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\begin{document}

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\section*{model structure on simplicial sets}

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

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

[[!include model category theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{classical_model_structure}{Classical Model Structure}\dotfill \pageref*{classical_model_structure} \linebreak
\noindent\hyperlink{characterisations_of_weak_homotopy_equivalences}{Characterisations of weak homotopy equivalences}\dotfill \pageref*{characterisations_of_weak_homotopy_equivalences} \linebreak
\noindent\hyperlink{relation_to_the_model_structure_on_strict_groupoids}{Relation to the model structure on strict $\infty$-groupoids}\dotfill \pageref*{relation_to_the_model_structure_on_strict_groupoids} \linebreak
\noindent\hyperlink{constructive_version}{Constructive version}\dotfill \pageref*{constructive_version} \linebreak
\noindent\hyperlink{joyals_model_structure}{Joyal's Model Structure}\dotfill \pageref*{joyals_model_structure} \linebreak
\noindent\hyperlink{comparison}{Comparison}\dotfill \pageref*{comparison} \linebreak
\noindent\hyperlink{fibrant_replacement}{Fibrant replacement}\dotfill \pageref*{fibrant_replacement} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{properness}{Properness}\dotfill \pageref*{properness} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

[[simplicial set|Simplicial sets]] are the archetypical combinatorial ``[[model category|model]]'' for the [[(∞,1)-category]] of (compactly generated weakly Hausdorff) [[topological space]]s and equivalently that of [[∞-groupoid]]s, as well as a standard model for the [[(∞,1)-category of (∞,1)-categories]] [[(∞,1)Cat]] itself.

This statement is made precise by the existence of the structure of a [[model category]] on [[sSet]], called the \textbf{[[classical model structure on simplicial sets]]} that is a [[presentable (infinity,1)-category|presentation]] for the [[(infinity,1)-category]] [[Top]], as well as the \textbf{Joyal model structure} which similarly is a presentation of the $(\infty,1)$-category $(\infty,1)Cat$.

\hypertarget{classical_model_structure}{}\subsection*{{Classical Model Structure}}\label{classical_model_structure}

The [[classical model structure on simplicial sets]], $sSet_{Quillen}$, has the following distinguished classes of morphisms:

\begin{defn}
\label{}\hypertarget{}{}
\begin{itemize}%
\item The \textbf{cofibrations} $C$ are simply the [[monomorphism]]s $f : X \to Y$ which are precisely the levelwise injections, i.e. the morphisms of simplicial sets such that $f_n : X_n \to Y_n$ is an injection of sets for all $n \in \mathbb{N}$.


\item The \textbf{weak equivalences} $W$ are \textbf{weak homotopy equivalences}, i.e., morphisms whose [[geometric realization]] is a weak homotopy equivalence of topological spaces.



\end{itemize}
\end{defn}
\begin{itemize}%
\item The \textbf{fibrations} $F$ are the \textbf{[[Kan fibration]]s}, i.e., maps $f : X \to Y$ which have the [[right lifting property]] with respect to all [[horn]] inclusions.

\begin{displaymath}
\itexarray{
    \Lambda^k[n] &\to& X
    \\
    \downarrow &{}^\exists\nearrow& \downarrow^f 
    \\
    \Delta[n] &\to& Y
  }
  \,.
\end{displaymath}

\item A morphism $f : X \to Y$ of fibrant simplicial sets / [[Kan complex]]es is a weak equivalence precisely if it induces an [[isomorphism]] on all [[simplicial homotopy groups]].


\item All simplicial sets are cofibrant with respect to this model structure.


\item The fibrant objects are precisely the [[Kan complex|Kan complexes]].



\end{itemize}
\begin{prop}
\label{}\hypertarget{}{}
The \textbf{acyclic fibrations} (i.e. the maps that are both fibrations as well as weak equivalences) between [[Kan complex]]es are precisely the morphisms $f : X \to Y$ that have the [[right lifting property]] with respect to all inclusions $\partial \Delta[n] \hookrightarrow \Delta[n]$ of boundaries of $n$-simplices into their $n$-simplices

\begin{displaymath}
\itexarray{
      \partial \Delta[n] &\to& X
      \\
      \downarrow &{}^\exists\nearrow& \downarrow^f 
      \\
      \Delta[n] &\to& Y
    }
    \,.
\end{displaymath}
\end{prop}
This appears spelled out for instance as (\hyperlink{GoerssJardine}{Goerss-Jardine, theorem 11.2}).

In fact:

\begin{prop}
\label{}\hypertarget{}{}
$sSet_{Quillen}$ is a [[cofibrantly generated model category]] with

\begin{itemize}%
\item generating cofibrations the [[boundary]] inclusions $\partial \Delta[n] \to \Delta[n]$;


\item generating acyclic cofibrations the [[horn]] inclusions $\Lambda^i[n] \to \Delta[n]$.



\end{itemize}
\end{prop}
\begin{theorem}
\label{}\hypertarget{}{}
The [[singular simplicial complex]]-functor and [[geometric realization]]

\begin{displaymath}
({\vert -\vert}\dashv Sing)
  : 
  Top_{Quillen}
  \stackrel{\overset{{\vert -\vert}}{\leftarrow}}{\underset{Sing}{\to}}
  sSet_{Quillen}
\end{displaymath}
constitutes a [[Quillen equivalence]] with the standard [[Quillen model structure on topological spaces]].

\end{theorem}
For more on this see [[homotopy hypothesis]].

\hypertarget{characterisations_of_weak_homotopy_equivalences}{}\subsubsection*{{Characterisations of weak homotopy equivalences}}\label{characterisations_of_weak_homotopy_equivalences}

\begin{theorem}
\label{}\hypertarget{}{}
Let $W$ be the smallest class of morphisms in $sSet$ satisfying the following conditions:

\begin{enumerate}%
\item The class of monomorphisms that are in $W$ is closed under [[pushout]], [[transfinite composition]], and [[retracts]].
\item $W$ has the [[two-out-of-three]] property in $sSet$ and contains all the [[isomorphisms]].
\item For all natural numbers $n$, the unique morphism $\Delta [n] \to \Delta [0]$ is in $W$.

\end{enumerate}
Then $W$ is the class of weak homotopy equivalences.

\end{theorem}
\begin{proof}
\begin{itemize}%
\item First, notice that the horn inclusions $\Lambda^0 [1] \hookrightarrow \Delta [1]$ and $\Lambda^1 [1] \hookrightarrow \Delta [1]$ are in $W$.
\item Suppose that the horn inclusion $\Lambda^k [m] \hookrightarrow \Delta [m]$ is in $W$ for all $m \lt n$ and all $0 \le k \le m$. Then for $0 \le l \le n$, the horn inclusion $\Lambda^l [n] \hookrightarrow \Delta [n]$ is also in $W$.
\item Quillen's [[small object argument]] then implies all the trivial cofibrations are in $W$.
\item If $p : X \to Y$ is a trivial Kan fibration, then its right lifting property implies there is a morphism $s : Y \to X$ such that $p \circ s = id_Y$, and the two-out-of-three property implies $s : Y \to X$ is a trivial cofibration. Thus every trivial Kan fibration is also in $W$.
\item Every weak homotopy equivalence factors as $p \circ i$ where $p$ is a trivial Kan fibration and $i$ is a trivial cofibration, so every weak homotopy equivalence is indeed in $W$.
\item Finally, noting that the class of weak homotopy equivalences satisfies the conditions in the theorem, we deduce that it is the \emph{smallest} such class.

\end{itemize}
\end{proof}
As a corollary, we deduce that the classical model structure on $sSet$ is the smallest (in terms of weak equivalences) model structure for which the cofibrations are the monomorphisms and the weak equivalences include the (combinatorial) homotopy equivalences.

\begin{prop}
\label{}\hypertarget{}{}
Let $\pi_0 : sSet \to Set$ be the connected components functor, i.e. the left adjoint of the constant functor $cst : Set \to sSet$. A morphism $f : Z \to W$ in $sSet$ is a weak homotopy equivalence if and only if the induced map

\begin{displaymath}
\pi_0 K^f : \pi_0 K^W \to \pi_0 K^Z
\end{displaymath}
is a bijection for all \emph{Kan complexes} $K$.

\end{prop}
\begin{proof}
One direction is easy: if $K$ is a Kan complex, then axiom SM7 for [[simplicial model categories]] implies the functor $K^{(-)} : sSet^{op} \to sSet$ is a right [[Quillen functor]], so Ken Brown's lemma implies it preserves all weak homotopy equivalences; in particular, $\pi_0 K^{(-)} : sSet^{op} \to Set$ sends weak homotopy equivalences to bijections.

Conversely, when $K$ is a Kan complex, there is a natural bijection between $\pi_0 K^X$ and the hom-set $Ho (sSet) (X, K)$, and thus by the [[Yoneda lemma]], a morphism $f : Z \to W$ such that the induced morphism $\pi_0 K^W \to \pi_0 K^Z$ is a bijection for all Kan complexes $K$ is precisely a morphism that becomes an isomorphism in $Ho (sSet)$, i.e. a weak homotopy equivalence.

\end{proof}
\hypertarget{relation_to_the_model_structure_on_strict_groupoids}{}\subsubsection*{{Relation to the model structure on strict $\infty$-groupoids}}\label{relation_to_the_model_structure_on_strict_groupoids}

\begin{quote}%
under construction


\end{quote}
Recall the [[model structure on strict omega-groupoids]] and the [[omega-nerve]] operation

\begin{displaymath}
N : Str \infty Grpd \to Kan Complx
  \,.
\end{displaymath}
\begin{quote}%
this ought to be a Quillen functor, but is it?


\end{quote}
As a warmup, let $C, D$ be ordinary [[groupoid]]s and $N(C)$, $N(D)$ their ordinary [[nerve]]s. We'd like to show in detail that

\begin{prop}
\label{}\hypertarget{}{}
A [[functor]] $F : C \to D$ is

\begin{itemize}%
\item [[k-surjective functor|k-surjective]] for all $k$ and hence a surjective [[equivalence of categories]] precisely if under the [[nerve]] $N(F) : N(C) \to N(D)$ it induces an acyclic fibration of Kan complexes;

\end{itemize}
\end{prop}
\begin{proof}
We know that both $N(C)$ and $N(D)$ are Kan complexes. By the above theorem it suffices to show that $N(f)$ being a surjective equivalence is the same as having all lifts

\begin{displaymath}
\itexarray{
    \partial \Delta[n] &\to& N(C)
    \\
    \downarrow &{}^\exists\nearrow& \downarrow^{N(F)}
    \\
    \Delta[n] &\to& N(D)
  }
  \,.
\end{displaymath}
We check successively what this means for increasing $n$:

\begin{itemize}%
\item $n= 0$. In degree 0 the boundary inclusion is that of the empty set into the [[point]] $\emptyset \hookrightarrow {*}$. The lifting property in this case amounts to saying that every point in $N(D)$ lifts through $N(F)$.

\begin{displaymath}
\itexarray{
    \emptyset &\to& N(C)
    \\
    \downarrow &{}^\exists\nearrow& \downarrow^{N(F)}
    \\
    {*} &\to& N(D)
  }
  \Leftrightarrow
  \itexarray{
    && N(C)
    \\
    &{}^\exists\nearrow& \downarrow^{N(F)}
    \\
    {*} &\to& N(D)
  }
  \,.
\end{displaymath}
This precisely says that $N(F)$ is surjective on 0-cells and hence that $F$ is surjective on objects.


\item $n=1$. In degree 1 the boundary inclusion is that of a pair of points as the endpoints of the interval $\{\circ, \bullet\} \hookrightarrow \{\circ \to \bullet\}$. The lifting property here evidently is equivalent to saying that for all objects $a,b \in Obj(C)$ all elements in $Hom(F(a),F(b))$ are hit. Hence that $F$ is a [[full functor]].


\item $n=2$. In degree 2 the boundary inclusion is that of the triangle as the boundary of a filled triangle. It is sufficient to restrict attention to the case that the map $\partial \Delta[2] \to N(C)$ sends the top left edge of the triangle to an identity. Then the lifting property here evidently is equivalent to saying that for all objects $a,b \in Obj(C)$ the map $F_{a,b} : Hom(a,b) \to Hom(F(a),F(b))$ is injective. Hence that $F$ is a [[faithful functor]].

\begin{displaymath}
\left(
  \itexarray{
    && b 
    \\
    & {}^{Id_a}\nearrow && \searrow^{f}
    \\
    a &&\stackrel{g}{\to}&& b
  }
  \right)
  \stackrel{N(F)}{\mapsto}
  \left(
  \itexarray{
    && b 
    \\
    & {}^{Id_a}\nearrow &\Downarrow^=& \searrow^{F(f)}
    \\
    a &&\stackrel{F(g)}{\to}&& b
  }
  \right)
\end{displaymath}


\end{itemize}
\end{proof}
\hypertarget{constructive_version}{}\subsubsection*{{Constructive version}}\label{constructive_version}

The original [[proofs]] of the existence of the [[classical model structure on simplicial sets]] are based in [[classical mathematics]] as they use the [[principle of excluded middle]] and the [[axiom of choice]], and are hence not valid in [[constructive mathematics]]. This becomes more than a philosophical issue with the relevance of this [[model category]]-[[structure]] in [[homotopy type theory]], where [[internalization]] into the [[type theory]] requires [[constructive mathematics|constructive]] methods for interpreting [[proofs as programs]].

A constructively valid model structure on simplicial sets and coinciding with the [[classical model structure on simplicial sets|classical model structure]] if [[excluded middle]] and [[axiom of choice]] are assumed was found in \href{constructive+model+structure+on+simplicial+sets#Henry19}{Henry 19}. Alternative simpler proofs were found in \href{constructive+model+structure+on+simplicial+sets#GambinoSattlerSzumilo19}{Gambino-Sattler-Szumiło 19}.

See at \emph{[[constructive model structure on simplicial sets]]}.

\hypertarget{joyals_model_structure}{}\subsection*{{Joyal's Model Structure}}\label{joyals_model_structure}

There is a second model structure on $sSet$ -- the \textbf{[[model structure for quasi-categories]]} $sSet_{Joyal}$ -- which is different (not [[Quillen equivalence|Quillen equivalent]]) to the classical one, due to [[Andre Joyal]], with the following distinguished classes of morphisms:

\begin{itemize}%
\item The cofibrations $C$ are monomorphisms, equivalently, levelwise injections.


\item The weak equivalences $W$ are \textbf{[[equivalence of quasi-categories|weak categorical equivalences]]}, which are morphisms $u : A \rightarrow B$ of simplicial sets such that the induced map $u^* : X^B \rightarrow X^A$ of internal-homs for all [[quasi-category|quasi-categories]] $X$ induces an isomorphism when applying the functor $\tau_0$ that takes a simplicial set to the set of isomorphism classes of objects of its fundamental category.


\item The fibrations $F$ are called variously \textbf{isofibrations} or \textbf{[[inner Kan fibration|quasi-fibration]]}. As always, these are determined by the classes $C$ and $W$. Quasi-fibrations between weak Kan complexes have a simple description; they are precisely the [[inner Kan fibrations]], the maps that have the right lifting property with respect to the inner [[horn]] inclusions and also the inclusion $j_0 : * \rightarrow J$ where $*$ is the terminal simplicial set and $J$ is the nerve of the groupoid on two objects with one non-trivial isomorphism.



\end{itemize}
All objects are cofibrant. The fibrant objects are precisely the [[quasi-categories]].

This model structure is cofibrantly generated. The generating cofibrations are the set $I$ described above. There is no known explicit description for the generating trivial cofibrations.

Importantly, this model structure is Quillen equivalent to several alternative model structures for the `'homotopy theory of homotopy theories`` such as that on the category of [[simplicially enriched category|simplicially enriched categories]].

\hypertarget{comparison}{}\subsubsection*{{Comparison}}\label{comparison}

Every weak categorical equivalence is a weak homotopy equivalence. Since both model structures have the same cofibrations, it follows that the classical model structure is a [[Bousfield localization]] of Joyal's model structure.

The Quillen model structure is the left [[Bousfield localization of model categories|Bousfield localization]] of $sSet_{Joyal}$ at the outer [[horn]] inclusions.

\hypertarget{fibrant_replacement}{}\subsection*{{Fibrant replacement}}\label{fibrant_replacement}

Fibrant replacement in $sSet_{Quillen}$ models the process of \emph{$\infty$-groupoidification}, of freely inverting all [[k-morphism]]s in a simplicial set. Techniques for fibrant replacements $sSet_{Quillen}$ are discussed at

\begin{itemize}%
\item [[Kan fibrant replacement]].

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

\begin{itemize}%
\item [[model structure on topological spaces]], [[model structure on simplicial presheaves]]


\item [[model structure on semi-simplicial sets]]



\end{itemize}
\hypertarget{properness}{}\subsection*{{Properness}}\label{properness}

The Quillen model structure is both left and right [[proper model category|proper]]. Left properness is automatic since all objects are cofibrant. Right properness follows from the following argument: it suffices to show that there is a functor $R$ which (1) preserves fibrations, (2) preserves pullbacks of fibrations, (3) preserves and reflects weak equivalences, and (4) lands in a category in which the pullback of a weak equivalence along a fibration is a weak equivalence. For if so, we can apply $R$ to the pullback of a fibration along a weak equivalence to get another such pullback in the codomain of $R$, which is a weak equivalence, and hence the original pullback was also a weak equivalence. Two such functors $R$ are

\begin{itemize}%
\item geometric realization $sSet \to Top$, where $Top$ denotes a sufficiently [[convenient category of topological spaces]] (e.g. the category of [[k-spaces]] suffices) and
\item $Ex^\infty : sSet \to Kan$, where $Kan$ is the category of [[Kan complexes]].

\end{itemize}
This can be found, for instance, in II.8.6--7 of \hyperlink{GoerssJardine}{Goerss-Jardine}. Another proof can be found in \hyperlink{Moss}{Moss}, and a different proof of properness can be found in \hyperlink{Cisinski06}{Cisinski, Prop. 2.1.5}.

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

Dan Quillen's original proof in

\begin{itemize}%
\item [[Dan Quillen]], \emph{Homotopical Algebra}, LNM 43, Springer, (1967)

\end{itemize}
of the [[classical model structure on simplicial sets]] is purely combinatorial (i.e. does not use topological spaces): he uses the theory of [[minimal Kan fibrations]], the fact that the latter are fiber bundles, as well as the fact that the [[classifying space]] of a [[simplicial group]] is a [[Kan complex]]. This proof has been rewritten several times in the literature: at the end of

\begin{itemize}%
\item [[Israel Gelfand]], [[Yuri Manin]], \emph{Methods of Homological Algebra}, Springer, 1996

\end{itemize}
as well as in

\begin{itemize}%
\item [[André Joyal]], [[Myles Tierney]] \emph{An introduction to simplicial homotopy theory}, 2005 (\href{http://hopf.math.purdue.edu/cgi-bin/generate?/Joyal-Tierney/JT-chap-01}{web})

\end{itemize}
A proof (in fact two variants of it) using the [[Kan fibrant replacement]] $Ex^\infty$ functor is given in section 2 of

\begin{itemize}%
\item [[Denis-Charles Cisinski]], \emph{[[joyalscatlab:Les préfaisceaux comme type d'homotopie]]}, Ast\'e{}risque, Volume 308, Soc. Math. France (2006), 392 pages (\href{http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf}{pdf})

\end{itemize}
which discusses the topic as a special case of a \emph{[[Cisinski model structure]]}.

The fun part is not that much about the existence of model structure, but to prove that the fibrations are precisely the [[Kan fibrations]] (and also to prove all the good properties of $Ex^\infty$ without using topological spaces); for two different proofs of this fact using $Ex^\infty$, see Prop. 2.1.41 as well as Scholium 2.3.21 for an alternative). For the rest, everything was already in the book of Gabriel and Zisman, for instance.

Another approach also using $Ex^\infty$ is in

\begin{itemize}%
\item Sean Moss, \emph{Another approach to the Kan-Quillen model structure}, \href{http://arxiv.org/abs/1506.04887}{arXiv}.

\end{itemize}
Other standard textbook references for the classical model structure are

\begin{itemize}%
\item [[Paul Goerss]], [[Rick Jardine]], \emph{[[Simplicial homotopy theory]]} (Birkh\"a{}user) (\href{http://www.maths.abdn.ac.uk/~bensondj/html/archive/goerss-jardine.html}{ps}).


\item [[Mark Hovey]], \emph{Model categories}



\end{itemize}
For references on the Joyal model structure see \emph{[[model structure for quasi-categories]]}.

As a [[categorical semantics]] for [[homotopy type theory]], the model structure on simplicial sets is considered in

\begin{itemize}%
\item [[Chris Kapulkin]], [[Peter LeFanu Lumsdaine]], [[Vladimir Voevodsky]], (\href{http://arxiv.org/abs/1203.2553}{arXiv:1203.2553})

\end{itemize}


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