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\begin{document}

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\section*{simplicial topological group}

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

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

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

[[!include homotopy - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\begin{defn}
\label{}\hypertarget{}{}
A \emph{simplicial topological group} is a [[simplicial object]] in the [[category]] of [[topological groups]].

\end{defn}
For various applications the ambient category [[Top]] of [[topological space]]s is taken specifically to be

\begin{itemize}%
\item the [[category]] of [[compactly generated space|compactly generated]] [[weakly Hausdorff space]]s, or


\item or the category of [[k-space]]s.



\end{itemize}
We take [[Top]] to be the category of [[k-space]]s in the following.

\begin{defn}
\label{WellPointedSimplicialTopologicalGroup}\hypertarget{WellPointedSimplicialTopologicalGroup}{}
A simplicial topological group $G$ is called \textbf{well-pointed} if for $*$ the trivial simplicial topological group and $i : * \to G$ the unique [[homomorphism]], all components $i_n : * \to G_n$ are [[closed cofibrations]].

\end{defn}
For $B \in Top$ a fixed base object, it is often desirable to work in ``$B$-parameterized families'', hence in the [[over-category]] $Top/B$ (see \hyperlink{MaySigurdson}{MaySigurdson}). There is the [[Strøm model structure|relative Strøm model structure]] on $Top/B$.

\begin{defn}
\label{}\hypertarget{}{}
A simplicial group in $G$ in $Top/B$ is called \textbf{well-sectioned} if for $B$ the trivial simplicial topological group over $B$ and $i : B \to G$ the unique [[homomorphism]], all components $i_n : B \to G_n$ are $\bar f$-cofibrations.

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

Recall for a [[discrete group|discrete]] [[simplicial group]] $G$ the notation $\bar W G \to W G$ for the [[Kan complex]] presentation of the [[universal principal infinity-bundle]] $\mathbf{E}G \to \mathbf{B}G$ from [[simplicial group]]. These constructions for discrete simplicial groups have immediate analogs for simplicial topological groups.

\begin{defn}
\label{}\hypertarget{}{}
Let $G$ be a simplicial topological group. Write $\bar W G \in Top^{\Delta^{op}}$ for the [[simplicial topological space]] whose topological space of $n$-[[simplices]] is the [[product]]

\begin{displaymath}
\bar W G_n := G_{n-1} \times G_{n-2} \cdots \times G_{0}
\end{displaymath}
in [[Top]], equipped wwith the evident (\ldots{}) face and degeneracy maps.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
We say a morphism $f : X \to Y$ of [[simplicial topological space]]s is a \textbf{global Kan fibration} if for all $n \in \mathbb{N}$ and $0 \leq k \leq n$ the canonical morphism

\begin{displaymath}
X_n \to Y_n \times_{sTop(\Lambda^n_k, Y)} sTop(\Lambda^n_k, X)
\end{displaymath}
in [[Top]] has a [[section]], where

\begin{itemize}%
\item $\Lambda^n_k \in$ [[sSet]] $\hookrightarrow Top^{\Delta^{op}}$ is the $k$th $n$-[[horn]] regarded as a [[discrete space|discrete]] [[simplicial topological space]]:


\item $sTop(-,-) : sTop^{op} \times sTop \to Top$ is the [[Top]]-[[hom object]].



\end{itemize}
We say a [[simplicial topological space]] $X_\bullet \in Top^{\Delta^{op}}$ is \textbf{(global) Kan simplicial space} if the unique morphism $X_\bullet \to *$ is a global Kan fibration, hence if for all $n \in \mathbb{N}$ and all $0 \leq i \leq n$ the canonical [[continuous function]]

\begin{displaymath}
X_n \to sTop(\Lambda^n_k, X)
\end{displaymath}
into the [[topological space]] of $k$th $n$-[[horn]]s admits a [[section]].

\end{defn}
This global notion of Kan simplicial spaces is considered for instance in (\hyperlink{BrownSzczarba}{BrownSzczarba}) and (\hyperlink{May}{May}).

\begin{prop}
\label{}\hypertarget{}{}
Let $G$ be a simplicial topological group. Then

\begin{enumerate}%
\item $G$ is a globally Kan simplicial topological space;


\item $\bar W G$ is a globally Kan simplicial topological space;


\item $W G \to \bar W G$ is a global Kan fibration.



\end{enumerate}
\end{prop}
\begin{proof}
The first statement appears as (\hyperlink{BrownSzczarba}{BrownSzczarba, theorem 3.8}), the second is noted in (\hyperlink{RobertsStevenson}{RobertsStevenson}), the third as (\hyperlink{BrownSzczarba}{BrownSzczarba, lemma 6.7}).

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
If $G$ is a \hyperlink{WellPointedSimplicialTopologicalGroup}{well-pointed} simplicial topological group, then

\begin{enumerate}%
\item $G$ is a [[nice simplicial topological space|good simplicial topological space]];


\item the [[geometric realization of simplicial topological spaces|geometric realization]] $|G|$ is well-pointed;


\item $\bar W G$ is a [[nice simplicial topological space|proper simplicial topological space]].



\end{enumerate}
\end{prop}
\begin{proof}
The statement about $\bar W G$ is proven in (\hyperlink{RobertsStevenson}{RobertsStevenson}). The other statements are referenced there.

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

\begin{itemize}%
\item [[simplicial topological space]], [[nice simplicial topological space]]


\item \textbf{simplicial topological group}


\item [[geometric realization of simplicial topological spaces]]



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

Basics theory of simplicial topological groups is in

\begin{itemize}%
\item E. H. Brown and R. H. Szczarba, \emph{Continuous cohomology and real homotopy type} , Trans. Amer. Math. Soc. 311 (1989), no. 1, 57 (\href{http://www.ams.org/journals/tran/1989-311-01/S0002-9947-1989-0929667-6/S0002-9947-1989-0929667-6.pdf}{pdf})

\end{itemize}
and

\begin{itemize}%
\item [[Peter May]], \emph{Geometry of iterated loop spaces} , SLNM 271, Springer-Verlag, 1972 (\href{http://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf}{pdf})

\end{itemize}
Their [[principal ∞-bundle]]s and [[geometric realization of simplicial topological spaces|geometric realization]] is discussed in

\begin{itemize}%
\item [[David Roberts]], [[Danny Stevenson]], \emph{Simplicial principal bundles in parametrized spaces}, \href{http://arxiv.org/abs/1203.2460}{arXiv:1203.2460}.

\end{itemize}
Discussion of [[homotopy theory]] over a base $B$ is in

\begin{itemize}%
\item [[Peter May]], J. Sigurdsson, \emph{Parametrized homotopy theory} (\href{http://www.math.uiuc.edu/K-theory/0716/}{web})

\end{itemize}
[[!redirects simplicial topological groups]]

[[!redirects topological simplicial group]] [[!redirects topological simplicial groups]]

[[!redirects well-pointed simplicial topological group]] [[!redirects well pointed simplicial topological group]] [[!redirects well-pointed simplicial topological groups]] [[!redirects well pointed simplicial topological groups]]

[[!redirects well-sectioned simplicial topological group]] [[!redirects well sectioned simplicial topological group]] [[!redirects well-sectioned simplicial topological groups]] [[!redirects well sectioned simplicial topological groups]]



\end{document}