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\section*{internal infinity-groupoid}

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

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

[[!include internal infinity-categories 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{KanInoo1Cat}{Internal $\infty$-groupoids in an $(\infty,1)$-category}\dotfill \pageref*{KanInoo1Cat} \linebreak
\noindent\hyperlink{KanIn1Cat}{Kan complexes in an ordinary category}\dotfill \pageref*{KanIn1Cat} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{InternalHornFillers}{Internal horn filler condition}\dotfill \pageref*{InternalHornFillers} \linebreak
\noindent\hyperlink{SimplicialSheaves}{In terms of simplicial sheaves}\dotfill \pageref*{SimplicialSheaves} \linebreak
\noindent\hyperlink{Comparison}{Comparison}\dotfill \pageref*{Comparison} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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 notion of \emph{internal [[∞-groupoid]]} is a [[vertical categorification]] of [[internal groupoid]]. As described at that entry on [[vertical categorification]], there is some flexibility possible in exactly what one may mean by this, depending for instance on which of several definitions of ordinary [[groupoid]]s one starts with, and how one deals with the higher coherences that are introduced upon categorification.

One very general notion of internal $\infty$-groupoid is given by taking some standard definition of $\infty$-groupoid and writing it down internally to an [[(∞,1)-category]]. This is described below in the section

\begin{itemize}%
\item \hyperlink{KanInoo1Cat}{∞-groupoids internal to an (∞,1)-category}.

\end{itemize}
For various applications somewhat stricter or at least more rigidified models for this may be useful. Notably one may wish to speak of $\infty$-groupoids internal to an ordinary category. In terms of [[Kan complexes]] as a model for ordinary [[∞-groupoids]], this is discussed below in the section

\begin{itemize}%
\item \hyperlink{KanIn1Cat}{Kan complexes internal to an ordinary category}.

\end{itemize}
Notice from the discussion there that these two aspects may and often do interplay: simplicial objects in [[Grothendieck topos|sheaf topos]]es may be used to [[presentable (infinity,1)-category|present]] [[(∞,1)-category|(∞,1)-categories]] inside of which one may be interested in $\infty$-groupoid objects in the first sense. For instance a Lie $\infty$-group object is usefully thought of as a \emph{group object} internal to the [[(∞,1)-topos]] presented by [[model structure on simplicial presheaves|a model of simplicial sets in]] the [[category of sheaves]] on [[Diff]].

Finally, a simplified version of [[∞-groupoid]] is often sufficient and useful: that given by [[strict omega-groupoid|strict ∞-groupoid]]s, equivalently [[crossed complexes]]. These may be straightforwardly [[internalization|internalized]] in any ordinary category with [[pullback]]s. This is discussed in

\begin{itemize}%
\item \hyperlink{Strict}{Internal strict $\infty$-groupoids}

\end{itemize}
Under the internal [[omega-nerve]] operation this will embed into the definition of Kan complexes in an ordinary category mentioned before. Even if one is interested just in strict $\infty$-groupoids, this embedding is useful in order to understand what the right notion of morphisms between these objects is, and what the extra [[descent]] conditions are that a proper internal formulation turns out to require on top of the conditions known from the external formulation (all this is discussed below). For that purpose [[Dominic Verity]]`s theorem described at

\begin{itemize}%
\item [[Verity on descent for strict omega-groupoid valued presheaves]]

\end{itemize}
is very useful.

\hypertarget{KanInoo1Cat}{}\subsection*{{Internal $\infty$-groupoids in an $(\infty,1)$-category}}\label{KanInoo1Cat}

The special case of internal \emph{1-groupoids} in an (∞,1)-category, which are still only associative and unital up to higher homotopy, but which do not include ``higher cells'' as additional data, is discussed in detail at:

\begin{itemize}%
\item [[groupoid object in an (∞,1)-category]].

\end{itemize}
The general case remains to be explored.

\hypertarget{KanIn1Cat}{}\subsection*{{Kan complexes in an ordinary category}}\label{KanIn1Cat}

A standard model for general [[∞-groupoid]]s is given by [[simplicial set]]s that are [[Kan fibration|Kan fibrant]]: [[Kan complex]]es. It is straightforward to [[internalization|internalize]] the [[horn]]-filler condition that characterizes [[Kan complex]]es from the [[category]] [[SSet]] of [[simplicial object]]s in [[Set]] to one of [[simplicial objects]] in any category $C$ with a bit of suitable structure.

\begin{itemize}%
\item This is described in \hyperlink{InternalHornFillers}{Internal horn filler condition}, below.

\end{itemize}
But for the [[simplicial object]]s in question to be usefully identified as [[∞-groupoid]]s internal to $C$, the horn filler condition is -- while necessary -- typically not sufficient. At least one has to be a bit careful about what it is one wants to model.

A very well understood special case that serves to give some feeling for the general situation is that where $C$ is a [[Grothendieck topos]]. In that case [[simplicial object]]s in $C$ are [[simplicial presheaf|simplicial sheaves]]. There is a well developed theory of [[model structure on simplicial presheaves|model structures on simplicial sheaves]] that are known to be [[models for ∞-stack (∞,1)-toposes]].

\begin{itemize}%
\item This is described in \hyperlink{SimplicialSheaves}{Simplicial sheaves}, below.

\end{itemize}
In terms of this the true $\infty$-groupoids internal to $C$ are those simplicial sheaves that are fibrant objects in the given [[model category]]. These fibrant simplicial sheaves in particular satisfy the internal Kan filler conditon, but they are characterized by a much stronger condition: in addition they satisfy a [[descent]] condition.

\begin{itemize}%
\item This is described in \hyperlink{Comparison}{Comparison}, below.

\end{itemize}
Another issue highlighted by this example is that the right notion of morphisms of internal Kan complexes are not just the internal morphisms of simplicial objects: for the case of $C$ a Grothendieck topos and using the model structure on simplicial sheaves as above, the right notion of morphism between two internal Kan complexes $X$ and $A$ is a morphism of simplicial objects $\hat X \to A$ out of a cofibrant replacement of $X$, as discussed at [[derived hom space]]. So the right notion of morphisms of internal $\infty$-groupoids are $\infty$-[[anafunctor]]s.

A little bit of theory for this exists for the slightly more general case that $C$ is an arbitrary [[elementary topos]]. See [[model structure on simplicial objects in a topos]].

For more general $C$ not much is known.

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

\hypertarget{InternalHornFillers}{}\paragraph*{{Internal horn filler condition}}\label{InternalHornFillers}

The [[Kan complex]] definition of [[∞-groupoid]] may be [[internalization|internalized]] to more general categories. Namely:

\begin{itemize}%
\item We can define a [[simplicial object]] $X$ in any [[category]] $C$, as a [[contravariant functor]] to $C$ from the [[simplex category]] $\Delta$;
\item Given a simplicial object $X$, we define the object $X^{\Lambda^n_k}$ of $k$-horns of $n$-simplices of $X$ (if it exists) as the [[weighted limit]] of $X\colon \Delta^{op} \to C$ weighted by the standard horn (thought of as a simplicial set) $\Lambda^n_k\colon \Delta^{op} \to Set$.
\item If we have a distinguished class of `surjections', then we require that each morphism $X_n \to X^{\Lambda^n_k}$ is `surjective'.

\end{itemize}
In particular, if $C$ is a [[left exact category]] equipped with a [[Grothendieck pretopology]], then the limits in (2) exist and one can choose single arrow coverings as distinguished `surjections'. So we define a \textbf{[[Kan object]]} in such a $C$ to be a simplicial object in $C$ satisfying the Kan conditions (3). Then we may define an \textbf{$\infty$-groupoid in $C$} to be simply a Kan object in $C$. (Recall that one way to define an ordinary $\infty$-groupoid is as a [[Kan complex]], that is a Kan object in [[Set]].)

Another, almost equivalent, way to describe the Kan conditions expressed in this way is by saying that ``the Kan conditions are satisfied'' is true in the [[internal logic]] of the [[site]] $C$, or equivalently the internal logic of its [[Grothendieck topos]] of sheaves (after applying the [[Yoneda embedding]] to obtain a simplicial object in this topos). Of course, if $C$ is itself a Grothendieck topos, then it is equivalent to the topos of sheaves on itself for its [[canonical topology]]. (This is only ``almost equivalent'' if the Grothendieck pretopology is in fact a topology.)

\hypertarget{SimplicialSheaves}{}\paragraph*{{In terms of simplicial sheaves}}\label{SimplicialSheaves}

If the category $C$ is a [[Grothendieck topos]], i.e. a [[category of sheaves]] (of sets) $C = Sh(S)= Sh(S,Set)$ on some small [[site]], then [[simplicial object]]s in $C$ are the same as [[simplicial sheaves]] on $S$

\begin{displaymath}
[\Delta^{op}, C)] \simeq Sh(S, SSet)
  \,.
\end{displaymath}
There are various ([[Quillen equivalence|Quillen equivalent]]) [[model category]] structures on the categories of simplicial sheaves or presheaves on a small site; see [[model structure on simplicial presheaves]] and [[model structure on simplicial sheaves]] for more. An $\infty$-groupoid in $C$ may be taken to be a fibrant object with respect to one of these model category structures.

\hypertarget{Comparison}{}\paragraph*{{Comparison}}\label{Comparison}

The two definitions are \emph{not} equivalent even when $C$ is a Grothendieck topos; the second is strictly stronger. Consider, for instance, a Grothendieck topos $C = Sh(S)$ and an [[internal groupoid]] in $C$, i.e. a sheaf of groupoids on $S$, but which is not a [[stack]]. Then the [[nerve]] of this internal groupoid will satisfy the Kan conditions in the sense of the first definition (by repeating the usual proof that the nerve of a groupoid is a Kan complex in the internal logic), but it will not be fibrant as a simplicial sheaf (since it is not a stack).

On the other hand, it is true that every fibrant simplicial sheaf satisfies the Kan conditions in the internal logic, by the following argument:

\begin{enumerate}%
\item The fibrant objects in any local [[model structure on simplicial sheaves]] in particular have the property that they are sheaves with values in Kan complexes.

Here a reminder on why this is so. The local [[model structure on simplicial sheaves|model structures on simplicial sheaves]] are left [[Bousfield localization]]s of the injective or projective [[global model structure on functors]]. So their fibrant objects are in particular fibrant in $[S^{op},SSet]_{proj/inj}$. The fibrant objects in $[S^{op}, SSet]_{proj}$ are (by definition of projective fibrations) precisely the objectwise Kan complexes. The fibrant objects in $[S^{op},SSet]_{inj}$ are less, but still contained in the collection of objectwise Kan complexes. So also the fibrant objects in $Sh(S,SSet)_{inj,proj}^{loc}$ are in particular Kan complex valued sheaves (that in addition satisfy a [[descent]] condition).


\item The [[weighted limit]]s over simplicial sheaves, or analogously the [[power]]ing of $Sh(S,SSet)$ over [[SSet]] works objectwise, so that for $X \in Sh(S,SSet)$ we have

\begin{displaymath}
X^{\Lambda_k[n]} : U \mapsto SSet(\Lambda_k[n],X(U))
\end{displaymath}
and of course

\begin{displaymath}
X_n = X^{\Delta[n]} : U \mapsto SSet(\Delta[n],X(U)) = X(U)_n
  \,.
\end{displaymath}


\end{enumerate}
Therefore fibrant $X \in Sh(S,SSet)$ have the property that for all $n, k$ the canonical morphism $X_n \to X^{\Lambda_k[n]}$ is a surjection when pulled back to any representable $U \to X$. In particular, the morphism is a [[stalk]]wise epimorphism, hence an [[epimorphism]] of sheaves.

Note that the local model structure on simplicial sheaves also contains information about the \emph{cofibrant} objects, which (it can be argued) are necessary to get the right notion of \emph{morphism} between ``internal $\infty$-groupoids.'' Every Kan complex in $Set$ is cofibrant, but the same cannot be expected to be true everywhere, and in general a (weak) ``internal $\infty$-functor'' should be expected to be a map out of a cofibrant replacement.

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

Here are some examples of internal $\infty$-groupoids according to the first definition (that is, internal Kan complexes).

\begin{itemize}%
\item A classical example consists of the \textbf{topological $\infty$-groupoids}. The [[nerve]] construction makes a topological $\infty$-groupoid from a [[topological groupoid]]. This is actually a characterization of topological groupoids among topological categories.


\item In particular, if $G$ is a [[topological group]], then $N(\mathbf{B}G)$ is a topological $\infty$-groupoid. This is relevant to the construction of the [[classifying spaces]] for continuous [[principal bundles]].


\item Another classical example consists of the [[∞-Lie groupoids]].


\item [[topological infinity-groupoid]]



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

\begin{itemize}%
\item [[groupoid object in an (infinity,1)-category]]


\item [[internal (infinity,1)-category]]


\item [[Kan-fibrant simplicial manifold]]


\item [[geometric infinity-stack]]



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

Models for [[∞-stacks]]/[[(∞,1)-presheaves]] in [[higher geometry]] by local Kan complexes of objects in a given site (for instance locally Kan [[simplicial manifolds]] for [[higher differential geometry]]) are discussed in

\begin{itemize}%
\item [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], \emph{Principal $\infty$-bundles - Presentations} (\href{http://arxiv.org/abs/1207.0249}{arXiv:1207.0249})

\end{itemize}
[[!redirects Internal infinity-groupoid]] [[!redirects Internal infinity-groupoids]]

[[!redirects Internal ∞-groupoid]] [[!redirects internal ∞-groupoid]] [[!redirects infinity-groupoid object]] [[!redirects ∞-groupoid object]] [[!redirects internal Kan complex]] [[!redirects internal-infinity groupoids]]

[[!redirects internal infinity-groupoids]]



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