hypertarget{}{}
For $K \in$ [[sSet]] a [[simplicial set]], write $\Delta_{/K}$ for its [[category of simplices]]. For $X_\bullet \in \mathcal{C}^{\Delta^{op}}$ a simplicial object, write

\begin{displaymath}
X[K] \colon \Delta^{op}_{/K} \to \Delta^{op} \stackrel{X}{\to} \mathcal{C}
\end{displaymath}
for the precomposition of $X_\bullet$ with the canonical projection. Moreover, write

\begin{displaymath}
X(K) \coloneqq \underset{\leftarrow}{\lim} X[K]
\end{displaymath}
for the [[(∞,1)-limit]] over this composite [[(∞,1)-functor]] in $\mathcal{C}$ (if it exists). (Notice: square brackets for the composite functor, round brackets for its $(\infty,1)$-limit.)

\end{defn}
\begin{remark}
\label{EquivalenceOfConeCategoriesAndLimits}\hypertarget{EquivalenceOfConeCategoriesAndLimits}{}
For $X_\bullet \in \mathcal{C}^{\Delta^{op}}$ and $K \to K'$ the following are equivalent

\begin{enumerate}%
\item the induced morphism of cone $(\infty,1)$-categoris $\mathcal{C}_{X[K]} \to \mathcal{C}_{X[K']}$ is an [[equivalence of (∞,1)-categories]];


\item the induced morphism of [[(∞,1)-limits]] $X(K) \to X(K')$ is an [[equivalence in an (∞,1)-category|equivalence]].



\end{enumerate}
\end{remark}
(The first perspective is used in (\hyperlink{Lurie}{Lurie}), the second in (\hyperlink{Lurie2}{Lurie2}).)

\begin{proof}
In one direction: the limit is the [[terminal object in an (∞,1)-category|terminal object]] in the cone category, and so is preserved by equivalences of cone categories. (This direction appears as (\hyperlink{Lurie}{Lurie, prop. 4.1.1.8})). Conversely, the limits is the object representing cones, and hence an equivalence of limits induces an equivalence of cone categories.

\end{proof}
\begin{prop}
\label{EquivalentCharacterizationsViaSlicesAndPowering}\hypertarget{EquivalentCharacterizationsViaSlicesAndPowering}{}
Let $\mathcal{C}$ be an $(\infty,1)$-category incarnated explicitly as a [[quasi-category]]. Then a [[simplicial object in an (∞,1)-category|simplicial object in]] $\mathcal{C}$ is a groupoid object if the following equivalent conditions hold.

\begin{enumerate}%
\item If $K \to K'$ is a morphism in [[sSet]] which is a [[weak homotopy equivalence]] and a [[bijection]] on [[vertices]], then the induced morphism on [[slice-(∞,1)-categories]]

\begin{displaymath}
\mathcal{C}_{/X[K]}
  \to 
  \mathcal{C}_{/X[K']}
\end{displaymath}
is an [[equivalence of (∞,1)-categories]] (a [[weak equivalence]] in the [[model structure for quasi-categories]]).


\item For every $n \geq 2$ and every $0 \leq i \leq n$, the morphism $\mathcal{C}_{/X[\Delta^n]} \to \mathcal{C}_{/X[\Lambda^n_i]}$ is an weak equivalence in the [[model structure for quasi-categories]]


\item (\ldots{})



\end{enumerate}
Using remark \ref{EquivalenceOfConeCategoriesAndLimits} this means equivalently that the simplicial object $X_\bullet$ is a groupoid precisely if the following

\begin{enumerate}%
\item $X_\bullet$ satisfies the ordinary [[Segal conditions]] and the morphism $X(\Delta^2) \to X(\Lambda^2_0)$ is an [[equivalence in an (∞,1)-category|equivalence]].


\item (\ldots{})



\end{enumerate}
\end{prop}
The first items appear as (\hyperlink{Lurie}{Lurie, prop. 6.1.2.6}). The second ones appear in the proofs of (\hyperlink{Lurie2}{Lurie2, prop. 1.1.8, lemma 1.2.25}).

\hypertarget{the_category_of_groupoid_objects}{}\subsubsection*{{The $(\infty,1)$-category of groupoid objects}}\label{the_category_of_groupoid_objects}

\begin{prop}
\label{}\hypertarget{}{}
The $(\infty,1)$-category of groupoid objects in $C$ is a [[reflective sub-(∞,1)-category]]

\begin{displaymath}
Grpd(C)
  \stackrel{\overset{}{\leftarrow}}{\hookrightarrow}
  Func(\Delta^{op}, C)
  \,.
\end{displaymath}
\end{prop}
This is [[Higher Topos Theory|HTT, prop. 6.1.2.9]]. In nice cases the image of this reflective subcategory are the effective epimorphisms:

\begin{prop}
\label{}\hypertarget{}{}
If $C = \mathbf{H}$ in an [[(∞,1)-semitopos]] there is a natural [[equivalence of (∞,1)-categories]]

\begin{displaymath}
Grpd(\mathbf{H})
  \simeq
  (\mathbf{H}^I)_{eff}
\end{displaymath}
between the $(\infty,1)$-category of groupoid objects in $\mathbf{H}$ and the full [[sub-(∞,1)-category]] of the [[arrow category]] of $\mathbf{H}$ (the [[(∞,1)-functor (∞,1)-category]] $Func(\Delta[1], \mathbf{H})$) on the [[effective epimorphism in an (∞,1)-category|effective epimorphisms]].

\end{prop}
This appears below [[Higher Topos Theory|HTT, cor. 6.2.3.5]].

\hypertarget{cech_nerves}{}\subsubsection*{{Cech nerves}}\label{cech_nerves}

Write $\Delta_a$ for the augmented [[simplex category]] (including the object $[-1]$).

\begin{prop}
\label{}\hypertarget{}{}
An augmented simplicial object $A^+ : \Delta_a^{op} \to C$ is the right [[Kan extension]] of its restriction to $[-1]$ and $[0]$

\begin{displaymath}
\itexarray{
    \{[-1] \leftarrow [0]\} &\stackrel{A^+|_{\leq 0}}{\to}& C
    \\
    \downarrow & \nearrow_{\mathrlap{A^+}}
    \\
    \Delta_a^{op}
  }
\end{displaymath}
precisley if $A^+|_{\geq 0}$ is a groupoid object in $C$ and the [[diagram]]

\begin{displaymath}
\itexarray{
    A_1 &\to& A_0
    \\
    \downarrow && \downarrow
    \\
    A_0 &\to& A_{-1}
  }
\end{displaymath}
is a [[(∞,1)-pullback]] in $C$.

\end{prop}
$A$ is called the \textbf{[[Cech nerve]]} of $A_0 \to A_{-1}$ if the equivalent conditions of this proposition are satisfied.

\hypertarget{Effective}{}\subsubsection*{{Effective quotients}}\label{Effective}

\begin{prop}
\label{}\hypertarget{}{}
In $C =$ [[∞Grpd]] every groupoid object is an effective quotient, def. \ref{EffectiveGroupoid}.

\end{prop}
This is [[Higher Topos Theory|HTT, below remark 6.1.2.15]] and [[Higher Topos Theory|HTT, cor. 6.1.3.20]].

More generally, this is true for every [[(∞,1)-topos]].

\begin{prop}
\label{}\hypertarget{}{}
In $C$ is an [[(∞,1)-topos]], then every groupoid object in $C$ is an effective quotient, def. \ref{EffectiveGroupoid}.

\end{prop}
This is [[Higher Topos Theory|HTT, theorem 6.1.0.6 (4) iv)]].

\hypertarget{Delooping}{}\subsubsection*{{Delooping}}\label{Delooping}

For $\mathcal{X}$ an [[(∞,1)-category]] with [[(∞,1)-pullback]]s and for $x : * \to X$ a [[pointed object|pointed]] object in $\mathcal{X}$, its [[loop space object]] at $x$ is the [[(∞,1)-pullback]]

\begin{displaymath}
\Omega_x X := {*} \prod_{X} {*}
\end{displaymath}
hence the object universally filling the diagram

\begin{displaymath}
\itexarray{
    \Omega_x X &\to& *
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{x}}
    \\
    * &\stackrel{x}{\to}& X
  }
  \,.
\end{displaymath}
Since this is the beginning of the [[Cech nerve]] of $* \to X$, $\Omega_x X$ is naturally equipped with the structure of an $\infty$-group object in $\mathcal{X}$.

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{X}$ be an [[(∞,1)-topos]]. Then the operation of forming [[loop space]] objects constitutes an [[equivalence of (∞,1)-categories]]

\begin{displaymath}
\Omega : PointedConnected(\mathcal{X}) \stackrel{\simeq}{\to}
   Grp(\mathcal{X})
\end{displaymath}
from the full [[sub-(∞,1)-category]] of the [[over-(∞,1)-category|under-(∞,1)-category]] $*/\mathcal{X}$ of [[pointed object]]s on those that are also [[0-connected]] (hence those that have an essentially unique point) with the $(\infty,1)$-category of group objects in $\mathcal{X}$.

\end{prop}
This is [[Higher Topos Theory|HTT, lemma 7.2.2.11 (1)]]

The inverse to $\Omega$ we write

\begin{displaymath}
\mathbf{B} : Grp(\mathcal{X}) \to PointedConnected(\mathcal{X})
  \,.
\end{displaymath}
For $G \in Grp(\mathcal{X})$ we call $\mathbf{B}G$ its [[delooping]].

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

\hypertarget{group_objects_in_an_topos}{}\subsubsection*{{Group objects in an $(\infty,1)$-topos}}\label{group_objects_in_an_topos}

When the ambient [[(∞,1)-category]] is an [[(∞,1)-topos]] then -- by the $\infty$-Giraud axioms -- all groupoid objects are [[quotient object|effective]], meaning that for

\begin{displaymath}
\mathbf{B}G = \lim_{\to} U_\bullet
\end{displaymath}
the [[(∞,1)-colimit]] over the group object $U_\bullet$ we have that $U_\bullet$ is reproduced as the Cech nerve of $* \to \mathbf{B}G$

\begin{displaymath}
\left(
      \cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} *
  \right)
  \simeq
  \left(
      \cdots 
   {*}\times_{\mathbf{B}G}{*}\times_{\mathbf{B}G}{*} \stackrel{\to}{\stackrel{\to}{\to}} {*}\times_{\mathbf{B}G}{*} \stackrel{\to}{\to} *
  \right) 
  \,.
\end{displaymath}
The object $\mathbf{B}G$ is the [[delooping]] object of the group object $G$.

For more on this see also [[principal ∞-bundle]].

\hypertarget{ModelsInInfGrpd}{}\subsubsection*{{Models for group objects in $\infty Grpd$}}\label{ModelsInInfGrpd}

There is a [[model category]] structure that presents the [[(∞,1)-category]] of group objects in [[∞Grpd]]: the [[∞-group]]s.

\begin{itemize}%
\item The group objects $G$ themselves are modeled by a model structure on the category $sGrp$ of [[simplicial group]]s.


\item Their delooping spaces $\mathbf{B}G$ are modeled by a model structure on the category $sSet_0$ of [[simplicial set]]s with a single vertex.



\end{itemize}
The operation of forming [[loop space object]]s constitutes a [[Quillen equivalence]] between these two model structures

\begin{displaymath}
\Omega : sSet_0 \stackrel{\simeq_{Quillen}}{\to} sGrp
  \,.
\end{displaymath}
The Quillen equivalence itself is in section 6 there.

\begin{prop}
\label{}\hypertarget{}{}
There exists the [[transferred model structure]] on the category $sGrp$ of [[simplicial group]]s along the [[forgetful functor]]

\begin{displaymath}
U : sGrp \to sSet_{Quillen}
\end{displaymath}
to the standard [[model structure on simplicial sets]].

This means that a morphism in $sGrp$ is a

\begin{itemize}%
\item weak equivalences


\item or fibration



\end{itemize}
precisely if it is so in $sSet_{Quillen}$.

\end{prop}
This appears as (\href{GoerssJardine}{GoerssJardine, ch V, theorem. 2.3}).

\begin{prop}
\label{}\hypertarget{}{}
There is a [[model structure on reduced simplicial sets]] $sSet_0$ ([[simplicial sets]] with a single vertex) whose

\begin{itemize}%
\item weak equivalences


\item and cofibrations



\end{itemize}
are those in the standard [[model structure on simplicial sets]].

\end{prop}
This appears as (\href{GoerssJardine}{GoerssJardine, ch V, prop. 6.2}).

\begin{prop}
\label{}\hypertarget{}{}
The simplicial loop space functor $G$ and the delooping functor $\bar W(-)$ (discussed at [[simplicial group]]) constitute a [[Quillen equivalence]]

\begin{displaymath}
(G \dashv \bar W) : sGr \stackrel{\overset{G}{\leftarrow}}{\underset{\bar W}{\to}}
  sSet_0
  \,.
\end{displaymath}
The $(G \dashv \bar W)$-[[unit of an adjunction|unit]] and counit are weak equivalences:

\begin{displaymath}
X \stackrel{\simeq}{\to} \bar W G X
\end{displaymath}
\begin{displaymath}
G \bar W G \stackrel{\simeq}{\to} G
  \,.
\end{displaymath}
\end{prop}
This appears as (\hyperlink{GoerssJardine}{GoerssJardine, ch. V prop. 6.3}).

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

\begin{itemize}%
\item [[∞-group]]


\item [[monoid]]. [[monoid object]], [[monoid object in an (∞,1)-category]]


\item [[group]], [[group object]]


\item [[ring]], [[ring object]]


\item [[looping and delooping]]



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

Groupoid objects in $(\infty,1)$-categories are the topic of section 6.1.2 in

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

\end{itemize}
Model category presentations of groupoid objects in $\infty Grpd$ by groupoidal [[complete Segal spaces]] are discussed in

\begin{itemize}%
\item [[Julia Bergner]],

\emph{Adding inverses to diagrams encoding algebraic structures}, Homology, Homotopy and Applications 10 (2008), no. 2, 149--174. (\href{http://arxiv.org/abs/math/0610291}{arXiv:0610291})

\emph{Adding inverses to diagrams II: Invertible homotopy theories are spaces}, Homology, Homotopy and Applications, Vol. 10 (2008), No. 2, pp.175-193. (\href{http://www.intlpress.com/hha/v10/n2/a9/}{web}, \href{http://arxiv.org/abs/0710.2254}{arXiv:0710.2254})



\end{itemize}
A standard textbook reference on $\infty$-groups in the [[classical model structure on simplicial sets]] is

\begin{itemize}%
\item [[Paul Goerss]], [[Rick Jardine]], chapter V of \emph{[[Simplicial homotopy theory]]} \href{http://www.maths.abdn.ac.uk/~bensondj/papers/g/goerss-jardine/ch-5.dvi}{chapter V}.

\end{itemize}
Discussion from the point of view of [[category objects in an (∞,1)-category]] is in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[(∞,2)-Categories and the Goodwillie Calculus]]} (\href{http://arxiv.org/abs/0905.0462}{arXiv:0905.0462})

\end{itemize}
[[!redirects groupoid object in an (∞,1)-category]] [[!redirects groupoid objects in an (∞,1)-category]] [[!redirects groupoid objects in an (infinity,1)-category]] [[!redirects group object in an (infinity,1)-category]] [[!redirects group object in an (∞,1)-category]] [[!redirects group objects in an (infinity,1)-category]] [[!redirects group objects in an (∞,1)-category]]



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