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\begin{document}

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\section*{classical model structure on simplicial sets}

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

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

[[!include topology - contents]]

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

[[!include model category theory - 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{background_on_combinatorial_topology}{Background on combinatorial topology}\dotfill \pageref*{background_on_combinatorial_topology} \linebreak
\noindent\hyperlink{simplicial_sets}{Simplicial sets}\dotfill \pageref*{simplicial_sets} \linebreak
\noindent\hyperlink{simplicial_homotopy}{Simplicial homotopy}\dotfill \pageref*{simplicial_homotopy} \linebreak
\noindent\hyperlink{kan_fibrations}{Kan fibrations}\dotfill \pageref*{kan_fibrations} \linebreak
\noindent\hyperlink{geometric_realization}{Geometric realization}\dotfill \pageref*{geometric_realization} \linebreak
\noindent\hyperlink{TheClassicalModelStructure}{The classical model structure $sSet_{Quillen}$}\dotfill \pageref*{TheClassicalModelStructure} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{basic_properties}{Basic properties}\dotfill \pageref*{basic_properties} \linebreak
\noindent\hyperlink{Properness}{Properness}\dotfill \pageref*{Properness} \linebreak
\noindent\hyperlink{quillen_equivalence_with_}{Quillen equivalence with $Top_{Quillen}$}\dotfill \pageref*{quillen_equivalence_with_} \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 \emph{classical model structure on simplicial sets} or \emph{Kan-Quillen model structure} , $sSet_{Quillen}$ (\hyperlink{Quillen67}{Quillen 67, II.3}) is a [[model category]] structure on the [[category]] [[sSet]] of [[simplicial sets]] which represents the standard classical [[homotopy theory]].

Its [[weak equivalences]] are the [[weak homotopy equivalences]] ([[isomorphisms]] on [[simplicial homotopy groups]]), its [[fibrations]] are the [[Kan fibrations]] and its [[cofibrations]] are the [[monomorphisms]] (degreewise injections).

The [[singular simplicial complex]]/[[geometric realization]] [[nerve and realization|adjunction]] constitutes a [[Quillen equivalence]] between $sSet_{Quillen}$ and $Top_{Quillen}$, the [[classical model structure on topological spaces]]. This is sometimes called part of the statement of the \emph{[[homotopy hypothesis]]} \href{homotopy+hypothesis#ForKanComplexes}{for Kan complexes}. In the language of [[(∞,1)-category theory]] this means that $sSet_{Quillen}$ and $Top_{Quillen}$ both are [[presentable (∞,1)-category|presentations]] of the [[(∞,1)-category]] [[∞Grpd]] of [[∞-groupoids]].

There are also other model structures on [[sSet]] itself, see at \emph{[[model structure on simplicial sets]]} for more. This entry here focuses on just the standard classical model structure.

\hypertarget{background_on_combinatorial_topology}{}\subsection*{{Background on combinatorial topology}}\label{background_on_combinatorial_topology}

This section reviews basics of the theory of [[simplicial sets]] (the modern version of the original ``combinatorial topology'') necessary to define, verify and analyse the classical model category structure on simplicial sets, \hyperlink{TheClassicalModelStructure}{below}. See also at \emph{[[simplicial homotopy theory]]}.

\hypertarget{simplicial_sets}{}\subsubsection*{{Simplicial sets}}\label{simplicial_sets}

The concept of [[simplicial sets]] is secretly well familiar already in basic [[algebraic topology]]: it reflects just the abstract structure carried by the [[singular simplicial complexes]] of [[topological spaces]], as in the definition of [[singular homology]] and [[singular cohomology]].

Conversely, every simplicial set may be [[geometric realization|geometrically realized]] as a topological space. These two [[adjoint]] operations turn out to exhibit the homotopy theory of simplicial sets as being equivalent ([[Quillen equivalence|Quillen equivalent]]) to the homotopy theory of topological spaces. For some purposes, working in [[simplicial homotopy theory]] is preferable over working with topological homotopy theory.

\begin{defn}
\label{TopologicalSimplex}\hypertarget{TopologicalSimplex}{}
For $n \in \mathbb{N}$, the \textbf{\href{simplex#TopologicalSimplex}{topological n-simplex}} is, up to [[homeomorphism]], the [[topological space]] whose underlying set is the subset

\begin{displaymath}
\Delta^n \coloneqq 
  \{
    \vec x \in \mathbb{R}^{n+1}
    |
    \sum_{i = 0 }^n x_i = 1 \; and \;
    \forall i . x_i \geq 0 
  \}
  \subset \mathbb{R}^{n+1}
\end{displaymath}
of the [[Cartesian space]] $\mathbb{R}^{n+1}$, and whose topology is the [[nLab:subspace topology]] induces from the canonical topology in $\mathbb{R}^{n+1}$.

\end{defn}
\begin{example}
\label{}\hypertarget{}{}
For $n = 0$ this is the [[point]], $\Delta^0 = *$.

For $n = 1$ this is the standard [[interval object]] $\Delta^1 = [0,1]$.

For $n = 2$ this is the filled triangle.

For $n = 3$ this is the filled tetrahedron.

\end{example}
\begin{defn}
\label{FaceInclusionInBarycentricCoords}\hypertarget{FaceInclusionInBarycentricCoords}{}
For $n \in \mathbb{N}$, $\n \geq 1$ and $0 \leq k \leq n$, the \textbf{$k$th $(n-1)$-face (inclusion)} of the topological $n$-simplex, def. \ref{TopologicalSimplex}, is the subspace inclusion

\begin{displaymath}
\delta_k : \Delta^{n-1} \hookrightarrow \Delta^n
\end{displaymath}
induced under the coordinate presentation of def. \ref{TopologicalSimplex}, by the inclusion

\begin{displaymath}
\mathbb{R}^n \hookrightarrow \mathbb{R}^{n+1}
\end{displaymath}
which ``omits'' the $k$th canonical coordinate:

\begin{displaymath}
(x_0, \cdots , x_{n-1}) \mapsto (x_0, \cdots, x_{k-1} , 0 , x_{k}, \cdots, x_n)
  \,.
\end{displaymath}
\end{defn}
\begin{example}
\label{}\hypertarget{}{}
The inclusion

\begin{displaymath}
\delta_0 : \Delta^0 \to \Delta^1
\end{displaymath}
is the inclusion

\begin{displaymath}
\{1\} \hookrightarrow [0,1]
\end{displaymath}
of the ``right'' end of the standard interval. The other inclusion

\begin{displaymath}
\delta_1 : \Delta^0 \to \Delta^1
\end{displaymath}
is that of the ``left'' end $\{0\} \hookrightarrow [0,1]$.

\end{example}


(graphics taken from \href{https://ncatlab.org/nlab/show/simplicial+set#Friedman08}{Friedman 08})

\begin{defn}
\label{DegeneracyProjectionsInBarycentricCoords}\hypertarget{DegeneracyProjectionsInBarycentricCoords}{}
For $n \in \mathbb{N}$ and $0 \leq k \lt n$ the \textbf{$k$th degenerate $(n)$-simplex (projection)} is the surjective map

\begin{displaymath}
\sigma_k : \Delta^{n} \to \Delta^{n-1}
\end{displaymath}
induced under the barycentric coordinates of def. \ref{TopologicalSimplex} under the surjection

\begin{displaymath}
\mathbb{R}^{n+1} \to \mathbb{R}^n
\end{displaymath}
which sends

\begin{displaymath}
(x_0, \cdots, x_n) \mapsto (x_0, \cdots, x_{k} + x_{k+1}, \cdots, x_n)
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{SingularSimplex}\hypertarget{SingularSimplex}{}
For $X \in$ [[Top]] and $n \in \mathbb{N}$, a \textbf{singular $n$-simplex} in $X$ is a [[continuous map]]

\begin{displaymath}
\sigma : \Delta^n \to X
\end{displaymath}
from the topological $n$-simplex, def. \ref{TopologicalSimplex}, to $X$.

Write

\begin{displaymath}
(Sing X)_n \coloneqq Hom_{Top}(\Delta^n , X)
\end{displaymath}
for the set of singular $n$-simplices of $X$.

\end{defn}


(graphics taken from \href{https://ncatlab.org/nlab/show/simplicial+set#Friedman08}{Friedman 08})

The sets $(Sing X)_\bullet$ here are closely related by an interlocking system of maps that make them form what is called a \emph{[[simplicial set]]}, and as such the collection of these sets of singular simplices is called the \emph{[[singular simplicial complex]]} of $X$. We discuss the definition of simplicial sets now and then come back to this below in def. \ref{SingularSimplicialComplex}.

Since the topological $n$-simplices $\Delta^n$ from def. \ref{TopologicalSimplex} sit inside each other by the face inclusions of def. \ref{FaceInclusionInBarycentricCoords}

\begin{displaymath}
\delta_k : \Delta^{n-1} \to \Delta^{n}
\end{displaymath}
and project onto each other by the degeneracy maps, def. \ref{DegeneracyProjectionsInBarycentricCoords}

\begin{displaymath}
\sigma_k : \Delta^{n+1} \to \Delta^n
\end{displaymath}
we dually have functions

\begin{displaymath}
d_k \coloneqq Hom_{Top}(\delta_k, X) : (Sing X)_n \to (Sing X)_{n-1}
\end{displaymath}
that send each singular $n$-simplex to its $k$-face and functions

\begin{displaymath}
s_k \coloneqq Hom_{Top}(\sigma_k,X) : (Sing X)_{n} \to (Sing X)_{n+1}
\end{displaymath}
that regard an $n$-simplex as beign a degenerate (``thin'') $(n+1)$-simplex. All these sets of simplices and face and degeneracy maps between them form the following structure.

\begin{defn}
\label{SimplicialSet}\hypertarget{SimplicialSet}{}
A \textbf{[[simplicial set]]} $S \in sSet$ is

\begin{itemize}%
\item for each $n \in \mathbb{N}$ a [[set]] $S_n \in Set$ -- the \textbf{set of $n$-[[simplices]]};


\item for each [[injective map]] $\delta_i : \overline{n-1} \to \overline{n}$ of [[nLab:totally ordered sets]] $\bar n \coloneqq \{ 0 \lt 1 \lt \cdots \lt n \}$

a [[function]] $d_i : S_{n} \to S_{n-1}$ -- the $i$th \textbf{face map} on $n$-simplices;


\item for each [[surjective map]] $\sigma_i : \overline{n+1} \to \bar n$ of [[totally ordered sets]]

a [[function]] $\sigma_i : S_{n} \to S_{n+1}$ -- the $i$th \textbf{degeneracy map} on $n$-simplices;



\end{itemize}
such that these functions satisfy the \emph{[[simplicial identities]]}.

\end{defn}
\begin{defn}
\label{SimplicialIdentities}\hypertarget{SimplicialIdentities}{}
The \textbf{simplicial identities} satisfied by face and degeneracy maps as above are (whenever these maps are composable as indicated):

\begin{enumerate}%
\item $d_i \circ d_j  = d_{j-1} \circ d_i$ if $i \lt j$,


\item $s_i \circ s_j  = s_j \circ s_{i-1}$ if $i \gt j$.


\item $d_i \circ s_j =  \left\{ \itexarray{ s_{j-1} \circ d_i &  if \;  i \lt j \\ id & if  \;  i = j \; or \; i = j+1 \\ s_j \circ d_{i-1} &  if i \gt j+1 } \right.$



\end{enumerate}
\end{defn}
It is straightforward to check by explicit inspection that the evident injection and restriction maps between the sets of [[singular simplices]] make $(Sing X)_\bullet$ into a simplicial set. However for working with this, it is good to streamline a little:

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{[[simplex category]]} $\Delta$ is the [[full subcategory]] of [[Cat]] on the free categories of the form

\begin{displaymath}
\begin{aligned}
    [0] & \coloneqq \{0\}
    \\
    [1] & \coloneqq \{0 \to 1\}
    \\
   [2] & \coloneqq \{0 \to 1 \to 2\}
   \\
   \vdots
   \end{aligned}
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This is called the ``simplex category'' because we are to think of the object $[n]$ as being the ``[[spine]]'' of the $n$-[[simplex]]. For instance for $n = 2$ we think of $0 \to 1 \to 2$ as the ``spine'' of the triangle. This becomes clear if we don't just draw the morphisms that \emph{generate} the category $[n]$, but draw also all their composites. For instance for $n = 2$ we have\_

\begin{displaymath}
[2]  
  = 
  \left\{
  \itexarray{
     && 1
     \\
     & \nearrow && \searrow
     \\
    0 &&\to&& 2
  }
  \right\}
  \,.
\end{displaymath}
\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
A [[functor]]

\begin{displaymath}
S : \Delta^{op} \to Set
\end{displaymath}
from the [[opposite category]] of the [[nLab:simplex category]] to the category [[Set]] of sets is canonically identified with a [[simplicial set]], def. \ref{SimplicialSet}.

\end{prop}
\begin{proof}
One checks by inspection that the simplicial identities characterize precisely the behaviour of the morphisms in $\Delta^{op}([n],[n+1])$ and $\Delta^{op}([n],[n-1])$.

\end{proof}
This makes the following evident:

\begin{example}
\label{StandardCosimplicialTopologicalSpace}\hypertarget{StandardCosimplicialTopologicalSpace}{}
The [[topological simplices]] from def. \ref{TopologicalSimplex} arrange into a \emph{[[cosimplicial object]] in [[Top]]}, namely a [[functor]]

\begin{displaymath}
\Delta^\bullet : \Delta \to Top
  \,.
\end{displaymath}
\end{example}
With this now the structure of a simplicial set on $(Sing X)_\bullet$, def. \ref{SingularSimplex}, is manifest: it is just the \emph{[[nerve]]} of $X$ with respect to $\Delta^\bullet$, namely:

\begin{defn}
\label{SingularSimplicialComplex}\hypertarget{SingularSimplicialComplex}{}
For $X$ a [[topological space]] its \textbf{[[singular simplicial complex|simplicial set of singular simplicies]]} (often called the \textbf{[[singular simplicial complex]]})

\begin{displaymath}
(Sing X)_\bullet : \Delta^{op} \to Set
\end{displaymath}
is given by composition of the functor from example \ref{StandardCosimplicialTopologicalSpace} with the [[nLab:hom functor]] of [[nLab:Top]]:

\begin{displaymath}
(Sing X) : [n] \mapsto Hom_{Top}( \Delta^n , X )
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
It turns out -- this is the content of the \emph{[[nLab:homotopy hypothesis]]-theorem} (\href{model+structure+on+simplicial+sets#Quillen67}{Quillen 67}) -- that [[homotopy type]] of the topological space $X$ is entirely captured by its singular simplicial complex $Sing X$. Moreover, the [[geometric realization]] of $Sing X$ is a model for the same [[homotopy type]] as that of $X$, but with the special property that it is canonically a [[cell complex]] -- a [[CW-complex]]. Better yet, $Sing X$ is itself already good cell complex, namely a [[Kan complex]]. We come to this below.

\end{remark}
\hypertarget{simplicial_homotopy}{}\subsubsection*{{Simplicial homotopy}}\label{simplicial_homotopy}

The concept of [[homotopy]] of morphisms between simplicial sets proceeds in direct analogy with that in [[topological spaces]].

\begin{defn}
\label{LeftHomotopyOfSimplicialSets}\hypertarget{LeftHomotopyOfSimplicialSets}{}
For $X$ a [[simplicial set]], def. \ref{SimplicialSet}, its \emph{simplicial [[cylinder object]]} is the [[Cartesian product]] $X\times \Delta[1]$ (formed in the [[category]] [[sSet]]).

A \emph{[[left homotopy]]}

\begin{displaymath}
\eta \;\colon\; f \Rightarrow g
\end{displaymath}
between two morphisms

\begin{displaymath}
f,g\;\colon\; X \longrightarrow Y
\end{displaymath}
of [[simplicial sets]] is a morphism

\begin{displaymath}
\eta \;\colon\; X \times \Delta[1] \longrightarrow Y
\end{displaymath}
such that the following [[commuting diagram|diagram commutes]]

\begin{displaymath}
\itexarray{
     X 
     \\
     {}^{\mathllap{(id_X,d_1)}}\downarrow & \searrow^{\mathllap{f}}
     \\
     X \times \Delta^1 &\stackrel{\eta}{\longrightarrow}& Y
     \\
     {}^{\mathllap{(id_x, d_0)}}\uparrow & \nearrow_{\mathllap{g}}
     \\
     X
  }
  \,.
\end{displaymath}
For $Y$ a [[Kan complex]], def. \ref{SimplicialSet}, its \emph{simplicial [[path space object]]} is the [[function complex]] $X^{\Delta[1]}$ (formed in the [[category]] [[sSet]]).

A \emph{[[right homotopy]]}

\begin{displaymath}
\eta \;\colon\; f \Rightarrow g
\end{displaymath}
between two morphisms

\begin{displaymath}
f,g\;\colon\; X \longrightarrow Y
\end{displaymath}
of [[simplicial sets]] is a morphism

\begin{displaymath}
\eta \colon X \longrightarrow Y^{\Delta[1]}
\end{displaymath}
such that the following [[commuting diagram|diagram commutes]]

\begin{displaymath}
\itexarray{
    && Y
    \\
    & {}^{\mathllap{f}}\nearrow & \uparrow^{\mathrlap{Y^{d_1}}}
    \\
    X &\stackrel{\eta}{\longrightarrow}& Y^{\Delta[1]}
    \\
    & {}_{\mathllap{g}}\searrow & \downarrow^{\mathrlap{Y^{d_0}}}
    \\
    && 
    Y
  }
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{LeftHomotopyIsEquivalence}\hypertarget{LeftHomotopyIsEquivalence}{}
For $Y$ a [[Kan complex]], def. \ref{KanComplexes}, and $X$ any [[simplicial set]], then left homotopy, def. \ref{LeftHomotopyOfSimplicialSets}, regarded as a [[relation]]

\begin{displaymath}
(f\sim g) \Leftrightarrow (f \stackrel{\exists}{\Rightarrow} g)
\end{displaymath}
on the [[hom set]] $Hom_{sSet}(X,Y)$, is an [[equivalence relation]].

\end{prop}
\begin{defn}
\label{HomotopyEquivalence}\hypertarget{HomotopyEquivalence}{}
A morphism $f \colon X \longrightarrow Y$ of [[simplicial sets]] is a left/right [[homotopy equivalence]] if there exists a morphisms $X \longleftarrow Y \colon g$ and left/right homotopies (def. \ref{LeftHomotopyOfSimplicialSets})

\begin{displaymath}
g \circ f \Rightarrow id_X\,,\;\;\;\; f\circ g \Rightarrow id_Y
\end{displaymath}
\end{defn}
The the basic invariants of [[simplicial sets]]/[[Kan complexes]] in [[simplicial homotopy theory]] are their [[simplicial homotopy groups]], to which we turn now.

Given that a [[Kan complex]] is a special [[simplicial set]] that [[homotopy hypothesis|behaves like]] a combinatorial model for a [[topological space]], the \emph{simplicial homotopy groups} of a Kan complex are accordingly the combinatorial analog of the [[homotopy groups]] of [[topological spaces]]: instead of being maps from topological [[spheres]] modulo maps from topological disks, they are maps from the [[boundary of a simplex]] modulo those from the [[simplex]] itself.

Accordingly, the definition of the discussion of simplicial homotopy groups is essentially literally the same as that of [[homotopy groups]] of topological spaces. One technical difference is for instance that the definition of the group structure is slightly more non-immediate for simplicial homotopy groups than for topological homotopy groups (see below).

\begin{defn}
\label{UnderlyingSetsOfSimplicialHomotopyGroups}\hypertarget{UnderlyingSetsOfSimplicialHomotopyGroups}{}
For $X$ a [[Kan complex]], then its \textbf{0th [[simplicial homotopy group]]} (or \textbf{set of [[connected components]]}) is the set of [[equivalence classes]] of vertices modulo the [[equivalence relation]] $X_1 \stackrel{(d_1,d_0)}{\longrightarrow} X_0 \times X_0$

\begin{displaymath}
\pi_0(X) \colon X_0/X_1
  \,.
\end{displaymath}
For $x \in X_0$ a vertex and for $n \in \mathbb{N}$, $n \geq 1$, then the underlying [[set]] of the \textbf{$n$th [[simplicial homotopy group]]} of $X$ at $x$ -- denoted $\pi_n(X,x)$ -- is, the set of [[equivalence classes]] $[\alpha]$ of morphisms

\begin{displaymath}
\alpha \colon \Delta^n \to X
\end{displaymath}
from the simplicial $n$-[[simplex]] $\Delta^n$ to $X$, such that these take the [[boundary of a simplex|boundary of the simplex]] to $x$, i.e. such that they fit into a [[commuting diagram]] in [[sSet]] of the form

\begin{displaymath}
\itexarray{
     \partial \Delta[n] & \longrightarrow &  \Delta[0]
     \\
     \downarrow && \downarrow^{\mathrlap{x}}
     \\
     \Delta[n] &\stackrel{\alpha}{\longrightarrow}& X
   }
  \,,
\end{displaymath}
where two such maps $\alpha, \alpha'$ are taken to be equivalent is they are related by a [[simplicial homotopy]] $\eta$

\begin{displaymath}
\itexarray{
     \Delta[n]
     \\
     \downarrow^{i_0} & \searrow^{\alpha}
     \\
     \Delta[n] \times \Delta[1]
     &\stackrel{\eta}{\longrightarrow}&
     X
     \\
     \uparrow^{i_1} & \nearrow_{\alpha'}
     \\
     \Delta[n]
  }
\end{displaymath}
that fixes the boundary in that it fits into a [[commuting diagram]] in [[sSet]] of the form

\begin{displaymath}
\itexarray{
     \partial \Delta[n] \times \Delta[1]
     & \longrightarrow &
     \Delta[0]
     \\
     \downarrow && \downarrow^{\mathrlap{x}}
     \\
     \Delta[n]  \times \Delta[1]
     &\stackrel{\eta}{\longrightarrow}&
     X
  }
  \,.
\end{displaymath}
\end{defn}
These sets are taken to be equipped with the following group structure.

\begin{defn}
\label{ProductOnSimplicialHomotopyGroups}\hypertarget{ProductOnSimplicialHomotopyGroups}{}
For $X$ a [[Kan complex]], for $x\in X_0$, for $n \geq 1$ and for $f,g \colon \Delta[n] \to X$ two representatives of $\pi_n(X,x)$ as in def. \ref{UnderlyingSetsOfSimplicialHomotopyGroups}, consider the following $n$-simplices in $X_n$:

\begin{displaymath}
v_i 
  \coloneqq
  \left\{
    \itexarray{
      s_0 \circ s_0 \circ \cdots \circ s_0 (x) & for \; 0 \leq i \leq  n-2
      \\
      f & for \; i = n-1
      \\
      g & for \; i = n+1
    }
  \right.
\end{displaymath}
This corresponds to a morphism $\Lambda^{n+1}[n] \to X$ from a [[horn]] of the $(n+1)$-[[simplex]] into $X$. By the [[Kan complex]] property of $X$ this morphism has an [[extension]] $\theta$ through the $(n+1)$-[[simplex]] $\Delta[n]$

\begin{displaymath}
\itexarray{
     \Lambda^{n+1}[n] & \longrightarrow & X
     \\
     \downarrow & \nearrow_{\mathrlap{\theta}}
     \\
     \Delta[n+1]
  }
\end{displaymath}
From the [[simplicial identities]] one finds that the boundary of the $n$-simplex arising as the $n$th boundary piece $d_n \theta$ of $\theta$ is constant on $x$

\begin{displaymath}
d_i d_{n} \theta = d_{n-1} d_i \theta = x
\end{displaymath}
So $d_n \theta$ represents an element in $\pi_n(X,x)$ and we define a product operation on $\pi_n(X,x)$ by

\begin{displaymath}
[f]\cdot [g] \coloneqq [d_n \theta]
  \,.
\end{displaymath}
\end{defn}
(e.g. \hyperlink{GoerssJardine99}{Goerss-Jardine 99, p. 26})

\begin{remark}
\label{}\hypertarget{}{}
All the degenerate $n$-simplices $v_{0 \leq i \leq n-2}$ in def. \ref{ProductOnSimplicialHomotopyGroups} are just there so that the gluing of the two $n$-cells $f$ and $g$ to each other can be regarded as forming the boundary of an $(n+1)$-simplex except for one face. By the Kan extension property that missing face exists, namely $d_n \theta$. This is a choice of gluing composite of $f$ with $g$.

\end{remark}
\begin{lemma}
\label{}\hypertarget{}{}
The product on homotopy group elements in def. \ref{ProductOnSimplicialHomotopyGroups} is well defined, in that it is independent of the choice of representatives $f$, $g$ and of the extension $\theta$.

\end{lemma}
e.g. (\hyperlink{GoerssJardine99}{Goerss-Jardine 99, lemma 7.1})

\begin{lemma}
\label{}\hypertarget{}{}
The product operation in def. \ref{ProductOnSimplicialHomotopyGroups} yields a [[group]] structure on $\pi_n(X,x)$, which is [[abelian group|abelian]] for $n \geq 2$.

\end{lemma}
e.g. (\hyperlink{GoerssJardine99}{Goerss-Jardine 99, theorem 7.2})

\begin{remark}
\label{}\hypertarget{}{}
The first homotopy group, $\pi_1(X,x)$, is also called the \emph{[[fundamental group]]} of $X$.

\end{remark}
\begin{defn}
\label{WeakHomotopyEquivalence}\hypertarget{WeakHomotopyEquivalence}{}
For $X,Y \in KanCplx \hookrightarrow sSet$ two [[Kan complexes]], then a morphism

\begin{displaymath}
f \colon X \longrightarrow Y
\end{displaymath}
is called a \textbf{[[weak homotopy equivalence]]} if it induces [[isomorphisms]] on all [[simplicial homotopy groups]], i.e. if

\begin{enumerate}%
\item $\pi_0(f) \colon \pi_0(X) \longrightarrow \pi_0(Y)$ is a [[bijection]] of sets;


\item $\pi_n(f,x) \colon \pi_n(X,x) \longrightarrow \pi_n(Y,f(x))$ is an [[isomorphism]] of [[groups]] for all $x\in X_0$ and all $n \in \mathbb{N}$; $n \geq 1$.



\end{enumerate}
\end{defn}
\hypertarget{kan_fibrations}{}\subsubsection*{{Kan fibrations}}\label{kan_fibrations}

\begin{defn}
\label{Horn}\hypertarget{Horn}{}
For each $i$, $0 \leq i \leq n$, the \textbf{$(n,i)$-horn} is the subsimplicial set

\begin{displaymath}
\Lambda^i[n]
  \hookrightarrow
  \Delta[n]
\end{displaymath}
of the simplicial $n$-[[simplex]], which is the [[union]] of all faces \emph{except} the $i^{th}$ one.

This is called an \textbf{outer horn} if $k = 0$ or $k = n$. Otherwise it is an \textbf{inner horn}.

\end{defn}


(graphics taken from \href{https://ncatlab.org/nlab/show/simplicial+set#Friedman08}{Friedman 08})

\begin{remark}
\label{}\hypertarget{}{}
Since [[sSet]] is a [[presheaf category]], [[unions]] of [[subobjects]] make sense and they are calculated objectwise, thus in this case dimensionwise. This way it becomes clear what the structure of a horn as a functor $\Lambda^k[n]: \Delta^{op} \to Set$ must therefore be: it takes $[m]$ to the collection of ordinal maps $f: [m] \to [n]$ which do not have the element $k$ in the image.

\end{remark}
\begin{defn}
\label{KanComplexes}\hypertarget{KanComplexes}{}
A \emph{[[Kan complex]]} is a [[simplicial set]] $S$ that satisfies the \emph{Kan condition},

\begin{itemize}%
\item which says that all [[horns]] of the simplicial set have \emph{fillers}/extend to [[simplices]];


\item which means equivalently that the unique homomorphism $S \to pt$ from $S$ to the [[point]] (the [[terminal object|terminal]] [[simplicial set]]) is a [[Kan fibration]];


\item which means equivalently that for all [[diagrams]] of the form

\begin{displaymath}
\itexarray{
  \Lambda^i[n] &\to& S
  \\
  \downarrow && \downarrow
  \\
  \Delta[n] &\to& pt
}
\;\;\;
\leftrightarrow
\;\;\;
\itexarray{
  \Lambda^i[n] &\to& S
  \\
  \downarrow && 
  \\
  \Delta[n] 
}
\end{displaymath}
there exists a diagonal morphism

\begin{displaymath}
\itexarray{
  \Lambda^i[n] &\to& S
  \\
  \downarrow &\nearrow& \downarrow
  \\
  \Delta[n] &\to& pt
}
\;\;\;
\leftrightarrow
\;\;\;
\itexarray{
  \Lambda^i[n] &\to& S
  \\
  \downarrow &\nearrow& 
  \\
  \Delta[n] 
}
\end{displaymath}
completing this to a [[commuting diagram]];


\item which in turn means equivalently that the map from $n$-simplices to $(n,i)$-horns is an [[epimorphism]]

\begin{displaymath}
[\Delta[n], S]\, \twoheadrightarrow \,[\Lambda^i[n],S]
\,.
\end{displaymath}


\end{itemize}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
For $X$ a [[topological space]], its [[singular simplicial complex]] $Sing(X)$, def. \ref{SingularSimplicialComplex}, is a Kan complex, def. \ref{KanComplexes}.

\end{prop}
\begin{proof}
The inclusions ${{\Lambda^n}_{Top}}_k \hookrightarrow \Delta^n_{Top}$ of topological horns into topological simplices are [[retracts]], in that there are [[continuous maps]] $\Delta^n_{Top} \to {{\Lambda^n}_{Top}}_k$ given by ``squashing'' a topological $n$-simplex onto parts of its boundary, such that

\begin{displaymath}
({{\Lambda^n}_{Top}}_k \to \Delta^n_{Top} \to
  {{\Lambda^n}_{Top}}_k)
  =
  Id
  \,.
\end{displaymath}
Therefore the map $[\Delta^n, \Pi(X)] \to [\Lambda^n_k,\Pi(X)]$ is an epimorphism, since it is equal to to $Top(\Delta^n, X) \to Top(\Lambda^n_k, X)$ which has a right inverse $Top(\Lambda^n_k, X) \to Top(\Delta^n, X)$.

\end{proof}
More generally:

\begin{defn}
\label{KanFibration}\hypertarget{KanFibration}{}
A morphism $\phi \colon S \longrightarrow T$ in [[sSet]] is called a \emph{[[Kan fibration]]} if it has the [[right lifting property]] again all [[horn]] inclusions, def. \ref{Horn}, hence if for every [[commuting diagram]] of the form

\begin{displaymath}
\itexarray{
    \Lambda^i[n] &\longrightarrow& S
    \\
    \downarrow && \downarrow^{\mathrlap{\phi}}
    \\
    \Delta[n] &\longrightarrow& T
  }
\end{displaymath}
there exists a lift

\begin{displaymath}
\itexarray{
    \Lambda^i[n] &\longrightarrow& S 
    \\
    \downarrow &\nearrow& \downarrow^{\mathrlap{\phi}}
    \\
    \Delta[n] &\longrightarrow& T
  }
  \,.
\end{displaymath}
\end{defn}
This is the simplicial incarnation of the concept of [[Serre fibrations]] of topological spaces:

\begin{defn}
\label{SerreFibration}\hypertarget{SerreFibration}{}
A [[continuous function]] $f \colon X \longrightarrow Y$ between [[topological spaces]] is a [[Serre fibration]] if for all [[CW-complexes]] $C$ and for every [[commuting diagram]] in [[Top]] of the form

\begin{displaymath}
\itexarray{
    C &\longrightarrow& X 
    \\
    \downarrow && \downarrow^{\mathrlap{f}}
    \\
    C \times I &\longrightarrow& Y
  }
\end{displaymath}
there exists a lift

\begin{displaymath}
\itexarray{
    C &\longrightarrow& X 
    \\
    \downarrow &\nearrow& \downarrow^{\mathrlap{f}}
    \\
    C \times I &\longrightarrow& Y
  }
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{SingDetextsAndReflectsFibrations}\hypertarget{SingDetextsAndReflectsFibrations}{}
A [[continuous function]] $f \colon X \longrightarrow Y$ is a [[Serre fibration]], def. \ref{SerreFibration}, precisely if $Sing(f) \colon Sing(X) \longrightarrow Sing(Y)$ (def. \ref{SingularSimplicialComplex}) is a [[Kan fibration]], def. \ref{KanFibration}.

\end{prop}
The proof uses the basic tool of [[nerve and realization]]-[[adjunction]] to which we get to below in prop. \ref{NerveAndRealizationAdjunction}.

\begin{proof}
First observe that the left [[lifting property]] against all $C \hookrightarrow C \times I$ for $C$ a [[CW-complex]] is equivalent to left lifting against [[geometric realization]] ${\vert \Lambda^i[n]\vert} \hookrightarrow {\vert \Delta[n]\vert}$ of [[horn]] inclusions. Then apply the [[natural isomorphism]] $Top({\vert-\vert},-) \simeq sSet(-,Sing(-))$, given by the [[adjunction]] of prop. \ref{NerveAndRealizationAdjunction} and example \ref{TopologicalRealizationOfSimplicialSets}, to the lifting diagrams.

\end{proof}
\begin{lemma}
\label{PullbackOfKanFibrationSendsLeftHomotopyToFiberwiseHomotopyequivalence}\hypertarget{PullbackOfKanFibrationSendsLeftHomotopyToFiberwiseHomotopyequivalence}{}
Let $p \colon X \longrightarrow Y$ be a [[Kan fibration]], def. \ref{KanFibration}, and let $f_1,f_2 \colon A \longrightarrow X$ be two morphisms. If there is a [[left homotopy]] (def. \ref{LeftHomotopyOfSimplicialSets}) $f_1 \Rightarrow f_2$ between these maps, then there is a fiberwise [[homotopy equivalence]], def. \ref{HomotopyEquivalence}, between the [[pullback]] fibrations $f_1^\ast X \simeq f_2^\ast X$.

\end{lemma}
(e.g. \hyperlink{GoerssJardine99}{Goerss-Jardine 99, chapter I, lemma 10.6})

While [[simplicial sets]] have the advantage of being purely combinatorial structures, the [[singular simplicial complex]] of any given [[topological space]], def. \ref{SingularSimplicialComplex} is in general a huge simplicial set which does not lend itself to detailed inspection. The following is about small models.

\begin{defn}
\label{MinimalKanFibration}\hypertarget{MinimalKanFibration}{}
A [[Kan fibration]] $\phi \colon S \longrightarrow T$, def. \ref{KanFibration}, is called a \textbf{[[minimal Kan fibration]]} if for any two cells in the same fiber with the same [[boundary]] if they are homotopic relative their boundary, then they are already equal.

More formally, $\phi$ is minimal precisely if for every [[commuting diagram]] of the form

\begin{displaymath}
\itexarray{
    (\partial \Delta[n]) \times \Delta[1]
    &\stackrel{p_1}{\longrightarrow}&
    \partial \Delta[n]
    \\
    \downarrow && \downarrow
    \\
    \Delta[n] \times \Delta[1]
    &\stackrel{h}{\longrightarrow}&
    S
    \\
    \downarrow^{\mathrlap{p_1}} && \downarrow^{\mathrlap{\phi}}
    \\
    \Delta[n] &\longrightarrow& T
  }
\end{displaymath}
then the two composites

\begin{displaymath}
\Delta[n]
  \stackrel{\overset{d_0}{\longrightarrow}}{\underset{d_1}{\longrightarrow}} 
  \Delta[n] \times \Delta[1]
  \stackrel{h}{\longrightarrow}
  S
\end{displaymath}
are equal.

\end{defn}
\begin{prop}
\label{PullbackPreservesMinimalFibration}\hypertarget{PullbackPreservesMinimalFibration}{}
The [[pullback]] (in [[sSet]]) of a [[minimal Kan fibration]], def. \ref{MinimalKanFibration}, along any morphism is again a mimimal Kan fibration.

\end{prop}
\ldots{} [[anodyne extensions]]\ldots{}

(\hyperlink{GoerssJardine99}{Goerss-Jardine 99, chapter I, section 4}, \hyperlink{JoyalTierney05}{Joyal-Tierney 05, section 31})

\begin{prop}
\label{KanFibrationHasMinimalStrongDeformationRetract}\hypertarget{KanFibrationHasMinimalStrongDeformationRetract}{}
For every [[Kan fibration]], def. \ref{KanFibration}, there exists a fiberwise [[strong deformation retract]] to a [[minimal Kan fibration]], def. \ref{MinimalKanFibration}.

\end{prop}
(e.g. \hyperlink{GoerssJardine99}{Goerss-Jardine 99, chapter I, prop. 10.3}, \hyperlink{JoyalTierney05}{Joyal-Tierney 05, theorem 3.3.1, theorem 3.3.3}).

\begin{proof}
Choose representatives by [[induction]], use that in the induction step one needs lifts of [[anodyne extensions]] against a [[Kan fibration]], which exist.

\end{proof}
\begin{lemma}
\label{FiberwiseHomotopyEquivalenceOfMinimalFibrationsIsIso}\hypertarget{FiberwiseHomotopyEquivalenceOfMinimalFibrationsIsIso}{}
A morphism between [[minimal Kan fibrations]], def. \ref{MinimalKanFibration}, which is fiberwise a [[homotopy equivalence]], def. \ref{HomotopyEquivalence}, is already an [[isomorphism]].

\end{lemma}
(e.g. \hyperlink{GoerssJardine99}{Goerss-Jardine 99, chapter I, lemma 10.4})

\begin{proof}
Show the statement degreewise. In the [[induction]] one needs to lift [[anodyne extensions]] agains a [[Kan fibration]].

\end{proof}
\begin{lemma}
\label{MinimalKanFibrationAreFiberBundles}\hypertarget{MinimalKanFibrationAreFiberBundles}{}
Every [[minimal Kan fibration]], def. \ref{MinimalKanFibration}, over a [[connected]] base is a simplicial [[fiber bundle]], locally trivial over every simplex of the base.

\end{lemma}
(e.g. \hyperlink{GoerssJardine99}{Goerss-Jardine 99, chapter I, corollary 10.8})

\begin{proof}
By assumption of the base being connected, the classifying maps for the fibers over any two vertices are connected by a [[zig-zag]] of [[homotopies]], hence by lemma \ref{PullbackOfKanFibrationSendsLeftHomotopyToFiberwiseHomotopyequivalence} the fibers are connected by [[homotopy equivalences]] and then by prop. \ref{PullbackPreservesMinimalFibration} and lemma \ref{FiberwiseHomotopyEquivalenceOfMinimalFibrationsIsIso} they are already isomorphic. Write $F$ for this [[typical fiber]].

Moreover, for all $n$ the morphisms $\Delta[n] \to \Delta[0] \to \Delta[n]$ are [[left homotopy|left homotopic]] to $\Delta[n] \stackrel{id}{\to} \Delta[n]$ and so applying lemma \ref{PullbackOfKanFibrationSendsLeftHomotopyToFiberwiseHomotopyequivalence} and prop. \ref{FiberwiseHomotopyEquivalenceOfMinimalFibrationsIsIso} once more yields that the fiber over each $\Delta[n]$ is [[isomorphism|isomorphic]] to $\Delta[n]\times F$.

\end{proof}
\hypertarget{geometric_realization}{}\subsubsection*{{Geometric realization}}\label{geometric_realization}

So far we we have considered passing from [[topological spaces]] to [[simplicial sets]] by applying the [[singular simplicial complex]] functor of def. \ref{SingularSimplicialComplex}. Now we discuss a [[left adjoint]] of this functor, called [[geometric realization]], which turns a simplicial set into a topological space by identifying each of its abstract [[n-simplices]] with the standard topological $n$-simplex.

This is an example of a general abstract phenomenon:

\begin{prop}
\label{NerveAndRealizationAdjunction}\hypertarget{NerveAndRealizationAdjunction}{}
Let

\begin{displaymath}
\delta \;\colon\; D \longrightarrow \mathcal{C}
\end{displaymath}
be a [[functor]] from a [[small category]] $D$ to a [[locally small category]] $\mathcal{C}$ with all [[colimits]]. Then the [[nerve]]-functor

\begin{displaymath}
N \;\colon\; \mathcal{C} \longrightarrow [D^{op}, Set]
\end{displaymath}
\begin{displaymath}
N(X) \coloneqq \mathcal{C}(\delta(-),X)
\end{displaymath}
has a [[left adjoint]] functor ${\vert-\vert}$, called [[geometric realization]],

\begin{displaymath}
({\vert-\vert} \dashv N)
  \;\colon\;
  \mathcal{C}
  \stackrel{\overset{{\vert-\vert}}{\longleftarrow}}{\underset{N}{\longrightarrow}}
  [D^{op}, Set]
\end{displaymath}
given by the [[coend]]

\begin{displaymath}
{\vert S\vert}
  =
  \int^{d \in D} \delta(d) \cdot S(d)
  \,.
\end{displaymath}
\end{prop}
(\href{nerve+and+realization#Kan58}{Kan 58})

\begin{proof}
By basic propeties of [[ends]] and [[coends]]:

\begin{displaymath}
\begin{aligned}
     [D^{op}, Set](S,N(X))
     & =
     \int_{d \in D} Set(S(d), N(X)(d))
     \\
     & = \int_{d\in D} Set(S(d), \mathcal{C}(\delta(d),X))
     \\
     & \simeq \int_{d \in D} \mathcal{C}(\delta(d) \cdot S(d), X)
     \\
     & \simeq \mathcal{C}(\int^{d \in D} \delta(d) \cdot S(d), X)
     \\
     & = \mathcal{C}({\vert S\vert}, X)
     \,.
  \end{aligned}
\end{displaymath}
\end{proof}
\begin{example}
\label{TopologicalRealizationOfSimplicialSets}\hypertarget{TopologicalRealizationOfSimplicialSets}{}
The [[singular simplicial complex]] functor $Sing$ of def. \ref{SingularSimplicialComplex} has a [[left adjoint]] [[geometric realization]] functor

\begin{displaymath}
{\vert-\vert}
  \colon
  sSet \longrightarrow Top
\end{displaymath}
given by the [[coend]]

\begin{displaymath}
{\vert S \vert}
  =
  \int^{[n]\in \Delta}
  \Delta^n \cdot S_n
  \,.
\end{displaymath}
\end{example}
Topological geometric realization takes values in particularly nice topological spaces.

\begin{defn}
\label{TopologicalRealizationOfsSetLandsInCWComplexes}\hypertarget{TopologicalRealizationOfsSetLandsInCWComplexes}{}
The topological [[geometric realization]] of [[simplicial sets]] in example \ref{TopologicalRealizationOfSimplicialSets} takes values in [[CW-complexes]].

\end{defn}
(e.g. \hyperlink{GoerssJardine99}{Goerss-Jardine 99, chapter I, prop. 2.3})

Thus for a topological space $X$ the [[adjunction counit]] $\epsilon_X \colon {\vert Sing X\vert} \longrightarrow X$ of the [[nerve and realization]]-adjunction is a candidate for a replacement of $X$ by a CW-complex. For this, $\epsilon_X$ should be at least a [[weak homotopy equivalence]], i.e. induce [[isomorphisms]] on all [[homotopy groups]]. Since homotopy groups are built from maps into $X$ out of [[compact topological spaces]] it is plausible that this works if the topology of $X$ is entirely detected by maps out of compact topological spaces into $X$. Topological spaces with this property are called [[compactly generated topological spaces|compactly generated]].

We take \emph{[[compact topological space]]} to imply \emph{[[Hausdorff topological space]]}.

\begin{defn}
\label{kTop}\hypertarget{kTop}{}
A [[subspace]] $U \subset X$ of a [[topological space]] $X$ is called \textbf{compactly open} or \textbf{compactly closed}, respectively, if for every [[continuous function]] $f \colon K \longrightarrow X$ out of a [[compact topological space]] the [[preimage]] $f^{-1}(U) \subset K$ is open or closed, respectively.

A topological space $X$ is a \textbf{[[compactly generated topological space]]} if each of its compactly closed subspaces is already closed.

Write

\begin{displaymath}
Top_{cg} \hookrightarrow Top
\end{displaymath}
for the [[full subcategory]] of [[Top]] on the compactly generated topological spaces.

\end{defn}
Often the condition is added that a compactly closed topological space be also a [[weakly Hausdorff topological space]].

\begin{example}
\label{ExamplesOfCompactlyGeneratedTopologiclSpaces}\hypertarget{ExamplesOfCompactlyGeneratedTopologiclSpaces}{}
Examples of [[compactly generated topological spaces]], def. \ref{kTop}, include

\begin{itemize}%
\item every [[compact space]];


\item every [[locally compact space]];


\item every [[topological manifold]];


\item every [[CW-complex]];


\item every [[first countable space]]



\end{itemize}
\end{example}
\begin{cor}
\label{TopologicalRealizationOfSSetLandsInkTop}\hypertarget{TopologicalRealizationOfSSetLandsInkTop}{}
The topological [[geometric realization]] functor of [[simplicial sets]] in example \ref{TopologicalRealizationOfSimplicialSets} takes values in [[compactly generated topological spaces]]

\begin{displaymath}
{\vert - \vert}
  \;\colon\;
  sSet
  \longrightarrow Top_{cg}
\end{displaymath}
\end{cor}
\begin{proof}
By example \ref{ExamplesOfCompactlyGeneratedTopologiclSpaces} and prop. \ref{TopologicalRealizationOfsSetLandsInCWComplexes}.

\end{proof}
\begin{prop}
\label{kTopIsCoreflectiveInTop}\hypertarget{kTopIsCoreflectiveInTop}{}
The [[subcategory]] $Top_{cg} \hookrightarrow Top$ of def. \ref{kTop} has the following properties

\begin{enumerate}%
\item It is a [[coreflective subcategory]]

\begin{displaymath}
Top_{cg} \stackrel{\hookrightarrow}{\underset{k}{\longleftarrow}} Top
  \,.
\end{displaymath}
The coreflection $k(X)$ of a topological space is given by adding to the open subsets of $X$ all compactly open subsets, def. \ref{kTop}.


\item It has all small [[limits]] and [[colimits]].

The colimits are computed in $Top$, the limits are the image under $k$ of the limits as computed in $Top$.


\item It is a [[cartesian closed category]].

The [[cartesian product]] in $Top_{cg}$ is the image under $k$ of the Cartesian product formed in $Top$.



\end{enumerate}
\end{prop}
This is due to (\href{compactly+generated+topological+space#Steenrod67}{Steenrod 67}), expanded on in (\href{compactly+generated+topological+space#Lewis78}{Lewis 78, appendix A}). One says that prop. \ref{kTopIsCoreflectiveInTop} with example \ref{ExamplesOfCompactlyGeneratedTopologiclSpaces} makes $Top_{cg}$ a ``[[convenient category of topological spaces]]''.

\begin{prop}
\label{Timesk}\hypertarget{Timesk}{}
Regarded, via corollary \ref{TopologicalRealizationOfSSetLandsInkTop} as a functor ${\vert - \vert} \colon sSet \to Top_{cg}$, [[geometric realization]] preserves [[finite limits]].

\end{prop}
See at \emph{\href{geometric+realization#GeometricRealizationIsLeftExact}{Geometric realization is left exact}}.

\begin{proof}
The key step in the proof is to use the [[cartesian closed category|cartesian closure]] of $Top_{cg}$ (prop. \ref{kTopIsCoreflectiveInTop}). This gives that the [[Cartesian product]] is a [[left adjoint]] and hence preserves colimits in each variable, so that the [[coend]] in the definition of the geometric realization may be taken out of Cartesian products.

\end{proof}
\begin{lemma}
\label{}\hypertarget{}{}
The [[geometric realization]], example \ref{TopologicalRealizationOfSimplicialSets}, of a [[minimal Kan fibration]], def. \ref{MinimalKanFibration} is a [[Serre fibration]], def. \ref{SerreFibration}.

\end{lemma}
This is due to ([[Calculus of fractions and homotopy theory|Gabriel-Zisman 67]]). See for instance (\hyperlink{GoerssJardine99}{Goerss-Jardine 99, chapter I, corollary 10.8, theorem 10.9}).

\begin{proof}
By prop. \ref{MinimalKanFibrationAreFiberBundles} minimal Kan fibrations are simplicial [[fiber bundles]], locally trivial over each simplex in the base. By prop. \ref{Timesk} this property translates to their [[geometric realization]] also being a locally trivial [[fiber bundle]] of [[topological spaces]], hence in particular a [[Serre fibration]].

\end{proof}
\begin{prop}
\label{GeometricRealizationOfKanFibrationIsSerreFibration}\hypertarget{GeometricRealizationOfKanFibrationIsSerreFibration}{}
The [[geometric realization]], example \ref{TopologicalRealizationOfSimplicialSets}, of any [[Kan fibration]], def. \ref{KanFibration} is a [[Serre fibration]], def. \ref{SerreFibration}.

\end{prop}
This is due to (\href{Kan+fibration#Quillen68}{Quillen 68}). See for instance (\hyperlink{GoerssJardine99}{Goerss-Jardine 99, chapter I, theorem 10.10}).

\begin{prop}
\label{UnitOfSingularNerveAndRealizationIsWEOnKanComplexes}\hypertarget{UnitOfSingularNerveAndRealizationIsWEOnKanComplexes}{}
For $S$ a [[Kan complex]], then the [[adjunction unit|unit]] of the [[nerve and realization]]-[[adjunction]] (prop. \ref{NerveAndRealizationAdjunction}, example \ref{TopologicalRealizationOfSimplicialSets})

\begin{displaymath}
S \longrightarrow Sing {\vert S \vert}
\end{displaymath}
is a [[weak homotopy equivalence]], def. \ref{WeakHomotopyEquivalence}.

For $X$ any [[topological space]], then the [[adjunction counit]]

\begin{displaymath}
{\vert Sing X\vert} \longrightarrow X
\end{displaymath}
is a [[weak homotopy equivalence]]

\end{prop}
e.g. (\hyperlink{GoerssJardine99}{Goerss-Jardine 99, chapter I, prop. 11.1 and p. 63}).

\begin{proof}
Use prop. \ref{SingDetextsAndReflectsFibrations} and prop. \ref{GeometricRealizationOfKanFibrationIsSerreFibration} applied to the [[path fibration]] to proceed by [[induction]].

\end{proof}
\hypertarget{TheClassicalModelStructure}{}\subsection*{{The classical model structure $sSet_{Quillen}$}}\label{TheClassicalModelStructure}

\begin{defn}
\label{ClassesOfMorphismsOnsSetQuillen}\hypertarget{ClassesOfMorphismsOnsSetQuillen}{}
The classical model structure on [[simplicial sets]], $sSet_{Quillen}$, has the following distinguished classes of morphisms:

\begin{itemize}%
\item The classical \textbf{weak equivalences} $W$ are the morphisms whose [[geometric realization]], example \ref{TopologicalRealizationOfSimplicialSets}, is a [[weak homotopy equivalence]] of [[topological spaces]];


\item The classical \textbf{fibrations} $F$ are the \textbf{[[Kan fibrations]]}, def. \ref{KanFibration};


\item The classical \textbf{cofibrations} $C$ are the [[monomorphisms]] of simplicial sets, i.e. the degreewise [[injections]].



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

\hypertarget{basic_properties}{}\subsubsection*{{Basic properties}}\label{basic_properties}

\begin{prop}
\label{}\hypertarget{}{}
In model structure $sSet_{Quillen}$, def. \ref{ClassesOfMorphismsOnsSetQuillen}, the following holds.

\begin{itemize}%
\item The fibrant objects are precisely the [[Kan complexes]].


\item A morphism $f : X \to Y$ of fibrant simplicial sets / [[Kan complexes]] is a weak equivalence precisely if it induces an [[isomorphism]] on all [[simplicial homotopy groups]], def. \ref{UnderlyingSetsOfSimplicialHomotopyGroups}.


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



\end{itemize}
\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
The \textbf{acyclic fibrations} in $sSet_{Quillen}$(i.e. the maps that are both fibrations as well as weak equivalences) between [[Kan complexes]] 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{GoerssJardine99}{Goerss-Jardine 99, 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{}{}
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{Properness}{}\subsubsection*{{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 may be found, for instance, in II.8.6--7 of \href{model+structure+on+simplicial+sets#GoerssJardine}{Goerss-Jardine}. Another proof may be found in \href{model+structure+on+simplicial+sets#Moss}{Moss}, and a different proof of properness may be found in \href{model+structure+on+simplicial+sets#Cisinski06}{Cisinski, Prop. 2.1.5}.

\hypertarget{quillen_equivalence_with_}{}\subsubsection*{{Quillen equivalence with $Top_{Quillen}$}}\label{quillen_equivalence_with_}

\begin{theorem}
\label{}\hypertarget{}{}
The [[singular simplicial complex]]/[[geometric realization]]-[[nerve and realization|adjunction]] of example \ref{TopologicalRealizationOfSimplicialSets} constitutes a [[Quillen equivalence]] of the classical model structure $sSet_{Quillen}$ of def. \ref{ClassesOfMorphismsOnsSetQuillen} with the [[classical model structure on topological spaces]]:

\begin{displaymath}
({\vert -\vert}\dashv Sing)
  : 
  Top_{Quillen}
  \stackrel{\overset{{\vert -\vert}}{\leftarrow}}{\underset{Sing}{\to}}
  sSet_{Quillen}
\end{displaymath}
\end{theorem}
\begin{proof}
First of all, the adjunction is indeed a [[Quillen adjunction]]: prop. \ref{SingDetextsAndReflectsFibrations} says in particular that $Sing(-)$ takes [[Serre fibrations]] to [[Kan fibrations]] and prop. \ref{TopologicalRealizationOfsSetLandsInCWComplexes} gives that ${\vert-\vert}$ sends monomorphisms of simplicial sets to [[relative cell complexes]].

Now prop. \ref{UnitOfSingularNerveAndRealizationIsWEOnKanComplexes} says that the derived adjunction unit and counit are weak equivalences, and hence the Quillen adjunction is a Quillen equivalence.

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

\begin{itemize}%
\item [[model structure on simplicial sets]]


\item [[constructive model structure on simplicial sets]]


\item [[model structure on reduced simplicial sets]]


\item [[classical model structure on topological spaces]]



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

The original article is

\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}
The proof there 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


\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

\begin{itemize}%
\item [[Denis-Charles Cisinski]], section 2 of \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}
A standard textbook references for the classical model structure is

\begin{itemize}%
\item [[Paul Goerss]], [[Rick Jardine]], chapter 1 of \emph{[[Simplicial homotopy theory]]}, Birkh\"a{}user 1999, 2009 (\href{http://www.maths.abdn.ac.uk/~bensondj/html/archive/goerss-jardine.html}{ps}).

\end{itemize}
A proof of the model structure not relying on the [[classical model structure on topological spaces]] nor on explicit models for [[Kan fibrant replacement]] is givn in

\begin{itemize}%
\item [[Christian Sattler]], \emph{The Equivalence Extension Property and Model Structures} (\href{https://arxiv.org/abs/1704.06911}{arXiv:1704.06911})

\end{itemize}
[[!redirects Quillen model structure on simplicial sets]] [[!redirects Kan-Quillen model structure on simplicial sets]] [[!redirects Kan-Quillen model structure]]

[[!redirects classical model category of simplicial sets]]



\end{document}