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\begin{document}

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\section*{model structure on topological spaces}

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

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

[[!include model category theory - contents]]

\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{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{QuillenModelStructure}{Classical Quillen Model Structure}\dotfill \pageref*{QuillenModelStructure} \linebreak
\noindent\hyperlink{StromModelStructure}{Hurewicz (or Str\o{}m) Model Structure}\dotfill \pageref*{StromModelStructure} \linebreak
\noindent\hyperlink{mixed_model_structure}{Mixed Model Structure}\dotfill \pageref*{mixed_model_structure} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{OnNiceTopologicalSpaces}{Restriction to convenient categories of topological spaces}\dotfill \pageref*{OnNiceTopologicalSpaces} \linebreak
\noindent\hyperlink{relation_between__and_}{Relation between $Top_{Quillen}$ and $sSet_{Quillen}$}\dotfill \pageref*{relation_between__and_} \linebreak
\noindent\hyperlink{relation_between__and__2}{Relation between $Top_{Quillen}$ and $Top_{Strom}$}\dotfill \pageref*{relation_between__and__2} \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}

Philosophically, [[model category|model structures]] allow one to [[localization|localize]] a [[category]] at a particular collection of [[category with weak equivalences|weak equivalences]], which one would like to formally invert.

For [[topological space]]s, there are two natural candidates for the collection $W$ of weak equivalences:

\begin{enumerate}%
\item the [[weak homotopy equivalence]]s


\item and the [[homotopy equivalence]]s.



\end{enumerate}
Both of these have accompanying model structures. Interestingly, these two model structures can also be combined to form what's known as the \emph{mixed} model structure.

All of these model structures exist not only on the category of all topological spaces, but also on most [[convenient categories of topological spaces]]. Using a nice category instead is sometimes important, such as if we want the model structure to be [[monoidal model category|monoidal]].

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

\hypertarget{QuillenModelStructure}{}\subsubsection*{{Classical Quillen Model Structure}}\label{QuillenModelStructure}

The first, and most prevalent, [[model category]] structure on [[Top]], called the \emph{[[classical model structure on topological spaces]]} (\hyperlink{Quillen67}{Quillen67, II.3}) has

\begin{itemize}%
\item weak equivalences are the [[weak homotopy equivalences]];


\item fibrations are the [[Serre fibrations]], maps which have the [[right lifting property]] with respect to all inclusions of the form $i_0 : D^n \hookrightarrow D^n \times I$ that include the $n$-disk as $D^n \times \{0\}$.


\item cofibrations are the ``[[retracts]] of [[relative cell complexes]]''.



\end{itemize}
This is a [[cofibrantly generated model category]]. The cofibrations $C$ are generated by the set of [[boundary]] inclusions $S^{n-1} \hookrightarrow D^n$ for all $n \in \mathbb{N}$ in the sense that they are the smallest saturated class containing these morphisms. As a consequence, of Quillen's [[small object argument]], all cofibrations have the form described above, where a [[relative cell complex]] is a [[transfinite composition|transfinite composite]] of [[pushouts]] of [[coproducts]] of these generating maps.

This model structure is sometimes called the \textbf{[[Quillen model structure on topological spaces]]} or \textbf{$q$-model structure} on [[Top]].

\hypertarget{StromModelStructure}{}\subsubsection*{{Hurewicz (or Str\o{}m) Model Structure}}\label{StromModelStructure}

A second model structure has

\begin{itemize}%
\item weak equivalences are the [[homotopy equivalences]];


\item fibrations are the \textbf{[[Hurewicz fibration]]s}, which are defined to be maps that have the right lifting property with respect to \emph{all} inclusions $i_0 : A \hookrightarrow A \times I$ for any topological space $A$.


\item cofibrations are determined by these classes and are called the \textbf{closed [[Hurewicz cofibration]]s}.



\end{itemize}
This model structure is sometimes called the \textbf{Hurewicz model structure}, since it uses Hurewicz fibrations and cofibrations, or also the $h$-model structure, where $h$ can stand for either ``Hurewicz'' or ``homotopy equivalence.'' However, it is also sometimes called the [[Strøm model structure]], since it was first proven to exist by [[Arne Strøm]].

\hypertarget{mixed_model_structure}{}\subsubsection*{{Mixed Model Structure}}\label{mixed_model_structure}

From the definitions, [[Hurewicz fibration]]s are necessarily [[Serre fibration]]s. It is well-known that [[homotopy equivalence]]s are [[weak homotopy equivalence]]s. If we write $(C_1, F_1, W_1)$ for the classes of the first model structure and $(C_2, F_2, W_2)$ for the classes of the second, we have $W_2 \subset W_1$ and $F_2 \subset F_1$.

In general given two model structures with these inclusions, we get a third \textbf{[[mixed model structure]]} $(C_m, F_2, W_1)$ where the cofibrations $C_m$ are determined by the other two classes.

On topological spaces, this model structure has

\begin{itemize}%
\item weak equivalences the [[weak homotopy equivalence]]s


\item fibrations the [[Hurewicz fibration]]s;


\item cofibrant spaces (the [[m-cofibrant space]]s) are precisely those spaces that are homotopy equivalent to [[CW complex]]es.



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

\hypertarget{OnNiceTopologicalSpaces}{}\subsubsection*{{Restriction to convenient categories of topological spaces}}\label{OnNiceTopologicalSpaces}

For the discussion of the [[homotopy theory]] given by the model structure on topological spaces, it is necessary or at least useful to pass to [[convenient categories of topological spaces]].

\begin{defn}
\label{}\hypertarget{}{}
Write

\begin{itemize}%
\item $kTop \hookrightarrow Top$ for the [[full subcategory]] of [[k-spaces]];


\item $CGTop \hookrightarrow Top$ for the full subcategory of [[compactly generated spaces]].



\end{itemize}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
There is a [[model category]] structure $kTop_{Quillen}$ on $kTop$ in which a morphism is a cofibration, fibration or weak equivalence, respectively, precisely if it is so under the inclusion $kTop \hookrightarrow Top$. And this inclusion is the [[left adjoint]] in a [[Quillen equivalence]]

\begin{displaymath}
Top_{Quillen} \stackrel{\overset{}{\hookleftarrow}}{\underset{}{\to}}
  kTop_{Quillen}
  \,.
\end{displaymath}
\end{prop}
This appears for instance as (\hyperlink{Hovey}{Hovey, theorem 2.4.23})

\begin{prop}
\label{}\hypertarget{}{}
There is a [[model category]] structure $CGTop_{Quillen}$ on $CGTop$ in which a morphism is a cofibration, fibration or weak equivalence, respectively, precisely if it is so under the inclusion $CGTop \hookrightarrow Top$. And this inclusion is the [[right adjoint]] in a [[Quillen equivalence]]

\begin{displaymath}
CGTop_{Quillen} \stackrel{\overset{w}{\leftarrow}}{\underset{}{\hookrightarrow}}
  kTop_{Quillen}
  \,.
\end{displaymath}
\end{prop}
Notice that $Top_{Quillen}$ is \emph{not} a [[monoidal model category]], because $Top$ itself is not ([[cartesian closed category|cartesian]]) [[closed monoidal category|closed]].

\begin{prop}
\label{}\hypertarget{}{}
Both $kTop_{Quillen}$ and $CGTop_{Quillen}$ are [[symmetric monoidal category|symmetric]] [[monoidal model categories]].

\end{prop}
This appears as (\hyperlink{Hovey}{Hovey, prop. 4.2.11}).

\begin{prop}
\label{}\hypertarget{}{}
In fact $CGTop_{Quillen}$ is a [[cartesian closed model category]]. (see e.g \hyperlink{BergerMoerdijk03}{Berger-Moerdijk 03})

\end{prop}
\hypertarget{relation_between__and_}{}\subsubsection*{{Relation between $Top_{Quillen}$ and $sSet_{Quillen}$}}\label{relation_between__and_}

The \hyperlink{QuillenModelStructure}{Quillen model structure} $Top_{Qullen}$ is [[Quillen equivalence|Quillen equivalent]] to the standard (Quillen) [[model structure on simplicial sets]] via the total [[fundamental infinity-groupoid|singular complex]] and [[geometric realization]] functors.

\begin{displaymath}
(\vert-\vert \dashv Sing)
  : 
  Top_{Quillen} 
    \stackrel{\overset{|-|}{\leftarrow}}{\underset{Sing}{\to}} 
  sSet_{Quillen}
  \,.
\end{displaymath}
Since the standard [[model structure on simplicial sets]] is a [[presentable (infinity,1)-category|presentation]] of the [[(∞,1)-category]] [[∞Grpd]] of [[∞-groupoid]]s realized as [[Kan complex]]es, this identifies topological spaces with [[∞-groupoid]]s in an [[(∞,1)-category|(∞,1)-categorical]] sense. Notably it says that every $\infty$-groupoid is, up to equivalence, the [[fundamental ∞-groupoid]] of some topological space.

This statement is called the [[homotopy hypothesis]] (which here is a theorem). See there for more details.

\hypertarget{relation_between__and__2}{}\subsubsection*{{Relation between $Top_{Quillen}$ and $Top_{Strom}$}}\label{relation_between__and__2}

The identity functor constitutes a [[Quillen adjunction]]

\begin{displaymath}
(Id \dashv Id)
   : 
   Top_{Strom}
    \stackrel{\leftarrow}{\to}
   Top_{Quillen}
\end{displaymath}
between the \hyperlink{QuillenModelStructure}{Quillen model structure} and the \hyperlink{StromModelStructure}{Strom model structure} on $Top$. Here $Top_{Strom} \to Top_{Quillen}$ is the [[Quillen adjunction|right Quillen functor]].

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

\begin{itemize}%
\item [[homotopical structure on C\emph{-algebras]]}

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

The original ``Quillen'' or ``q-'' model structure is due to

\begin{itemize}%
\item [[Dan Quillen]], chapter II, section 3 of \emph{Homotopical algebra}, Lecture Notes in Mathematics \textbf{43}, Springer-Verlag 1967, iv+156 pp.

\end{itemize}
An expository, concise and comprehensive writeup is in

\begin{itemize}%
\item [[Philip Hirschhorn]], \emph{The Quillen model category of topological spaces}, 2015 (\href{http://arxiv.org/abs/1508.01942}{arXiv:1508.01942})

\end{itemize}
Standard textbooks references include

\begin{itemize}%
\item [[Mark Hovey]] \emph{Model categories}


\item Hirschhorn \emph{Model categories and their localizations}.


\item [[Kate Ponto]], [[Peter May]], section 17 of \emph{More concise algebraic topology} (\href{http://www.maths.ed.ac.uk/~aar/papers/mayponto.pdf}{pdf})



\end{itemize}
For the ``Hurewicz,'' ``Str\o{}m,'' or ``h-'' model structure:

\begin{itemize}%
\item [[Arne Strøm]], \emph{The homotopy category is a homotopy category}, Archiv der Mathematik 23 (1972)

\end{itemize}
For the ``mixed'' or ``m-'' model structure:

\begin{itemize}%
\item Michael Cole, \emph{Mixing model structures}, Topology Appl. 153 no. 7 (2006) \href{http://dx.doi.org/10.1016/j.topol.2005.02.004}{doi}.

\end{itemize}
The generalization to the [[model structure on topological operads]] is due to

\begin{itemize}%
\item [[Clemens Berger]], [[Ieke Moerdijk]], \emph{Axiomatic homotopy theory for operads}, Comment. Math. Helv. Vol. 78 (2003), no. 4 (\href{http://arxiv.org/abs/math/0206094}{arXiv:math/0206094})

\end{itemize}
[[!redirects model category of topological spaces]]

[[!redirects model structures on topological spaces]] [[!redirects model structures on Top]]



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