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\begin{document}

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\section*{simplicial homotopy group}

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

\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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_topological_homotopy_groups}{Relation to topological homotopy groups}\dotfill \pageref*{relation_to_topological_homotopy_groups} \linebreak
\noindent\hyperlink{relation_to_homotopy_equivalence}{Relation to homotopy equivalence}\dotfill \pageref*{relation_to_homotopy_equivalence} \linebreak
\noindent\hyperlink{relation_to_chain_homology_groups_of_associated_moore_complexes}{Relation to chain homology groups of associated Moore complexes}\dotfill \pageref*{relation_to_chain_homology_groups_of_associated_moore_complexes} \linebreak
\noindent\hyperlink{long_exact_sequences_of_a_kan_fibration}{Long exact sequences of a Kan fibration}\dotfill \pageref*{long_exact_sequences_of_a_kan_fibration} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Simplicial homotopy groups are the basic invariants of [[simplicial sets]]/[[Kan complexes]] in [[simplicial homotopy theory]].

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).

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

Recall the classical [[model structure on simplicial sets]]. Let $X$ be a fibrant simplicial set, i.e. a [[Kan complex]].

\begin{defn}
\label{UnderlyingSetsOfSimplicialHomotopyGroups}\hypertarget{UnderlyingSetsOfSimplicialHomotopyGroups}{}
For $X$ a [[Kan complex]], then its \textbf{0th 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 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{GoerssJardine96}{Goerss-Jardine 96, 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{GoerssJardine96}{Goerss-Jardine 96, 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{GoerssJardine96}{Goerss-Jardine 96, theorem 7.2})

Finally:

\begin{defn}
\label{}\hypertarget{}{}
The simplicial homotopy groups of any [[simplicial set]], not necessarily [[Kan complex|Kan]], are those of any of its [[Kan fibrant replacements]] according to def. \ref{UnderlyingSetsOfSimplicialHomotopyGroups}.

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

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

\hypertarget{relation_to_topological_homotopy_groups}{}\subsubsection*{{Relation to topological homotopy groups}}\label{relation_to_topological_homotopy_groups}

The simplicial homotopy groups of a Kan complex coincide with the homotopy groups of its [[geometric realization]], see e.g. (\hyperlink{GoerssJardine96}{Goerss-Jardine 96, page 60}).

\hypertarget{relation_to_homotopy_equivalence}{}\subsubsection*{{Relation to homotopy equivalence}}\label{relation_to_homotopy_equivalence}

A morphism of simplicial sets which induces an [[isomorphism]] on all simplicial homotopy groups is called a [[weak homotopy equivalence]]. If it goes between [[Kan complexes]] then it is actually a [[homotopy equivalence]].

\hypertarget{relation_to_chain_homology_groups_of_associated_moore_complexes}{}\subsubsection*{{Relation to chain homology groups of associated Moore complexes}}\label{relation_to_chain_homology_groups_of_associated_moore_complexes}

Another way to get the group structure on the homotopy groups of a Kan complex, $X$, is via its [[Dwyer-Kan loop groupoid]] and the [[Moore complex]]. This gives a [[simplicial groupoid|simplicially enriched groupoid]] $G(X)$, or if we restricted to the pointed case, and just look at the loops at the base vertex, a [[simplicial group]]. (We will assume for the sake of simplicity that $X$ is \emph{reduced}, that is to say, $X_0$ is a singleton, and thus that $G(X)$ is a simplicial group.)

The construction of $G(X)$ is then given by the free group functor on the various levels, shifted by 1, and with a twist in the zeroth face map (see [[Dwyer-Kan loop groupoid]] and simplify to the reduced case.)

\begin{prop}
\label{}\hypertarget{}{}
There is an isomorphism between $\pi_n(X)$ as defined above and $H_{n-1}(N G(X))$, the $(n-1)$th homology group of the [[Moore complex]] of the simplicial group, $G(X)$.

\end{prop}
\hypertarget{long_exact_sequences_of_a_kan_fibration}{}\subsubsection*{{Long exact sequences of a Kan fibration}}\label{long_exact_sequences_of_a_kan_fibration}

For $f \colon X \longrightarrow Y$ a [[Kan fibration]], for $x\in X_0$ any vertex, for $y \coloneqq f(x) \in Y$ its image and $F_x \coloneqq f^{-1}(y)$ the [[fiber]] at that point, then the induced [[homomorphism]] of simplicial homotopy groups form a [[long exact sequence of homotopy groups]]

\begin{displaymath}
\cdots
  \to
  \pi_{n+1}(Y,y)
  \stackrel{}{\longrightarrow}
  \pi_n(F,x)
  \stackrel{}{\longrightarrow}
  \pi_n(X,x)
  \stackrel{}{\longrightarrow}
  \pi_n(Y,y)
  \stackrel{}{\longrightarrow}
  \pi_{n-1}(F,x)
  \to
  \cdots
\end{displaymath}
\begin{displaymath}
\cdots 
  \to
  \pi_1(F,x) \longrightarrow
  \pi_0(F) \stackrel{}{\longrightarrow} \pi_0(X) \longrightarrow \pi_0(Y)
\end{displaymath}
i.e. a [[long exact sequence]] of [[groups]] ending in a long exact sequence of [[pointed sets]].

(e.g. \hyperlink{GoerssJardine96}{Goerss-Jardine 96, lemma 7.3})

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{lemma}
\label{}\hypertarget{}{}
Let $C$ be a [[groupoid]] and $\mathcal{N}(C)$ its [[nerve]].

Then

\begin{itemize}%
\item $\pi_0 \mathcal{N}(C,c)$ is the set of isomorphism classes of $C$ with the class of $c$ as base point


\item $\pi_1 \mathcal{N}(C,c)$ is the [[automorphism group]] $Aut_C(c)$ of $c$


\item $\pi_{n \geq 2} \mathcal{N}(C,c)$ is trivial



\end{itemize}
\end{lemma}
In particular a [[functor]] $f : C \to D$ of [[groupoids]] is a [[equivalence of categories]] if under the nerve it induces a weak equivalence $\mathcal{N}(f) : \mathcal{N}(C) \to \mathcal{N}(D)$ of [[Kan complexes]]:

\begin{itemize}%
\item that $\pi_0 \mathcal{N}(f,c) : \pi_0(C,c) \to \pi_0(D,f(c))$ is an isomorphism implies that $f$ is an [[essentially surjective functor]] and is implied by $f$'s being a [[full functor]];
\item that $\pi_1 \mathcal{N}(f,c) : \pi_1(C,c) \to \pi_1(D,f(c))$ is an isomorphism is equivalent to $f$'s being a [[full and faithful functor]].

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

Textbook accounts include

\begin{itemize}%
\item [[Paul Goerss]], [[Rick Jardine]], chapter I, section 7 of \emph{[[Simplicial homotopy theory]]}, Progress in Mathematics, Birkh\"a{}user (1996)

\end{itemize}
Originally homotopy groups of simplicial sets had been defined in terms of the ordinary [[homotopy groups]] of the [[topological spaces]] [[geometric realization|realizing]] them. Apparently the first or one of the first discussions of the purely combinatorial definition is

\begin{itemize}%
\item [[Dan Kan]], \emph{A combinatorial definition of homotopy groups}, Annals of Mathematics Second Series, Vol. 67, No. 2 (Mar., 1958), pp. 282-312 (\href{http://www.jstor.org/stable/1970006?seq=8}{jstor})

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



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