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\section*{geometrically discrete infinity-groupoid}

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

\hypertarget{cohesive_toposes}{}\paragraph*{{Cohesive $\infty$-Toposes}}\label{cohesive_toposes}

[[!include cohesive infinity-toposes - 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{structures_in_}{Structures in $Disc\infty Grpd$}\dotfill \pageref*{structures_in_} \linebreak
\noindent\hyperlink{TopAsCohesiveInfinTopos}{Geometric homotopy and Galois theory}\dotfill \pageref*{TopAsCohesiveInfinTopos} \linebreak
\noindent\hyperlink{cohomology_and_principal_bundles}{Cohomology and principal $\infty$-bundles}\dotfill \pageref*{cohomology_and_principal_bundles} \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}

The term \emph{discrete $\infty$-groupoid} or \emph{bare $\infty$-groupoid} or \emph{geometrically [[discrete object|discrete]] [[geometric homotopy type]]} is essentially synonymous to just \emph{[[∞-groupoid]]} or just \emph{[[homotopy type]]}. It is used for emphasis in contexts where one considers $\infty$-groupoids with extra [[geometric homotopy type|geometric structure]] (e.g. [[cohesive (∞,1)-topos|cohesive]] [[stuff, structure, property|structure]]) to indicate that this extra structure is being disregarded, or rather that the special case of [[discrete space|discrete]] such structure is considered.

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

\begin{uobservation}
The [[terminal object in an (∞,1)-category|terminal]] [[(∞,1)-sheaf (∞,1)-topos]] [[∞Grpd]] is trivially a [[cohesive (∞,1)-topos]], where each of the defining four [[(∞,1)-functor]]s $(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : \infty Grpd \to \infty Grpd$ is an [[equivalence of (∞,1)-categories]].

\end{uobservation}
\begin{udefn}
In the context of [[cohesive (∞,1)-topos]]es we say that [[∞Grpd]] defines \textbf{discrete cohesion} and refer to its objects as \textbf{discrete $\infty$-groupoids}.

More generally, given any other [[cohesive (∞,1)-topos]]

\begin{displaymath}
(\Pi \dashv Disc \dashv \Gamma \dashv codisc) : \mathbf{H}  \stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}}
  \infty Grpd
\end{displaymath}
the [[inverse image]] $Disc$ of the [[global section]] functor is a [[full and faithful (∞,1)-functor]] and hence embeds [[∞Grpd]] as a full [[sub-(∞,1)-category]] of $\mathbf{H}$. A general object in $\mathbf{H}$ is a \emph{cohesive $\infty$-groupoid} . We say $X \in \mathbf{H}$ is a \textbf{discrete $\infty$-groupoid} if it is in the [[image]] of $Disc$.

\end{udefn}
\begin{uremark}
This generalizes the traditional use of the terms [[discrete space]] and [[discrete group]]:

\begin{itemize}%
\item a [[discrete space]] is equivalently a 0-[[truncated]] discrete $\infty$-groupoid;


\item a [[discrete group]] is equivalently a 0-[[truncated]] [[group object in an (∞,1)-category|group object]] in discrete $\infty$-groupoids.



\end{itemize}
\end{uremark}
\hypertarget{structures_in_}{}\subsection*{{Structures in $Disc\infty Grpd$}}\label{structures_in_}

We discuss now some of the general abstract [[cohesive (∞,1)-topos -- structures|structures in a cohesive (∞,1)-topos]] realized in discrete $\infty$-groupoids.

\hypertarget{TopAsCohesiveInfinTopos}{}\subsubsection*{{Geometric homotopy and Galois theory}}\label{TopAsCohesiveInfinTopos}

We discuss the general absatract notion of geometric homotopy in cohesive $(\infty,1)$-toposes (see ) in the context of discrete cohesion.

By the [[homotopy hypothesis]]-theorem the [[(∞,1)-topos]]es [[Top]] and [[∞Grpd]] are [[equivalence of (∞,1)-categories|equivalent]], hence indistinguishable by general abstract constructions in [[(∞,1)-topos theory]]. However, in practice it can be useful to distinguish them as two different [[presentable (∞,1)-category|presentations]] for an equivalence class of $(\infty,1)$-toposes.

For that purposes consider the following

\begin{udefn}
Define the [[quasi-categories]]

\begin{displaymath}
Top := N(Top_{Quillen})^\circ
\end{displaymath}
and

\begin{displaymath}
\infty Grpd := N(sSet_{Quillen})^\circ 
  \,,
\end{displaymath}
where on the right we have the standard [[model structure on topological spaces]] $Top_{Quillen}$ and the standard [[model structure on simplicial sets]] $sSet_{Quillen}$ and $N((-)^\circ)$ denotes the [[homotopy coherent nerve]] of the [[simplicial category]] given by the full [[sSet]]-subcategory of these [[simplicial model categories]] on fibrant-cofibrant objects.

\end{udefn}
For

\begin{displaymath}
({|-| \dashv Sing}) : Top_{Quillen} \stackrel{\overset{{|-|}}{\leftarrow}}{\underset{Sing}{\to}} sSet_{Quillen}
\end{displaymath}
the standard [[Quillen equivalence]] of the [[homotopy hypothesis]]-theorem given by the [[singular simplicial complex]]-functor and [[geometric realization]], write

\begin{displaymath}
(\mathbb{L} {|-|} \dashv \mathbb{R}Sing) : 
  Top 
    \stackrel{\overset{\mathbb{L}{|-|}}{\leftarrow}}{\underset{\mathbb{R}Sing}{\to}}
  \infty Grpd
\end{displaymath}
for the corresponding [[derived functor]]s (the image under the [[homotopy coherent nerve]] of the restriction of ${|-|}$ and $Sing$ to fibrant-cofibrant objects followed by functorial fibrant-cofibrant replacement) that constitute a pair of [[adjoint (∞,1)-functor]]s modeled as morphisms of [[quasi-categories]].

Since this is an [[equivalence of (∞,1)-categories]] either functor serves as the [[left adjoint]] and [[right adjoint]] and so we have

\begin{uobservation}
[[Top]] is exhibited a [[cohesive (∞,1)-topos]] over [[∞Grpd]] by setting

\begin{displaymath}
(\Pi \dashv Disc \dashv \Gamma \dashv coDisc)
  :
  Top
    \stackrel{\overset{\mathbb{R}Sing}{\to}}{\stackrel{\overset{\mathbb{L}{|-|}}{\leftarrow}}{\stackrel{\overset{\mathbb{R}Sing}{\to}}{\underset{\mathbb{L}{|-|}}{\leftarrow}}}}
  \infty Grpd
  \,.
  \,.
\end{displaymath}
In particular a presentation of the intrinsic [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] is given by the familiar [[singular simplicial complex]] construction

\begin{displaymath}
\Pi(X) \simeq \mathbb{R} Sing X
  \,.
\end{displaymath}
\end{uobservation}
\begin{uremark}
While degenerate, it is sometimes useful to make this example of a [[cohesive (∞,1)-topos]] explicit. For instance it allows to think of simplicial models for topological fibrations in terms of topological [[higher parallel transport]]. Some remarks on this are in .

\end{uremark}
\begin{uremark}
Notice that the [[topology]] that enters the explicit construction of the objects in [[Top]] here does \emph{not} show up as [[cohesive (∞,1)-topos|cohesive structure]]. A [[topological space]] here is a model for a \emph{discrete} $\infty$-groupoid, the [[topology]] only serves to allow the construction of $Sing X$. For discussion of $\infty$-groupoids equipped with genuine \emph{topological cohesion} see [[Euclidean-topological ∞-groupoid]].

\end{uremark}
\hypertarget{cohomology_and_principal_bundles}{}\subsubsection*{{Cohomology and principal $\infty$-bundles}}\label{cohomology_and_principal_bundles}

We discuss the general abstract notion of cohomology and principal $\infty$-bundles a in cohesive $\infty$-toposes (see ) in the context of discrete cohesion.

\begin{defn}
\label{}\hypertarget{}{}
Write $\mathrm{sGrp} = \mathrm{Grp}(\mathrm{sSet})$ for the category of [[simplicial group]]s.

\end{defn}
A classical reference is section 17 of \hyperlink{May}{May}.

\begin{prop}
\label{}\hypertarget{}{}
The category $\mathrm{sGrpd}$ inherits a [[model category]] structure [[transferred model structure|transferred]] along the forgetful functor $F : \mathrm{sGrp} \to \mathrm{sSet}$.

The category $\mathrm{sSet}_0 \hookrightarrow \mathrm{sSet}$ of reduced simplicial sets (simplicial sets with a single vertex) carries a model category structure whose weak equivalences and cofibrations are those of $\mathrm{sSet}_{\mathrm{Quillen}}$.

There is a [[Quillen equivalence]]

\begin{displaymath}
(G \dashv \bar W)
  :
  sGrp
   \stackrel{\overset{G}{\leftarrow}}{\underset{\bar W}{\to}}
  sSet_{0}
\end{displaymath}
which [[presentable (infinity,1)-category|presents]] the abstract [[looping and delooping]] equivalence of $\infty$-categories

\begin{displaymath}
(\Omega \dashv \mathbf{B})
   :
   \infty Grpd
    \stackrel{\overset{\Omega}{\leftarrow}}{\underset{B}{\to}}
   \infty Grpd_{connected}
   \,,
\end{displaymath}
\end{prop}
The model structures and the Quillen equivalence are classical, discussed in (\hyperlink{GoerssJardine}{GoerssJardine, section V})

This means on abstract grounds that for $G$ a [[simplicial group]], $\bar W G \in \mathrm{sSet}$ is a model of the [[classifying space|classifying]] [[delooping]] object $\mathbf{B}G$ for discrte $G$-[[principal ∞-bundles]]. The following statements assert that these principal $\infty$-bundles themselves can be modeled as ordinary [[simplicial principal bundle]]s

\begin{defn}
\label{}\hypertarget{}{}
For $G$ a [[simplicial group]] and $\bar W G$ the model for $\mathbf{B}G$ given by the above proposition, write

\begin{displaymath}
W G \to \bar W G
\end{displaymath}
for the simplicial [[decalage]] on $\bar W G$.

\end{defn}
This characterization of the object going by the classical name $W G$ is made fairly explicit in (\hyperlink{Duskin}{Duskin, p. 85}).

\begin{prop}
\label{}\hypertarget{}{}
The morphism $W G \to \bar W G$ is a Kan fibration resolution of the point inclusion ${*} \to \bar W G$.

\end{prop}
This follows directly from the characterization of $W G \to \bar W G$ by [[decalage]]. Pieces of this statement appear in (\hyperlink{May}{May}): lemma 18.2 there gives the fibration property, prop. 21.5 the contractibility of $W G$.

\begin{cor}
\label{}\hypertarget{}{}
For $G$ a [[simplicial group]], the sequence of simplicial sets

\begin{displaymath}
G \to W G \to \bar W G
\end{displaymath}
is a presentation of the [[fiber sequence]]

\begin{displaymath}
G \to * \to \mathbf{B}G
  \,.
\end{displaymath}
Hence $W G \to \bar W G$ is a model for the universal $G$-principal discrete $\infty$-bundle (see [[universal principal ∞-bundle]]):

every $G$-principal discrete $\infty$-bundle $P \to X$ in $\infty \mathrm{Grpd}$, which by definition is a [[homotopy fiber]]

\begin{displaymath}
\itexarray{
    P &\to& * 
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow
    \\
    X &\to& \mathbf{B}G
  }
\end{displaymath}
in [[?Gpd]], is presented in the standard [[model structure on simplicial sets]] by the ordinary [[pullback]]

\begin{displaymath}
\itexarray{
    P &\to& W G
    \\
    \downarrow && \downarrow
    \\
    X &\to& \bar W G
  }
  \,.
\end{displaymath}
\end{cor}
The explicit statement that the sequence $G \to W G \to \bar W G$ is a model for the looping fiber sequence appears on p. 239 of \emph{[[Crossed Menagerie]]} . The universality of $W G \to \bar W G$ for $G$-principal simplicial bundles is the topic of section 21 in (\hyperlink{May}{May}), where however it is not made explicit that the ``[[twisted cartesian product]]s'' considered there are precisely the models for the pullbacks as above. This is made explicit on page 148 of \emph{[[Crossed Menagerie]]}.

In \emph{[[Euclidean-topological ∞-groupoid]]} we discuss how this model of discrete principal $\infty$-bundles by simplicial principal bundles lifts to a model of topological principal $\infty$-bundles by simplicial topological bundles principal over [[simplicial topological group]]s.

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

\begin{itemize}%
\item [[discrete space]]


\item [[discrete group]]


\item [[discrete groupoid]]



\end{itemize}
[[!include geometries of physics -- table]]

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

[[simplicial group|Simplicial groups]] and [[simplicial principal bundles]] are discussed in

\begin{itemize}%
\item [[Peter May]], \emph{Simplicial Objects in Algebraic Topology} (\href{http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu}{djvu}).

\end{itemize}
and section V of

\begin{itemize}%
\item [[Paul Goerss]] and [[Rick Jardine]], 1999, \emph{Simplicial Homotopy Theory}, number 174 in Progress in Mathematics, Birkhauser. (\href{http://www.maths.abdn.ac.uk/~bensondj/html/archive/goerss-jardine.html}{ps})

\end{itemize}
The relation of $W G \to \bar W G$ to [[decalage]] is mentioned on p. 85 of

\begin{itemize}%
\item [[John Duskin]], \emph{Simplicial methods and the interpretation of ``triple'' cohomology}, number 163 in Mem. Amer. Math. Soc., 3, Amer. Math. Soc. (1975)

\end{itemize}
Discrete cohesion is the topic of section 3.1 of

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]} ,

\end{itemize}
where much of the above material is taken from.

[[!redirects geometrically discrete infinity-groupoids]]

[[!redirects geometrically discrete ∞-groupoid]] [[!redirects geometrically discrete ∞-groupoids]]

[[!redirects discrete infinity-groupoid]]

[[!redirects discrete ∞-groupoid]]

[[!redirects discrete ∞-groupoids]] [[!redirects discrete infinity-groupoids]]

[[!redirects Disc∞Grpd]]

[[!redirects discrete ∞-group]] [[!redirects discrete ∞-groups]]

[[!redirects discrete infinity-group]] [[!redirects discrete infinity-groups]]

[[!redirects geometrically discrete homotopy type]] [[!redirects geometrically discrete homotopy types]]

[[!redirects Discrete∞Groupoid]] [[!redirects Discrete∞Groupoids]] [[!redirects Discrete∞Grpd]]



\end{document}