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\begin{document}

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\section*{Borel model structure}

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

\hypertarget{model_category_theory}{}\paragraph*{{Model category theory}}\label{model_category_theory}

[[!include model category theory - contents]]

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

[[!include group theory - 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_slicing_over_}{Relation to slicing over $\bar W G$}\dotfill \pageref*{relation_to_slicing_over_} \linebreak
\noindent\hyperlink{cofibrant_replacement_and_homotopy_quotientsfixed_points}{Cofibrant replacement and homotopy quotients/fixed points}\dotfill \pageref*{cofibrant_replacement_and_homotopy_quotientsfixed_points} \linebreak
\noindent\hyperlink{relation_to_the_fine_model_structure_of_equivariant_homotopy_theory}{Relation to the fine model structure of equivariant homotopy theory}\dotfill \pageref*{relation_to_the_fine_model_structure_of_equivariant_homotopy_theory} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Given a [[simplicial group]] $G_\bullet$, the \emph{Borel model structure} is a [[model category]] structure on the [[category]] of [[simplicial sets]] equipped with $G$-[[action]] which presents the [[(∞,1)-category]] of [[∞-actions]] of the [[∞-group]] (see there) presented by $G$.

In the context of [[equivariant homotopy theory]] this is also called the ``coarse model structure'' (e.g. \hyperlink{Guillou}{Guillou, section 5}), since it is not equivalent to the homotopy theory of [[G-spaces]] which enters [[Elmendorf's theorem]].

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

Writing $\mathbf{B}G_\bullet$ for the one-object [[sSet-enriched category]] ([[simplicial groupoid]]) whose [[hom-object]] is $G_\bullet$. Write $sSet^{\mathbf{B}G_\bullet}$ for the corresponding [[sSet]]-[[enriched functor category]].

This carries the projective global [[model structure on functors]] ([[model structure on simplicial presheaves]]) $(sSet^{\mathbf{B}G_\bullet})_{proj}$. This is the Borel model structure (\hyperlink{DDK80}{DDK 80}).

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

\hypertarget{relation_to_slicing_over_}{}\subsubsection*{{Relation to slicing over $\bar W G$}}\label{relation_to_slicing_over_}

The central theorem of (\hyperlink{DDK80}{DDK 80}) is that the Borel model structure is [[Quillen equivalence|Quillen euqivalent]] to the [[slice model structure]] of the standard [[model structure on simplicial sets]] over the model $\bar W G_\bullet$ (see at \emph{[[simplicial group]]} for notation and details) for the [[delooping]] of $G_\bullet$.

This kind of relation is discussed in more detail at \emph{[[∞-action]]}.

\hypertarget{cofibrant_replacement_and_homotopy_quotientsfixed_points}{}\subsubsection*{{Cofibrant replacement and homotopy quotients/fixed points}}\label{cofibrant_replacement_and_homotopy_quotientsfixed_points}

The cofibrations $i \colon X \to Y$ in $(sSet^{\mathbf{B}G_\bullet})_{proj}$ are precisely those maps such that

\begin{enumerate}%
\item the underlying homomorphism of [[simplicial sets]] is a [[monomorphism]];


\item the $G_\bullet$-action is relatively [[free action]], i.e. [[free action|free]] on all [[simplices]] not in the [[image]] of $i$.



\end{enumerate}
This is part of (\hyperlink{DDK80}{DDK 80, proposition 2.2. ii)}). Also (\hyperlink{Guillou}{Guillou, prop. 5.3}).

In particular this means that an object is cofibrant if the $G_\bullet$-[[action]] on it is [[free action|free]]. Hence cofibrant replacement is in particular given by forming the product with the model $W G_\bullet$ for the total space of the [[universal principal bundle]] over $G_\bullet$ (see at \emph{[[simplicial group]]} for notation and more details).

It follows that for $X,A\in (sSet^{\mathbf{B}G_\bullet})_{proj}$ the [[derived hom space]]

\begin{displaymath}
R Hom_G(X,A)
\end{displaymath}
models the $G$-[[equivariant cohomology]] of $X$ with [[coefficients]] in $A$.

In particular,if $A$ is fibrant (the underlying simplicial set is a [[Kan complex]]) then

\begin{enumerate}%
\item if the $G_\bullet$-action on $A$ is trivial, then

\begin{displaymath}
R Hom_G(X,A) \simeq Hom_G(W G \times X , A) \simeq Hom(W G \times_G X, A)
\end{displaymath}
is equivalently maps of [[simplicial sets]] out of the [[Borel construction]] on $X$;


\item if $X= \ast$ is the point then

\begin{displaymath}
R Hom_G(X,A) \simeq Hom_G(W G, A) \simeq Hom(\bar W G , A)  \simeq A^{h G}
\end{displaymath}
is the [[homotopy fixed points]] of $A$.



\end{enumerate}
\hypertarget{relation_to_the_fine_model_structure_of_equivariant_homotopy_theory}{}\subsubsection*{{Relation to the fine model structure of equivariant homotopy theory}}\label{relation_to_the_fine_model_structure_of_equivariant_homotopy_theory}

The identity functor [[Quillen adjunction]] between the Borel model structure and [[equivariant homotopy theory]] (\hyperlink{Guillou}{Guillou, section 5}).

The left adjoint is

\begin{displaymath}
L = id \;\colon\; G_\bullet Act_{coarse} \longrightarrow G_\bullet Act_{fine}
\end{displaymath}
from the Borel model structure to the genuine [[equivariant homotopy theory]].

Because:

First of all, by (\hyperlink{Guillou}{Guillou, theorem 3.12, example 4.2}) $sSet^{\mathbf{B}G_\bullet}$ does carry a fine model structure. By (\hyperlink{Guillou}{Guillou, last line of page 3}) the fibrations and weak equivalences here are those maps which are ordinary fibrations and weak equivalences, respectively, on $H$-[[fixed point]] simplicial sets, for all subgroups $H$. This includes in particular the trivial subgroup and hence the identity functor

\begin{displaymath}
R = id \;\colon\; G_\bullet Act_{fine} \longrightarrow G_\bullet Act_{coarse}
\end{displaymath}
is right Quillen.

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

The model structure, the characterization of its cofibrations, and its equivalence to the [[slice model structure]] of $sSet$ over $\bar W G$ is due to

\begin{itemize}%
\item E. Dror, [[William Dwyer]], and [[Daniel Kan]], \emph{Equivariant maps which are self homotopy equivalences}, Proc. Amer. Math. Soc. 80 (1980), no. 4, 670--672 (\href{http://www.jstor.org/stable/2043448}{JSTOR})

\end{itemize}
This is mentioned for instance as exercise 4.2in

\begin{itemize}%
\item [[William Dwyer]], \emph{Homotopy theory of classifying spaces}, Lecture notes Copenhagen (June, 2008) \href{http://www.math.ku.dk/~jg/homotopical2008/Dwyer.CopenhagenNotes.pdf}{pdf}

\end{itemize}
Discussion in relation to the ``fine'' model structure of [[equivariant homotopy theory]] which appears in [[Elmendorf's theorem]] is in

\begin{itemize}%
\item [[Bert Guillou]], \emph{A short note on models for equivariant homotopy theory} (\href{http://www.math.uiuc.edu/~bertg/EquivModels.pdf}{pdf})

\end{itemize}
Discussion with the model of [[∞-groups]] by [[simplicial groups]] replaced by groupal [[Segal spaces]] is in

\begin{itemize}%
\item [[Matan Prasma]], \emph{Segal Group Actions} (\href{http://arxiv.org/abs/1311.4749}{arXiv:1311.4749})

\end{itemize}
Discussion of a [[global equivariant homotopy theory|globalized]] model structure for actions of all simplicial groups is in

\begin{itemize}%
\item [[Yonatan Harpaz]], [[Matan Prasma]], section 6.2 of \emph{The Grothendieck construction for model categories} (\href{http://arxiv.org/abs/1404.1852}{arXiv:1404.1852})

\end{itemize}


\end{document}