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\begin{document}

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\section*{simplicial set}

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

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{content}{}\section*{{Content}}\label{content}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{simplicial_sets_as_spaces_built_of_simplices}{Simplicial sets as spaces built of simplices}\dotfill \pageref*{simplicial_sets_as_spaces_built_of_simplices} \linebreak
\noindent\hyperlink{visualisation}{Visualisation}\dotfill \pageref*{visualisation} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{simplices_yoneda_embeddings}{$n$-simplices (Yoneda embeddings)}\dotfill \pageref*{simplices_yoneda_embeddings} \linebreak
\noindent\hyperlink{cartesian_products_of_simplices}{Cartesian products of simplices}\dotfill \pageref*{cartesian_products_of_simplices} \linebreak
\noindent\hyperlink{simplicial_complexes}{Simplicial complexes}\dotfill \pageref*{simplicial_complexes} \linebreak
\noindent\hyperlink{directed_graphs}{Directed graphs}\dotfill \pageref*{directed_graphs} \linebreak
\noindent\hyperlink{nerve_of_a_category}{Nerve of a category}\dotfill \pageref*{nerve_of_a_category} \linebreak
\noindent\hyperlink{singular_simplicial_complex_of_a_topological_space}{Singular simplicial complex of a topological space}\dotfill \pageref*{singular_simplicial_complex_of_a_topological_space} \linebreak
\noindent\hyperlink{bar_construction}{Bar construction}\dotfill \pageref*{bar_construction} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{classifying_topos}{Classifying topos}\dotfill \pageref*{classifying_topos} \linebreak
\noindent\hyperlink{simplicial_sets}{Simplicial sets}\dotfill \pageref*{simplicial_sets} \linebreak
\noindent\hyperlink{cosimplicial_sets}{Cosimplicial sets}\dotfill \pageref*{cosimplicial_sets} \linebreak
\noindent\hyperlink{as_models_in_homotopy_theory}{As models in homotopy theory}\dotfill \pageref*{as_models_in_homotopy_theory} \linebreak
\noindent\hyperlink{relation_to_dendroidal_sets}{Relation to dendroidal sets}\dotfill \pageref*{relation_to_dendroidal_sets} \linebreak
\noindent\hyperlink{variants}{Variants}\dotfill \pageref*{variants} \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}

Simplicial sets generalize the idea of [[simplicial complexes]]: a \emph{simplicial set} is like a combinatorial space built up out of gluing abstract [[simplex|simplices]] to each other. Equivalently, it is an object equipped with a rule for how to consistently map the objects of the [[simplex category]] into it.

More concretely, a simplicial set $S$ is a collection of [[sets]] $S_n$ for $n \in \mathbb{N}$, so that elements in $S_n$ are to be thought of as $n$-[[simplex|simplices]], equipped with a rule that says:

\begin{itemize}%
\item which $(n-1)$-simplices in $S_{n-1}$ are faces of which elements of $S_n$;
\item which $(n+1)$-simplices are [[thin element|thin]] in that they are really just $n$-simplices regarded as degenerate $(n+1)$-simplices.

\end{itemize}
One of the main uses of simplicial sets is as combinatorial \emph{models} for the (weak) [[homotopy type]] of [[topological spaces]]. They can also be taken as models for [[∞-groupoids]]. This is encoded in the [[model structure on simplicial sets]]. For more reasons why simplicial sets see MathOverflow \href{http://mathoverflow.net/questions/58497/is-there-a-high-concept-explanation-for-why-simplicial-leads-to-homotopy-theor}{here}.

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

The quick abstract definition of a simplicial set goes as follows:

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{simplicial set} is a [[presheaf]] on the [[simplex category]] $\Delta$, that is, a [[functor]] $X : \Delta^{op} \to Sets$ from the [[opposite category]] of the [[simplex category]] to the category [[Set]] of [[sets]]; equivalently this a [[simplicial object]] in [[Set]].

Equipped with the standard [[homomorphisms]] of [[presheaf|presheaves]] as morphisms (namely [[natural transformation|natural transformations]] of the corresponding [[functors]]), simplicial sets form the category [[sSet]] (denoted both $SSet$ or $sSet$).

\end{defn}
Explicitly this means the following.

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(simplicial set)}

A \textbf{simplicial set} $X \in sSet$ is

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


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

a [[function]] $\;d_i :\: X_{n} \to X_{n-1}\;$ -- the $i$th \emph{[[face map]]} on $n$-simplices ($n \gt 0$ and $0 \leq i \leq n$);


\item for each [[surjective map]] $\;\sigma_i :\: [n+1] \to [n]\;$ of totally ordered sets

a function $\;s_i :\: X_{n} \to X_{n+1}\;$ -- the $i$th \emph{[[degeneracy map]]} on $n$-simplices ($n \geq 0$ and $0 \leq i \leq n$);



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

\end{defn}
 $\,$

\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

\hypertarget{simplicial_sets_as_spaces_built_of_simplices}{}\subsubsection*{{Simplicial sets as spaces built of simplices}}\label{simplicial_sets_as_spaces_built_of_simplices}

\begin{itemize}%
\item The definition is to be understood from the point of view of [[space and quantity]]: a \textbf{simplicial set} is a space characterized by the fact that and how it may be \emph{probed} by mapping standard simplices into it: the set $S_n$ assigned by a simplicial set to the standard $n$-simplex $[n]$ is the \textbf{set of $n$-simplices} in this space, hence the way of mapping a standard $n$-simplex into this space.


\item For $S$ a simplicial set, the \textbf{face map}

\begin{displaymath}
d_i \coloneqq S(\delta^i): S_n \rightarrow S_{n-1}
\end{displaymath}
is dual to the unique injection $\delta^i : [n-1] \rightarrow [n]$ in the category $\Delta$ whose image omits the element $i \in [n]$.


\item Similarly, the \textbf{degeneracy map}

\begin{displaymath}
s_i \coloneqq S(\sigma^i) : S_n \rightarrow S_{n+1}
\end{displaymath}
is dual to the unique surjection $\sigma^i : [n+1] \rightarrow [n]$ in $\Delta$ such that $i \in [n]$ has two elements in its preimage.


\item The maps $\delta^i$ and $\sigma^i$ satisfy certain obvious relations -- the [[simplicial identities]] -- dual to those spelled out at [[simplex category]].



\end{itemize}
\hypertarget{visualisation}{}\subsubsection*{{Visualisation}}\label{visualisation}

(based on [[cubical set]])

The \textbf{face maps} go from sets $S_{n+1}$ of $(n+1)$-dimensional simplices to the corresponding set $S_{n}$ of $n$-dimensional simplices and can be thought of as sending each simplex in the simplicial set to one of its faces, for instance for $n=1$ the set $S_2$ of 2-simplices would be sent in three different ways by three different face maps to the set of $1$-simplices, for instance one of the face maps would send

\begin{displaymath}
\left(
  \itexarray{
       &          & b
     \\
       & \nearrow & \Downarrow^F & \searrow
     \\
     a &          & \rightarrow  &          & c
  }
  \right)
  \;\;
  \mapsto
  \;\;
  \left(
  \itexarray{
       &          & b
     \\
       & \nearrow
     \\
     a
  }
  \right)
\end{displaymath}
another one would send

\begin{displaymath}
\left(
  \itexarray{
       &          & b
     \\
       & \nearrow & \Downarrow^F & \searrow
     \\
     a &          & \rightarrow  &          & c
  }
  \right)
  \;\;
  \mapsto
  \;\;
  \left(
  \itexarray{
     a &          & \rightarrow  &          & c
  }
  \right)
  \,.
\end{displaymath}
On the other hand, the \textbf{degeneracy maps} go the other way round and send sets $S_n$ of $n$-simplices to sets $S_{n+1}$ of $(n+1)$-simplices by regarding an $n$-simplex as a degenerate or ``thin'' $(n+1)$-simplex in the various different ways that this is possible. For instance, again for $n=1$, a degeneracy map may act by sending

\begin{displaymath}
\left(
  \itexarray{
     a
     &\stackrel{f}{\to}&
     b
  }
  \right)
  \;\;
  \mapsto
  \;\;
  \left(
  \itexarray{
       &            & b
     \\
       & \nearrow_f & \Downarrow^{Id} & \searrow^{Id}
     \\
     a &            & \stackrel{f}\to &               & b
  }
  \right)
  \,.
\end{displaymath}
Notice the $Id$-labels, which indicate that the edges and faces labeled by them are ``[[thin element|thin]]'' in much the same way as an [[identity morphism]] is thin. They depend on lower dimensional features, (notice however that a simplicial set by itself is not equipped with any notion of composition of simplices, nor really, therefore, of identities. See [[quasicategory]] for a kind of simplicial set which does have such notions and [[simplicial T-complex]] for more on the intuitions behind this idea of ``thinness'').

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

\hypertarget{simplices_yoneda_embeddings}{}\subsubsection*{{$n$-simplices (Yoneda embeddings)}}\label{simplices_yoneda_embeddings}

Let $[n]$ denote the object of the [[simplex category]] $\Delta$ corresponding to the totally ordered set $\{ 0, 1, 2,\ldots, n\}$. Then the represented presheaf $\Delta(-, [n])$, typically written as $\Delta[n]$ is an example of a simplicial set. In particular we have $\Delta[n]_m=Hom_\Delta([m],[n])$ and hence $\Delta[n]_m$ is a finite set with $\binom{n+m+1}{n}$ elements.

By the Yoneda lemma, the $n$-simplices of a simplicial set $X$ are in natural bijective correspondence to maps $\Delta[n] \rightarrow X$ of simplicial sets.

\hypertarget{cartesian_products_of_simplices}{}\subsubsection*{{Cartesian products of simplices}}\label{cartesian_products_of_simplices}

The non-degenerate [[simplices]] in the simplicial set which is the [[Cartesian product]]

\begin{displaymath}
\Delta[1] \times \Delta[2]
\end{displaymath}
of the [[1-simplex]] $\Delta[1]$ with the [[2-simplex]] $\Delta[2]$ (i.e. the canonical simplicial [[cylinder object]] over the [[2-simplex]]) is the simplicial set which looks as follows:



\begin{quote}%
graphics grabbed from \hyperlink{Friedman08}{Friedman 08, p. 33}


\end{quote}
\hypertarget{simplicial_complexes}{}\subsubsection*{{Simplicial complexes}}\label{simplicial_complexes}

Every [[simplicial complex]] can be viewed a simplicial set (in several different ways).



\begin{quote}%
graphics grabbed form \href{https://arxiv.org/abs/1710.06129}{arXiv:1710.06129}


\end{quote}


\begin{quote}%
graphics grabbed from Maletic, 2013


\end{quote}
In particular any graph is thought of as being built of vertices and edges and so is a (1-dimensional) simplicial complex. Choosing a direction on the edges then gives a directed graph and that gives a simplicial set, as follows.

\hypertarget{directed_graphs}{}\subsubsection*{{Directed graphs}}\label{directed_graphs}

A [[directed graph]] (with loops and multiple edges allowed, i.e., a [[quiver]]) $E \rightrightarrows V$ is essentially the same thing as a 1-dimensional simplicial set, by taking $S_0 \coloneqq V$ to be the set of vertices and $S_1 \coloneqq E \uplus V$ to be the disjoint union of the set of edges with the set of vertices (the latter corresponding to the degenerate 1-simplices).

\hypertarget{nerve_of_a_category}{}\subsubsection*{{Nerve of a category}}\label{nerve_of_a_category}

If $C$ is a small category, the \textbf{nerve} of $C$ is a simplicial set which we denote $NC$. If we intepret the poset $[n]$ defined above as a category, we define the $n$-simplices of $NC$ to be the set of functors $[n] \rightarrow C$. Equivalently, the $0$-simplices of $NC$ are the objects of $C$, the $1$-simplices are the morphisms, and the $n$-simplices are strings of $n$ composable arrows in $C$. Face maps are given by composition (or omission, in the case of $d_0$ and $d_n$) and degeneracy maps are given by inserting identity arrows.

\hypertarget{singular_simplicial_complex_of_a_topological_space}{}\subsubsection*{{Singular simplicial complex of a topological space}}\label{singular_simplicial_complex_of_a_topological_space}

Recall from [[simplex category]] or [[geometric realization]] the standard functor $\Delta \to Top$ which sends $[n] \in \Delta$ to the standard topological $n$-simplex $\Delta^n$. This functor induces for every [[topological space]] $X$ the simplicial set

\begin{displaymath}
S X : [n] \mapsto Hom_{Top}(\Delta^n, X)
\end{displaymath}
called the \textbf{simplicial singular complex} of $X$. This simplicial set is always a [[Kan complex]] and may be regarded as the [[fundamental ∞-groupoid]] of $X$.

Following up on the idea of `'thinness'', a singular simplex $f: \Delta^n \to X$ may be called \textbf{thin} if it factors through a [[retraction]] $r: \Delta^n \to \Lambda^{n-1}_i$ to some horn of $\Delta^n$, then the well known [[Kan complex|Kan]] condition on $S X$ can be strengthened to say that every horn in $S X$ has a \emph{thin} filler. This also helps to give some intuitive underpinning to the idea of [[thin element]] in this simplicial context.

\hypertarget{bar_construction}{}\subsubsection*{{Bar construction}}\label{bar_construction}

For the moment see \emph{[[bar construction]]}.

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

\hypertarget{classifying_topos}{}\subsubsection*{{Classifying topos}}\label{classifying_topos}

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

The category of simplicial sets is a [[presheaf category]], and so in particular a [[Grothendieck topos]]. In fact, it is the [[classifying topos]] of the theory of ``intervals'', meaning [[totally ordered sets]] equipped with distinct top and bottom elements.

Specifically, if $E$ is a topos containing such an interval $I$, then we obtain a functor $\Delta \to E$ sending $[n]$ to the subobject

\begin{displaymath}
\{ (x_1,x_2,\dots,x_n) \;|\; x_1 \le x_2 \le \dots \le x_n \} \hookrightarrow I^n
\end{displaymath}
The corresponding [[geometric realization]]/[[nerve]] [[adjunction]] $E \leftrightarrows Set^{\Delta^{op}}$ is the [[geometric morphism]] which classifies $I$. In particular, the generic such interval is the simplicial 1-simplex $\Delta^1$; see [[generic interval]] for more.

The usual geometric realization into [[topological spaces]] cannot be obtained in this way precisely, since [[Top]] is not a topos. However, there are [[Top]]-like categories which are toposes, such as [[Johnstone's topological topos]].

\hypertarget{cosimplicial_sets}{}\paragraph*{{Cosimplicial sets}}\label{cosimplicial_sets}

Similarly, also the category $Set^{\Delta}$ of cosimplicial sets is a classifying topos: for inhabited [[linear orders]]. See at \emph{[[classifying topos]]} the section \emph{\href{http://ncatlab.org/nlab/show/classifying+topos#ForLinearOrders}{For (inhabited) linear orders}}.

\hypertarget{as_models_in_homotopy_theory}{}\subsubsection*{{As models in homotopy theory}}\label{as_models_in_homotopy_theory}

(\ldots{}) [[homotopy theory]] (\ldots{}) [[Kan complex]] (\ldots{}) [[quasi-category]] (\ldots{})

\hypertarget{relation_to_dendroidal_sets}{}\subsubsection*{{Relation to dendroidal sets}}\label{relation_to_dendroidal_sets}

For the moment see at \emph{[[dendroidal set]]} the section \hyperlink{http://ncatlab.org/nlab/show/dendroidal+set#RelationToSimplicialSets}{Relation to simplicial sets}

\hypertarget{variants}{}\subsection*{{Variants}}\label{variants}

\begin{itemize}%
\item A [[symmetric set]] is a simplicial set equipped with additional \emph{transposition maps} $t^n_i: X_n \to X_n$ for $i=0,\ldots,n-1$. These transition maps generate an [[action]] of the [[symmetric group]] on $X_n$ and satisfy certain commutation relations with the face and degeneracy maps.

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

\begin{itemize}%
\item [[simplicial map]]


\item [[simplicial object]]

\begin{itemize}%
\item \textbf{simplicial set}


\item [[pointed simplicial set]]


\item [[simplicial object in an (∞,1)-category]]



\end{itemize}

\item [[semi-simplicial object]]

\begin{itemize}%
\item [[semisimplicial set]]

\end{itemize}

\item [[simplicial homotopy theory]]


\item [[simplicial group]], [[reduced simplicial set]]


\item [[bisimplicial set]]


\item [[symmetric set]], [[cyclic set]], [[skew-simplicial set]]


\item [[globular set]], [[cubical set]], [[cellular set]], [[dendroidal set]]



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

A pedagogical introduction to simplicial sets is

\begin{itemize}%
\item Greg Friedman, \emph{An elementary illustrated introduction to simplicial sets} (\href{http://arxiv.org/abs/0809.4221}{arXiv:0809.4221})

\end{itemize}
A very clear and explicit exposition on the basics of simplicial sets is

\begin{itemize}%
\item [[Emily Riehl]], \emph{A leisurely introduction to simplicial sets}, 2008, 14 pages (\href{http://www.math.jhu.edu/~eriehl/ssets.pdf}{pdf}).

\end{itemize}
Another clear exposition is in the classic

\begin{itemize}%
\item [[Pierre Gabriel]], [[Michel Zisman]], [[Calculus of fractions and homotopy theory]].

\end{itemize}
A useful (if old) survey article is:

\begin{itemize}%
\item Edward B. Curtis, \emph{Simplicial homotopy theory}, Advances in Math., \textbf{6} (1971) 107 -- 209 \href{http://www.ams.org/mathscinet-getitem?mr=279808}{MR279808} 

\end{itemize}
More advanced treatments include

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

\end{itemize}
Some more facts about homotopical aspects of simplicial sets are discussed 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}
category: topology

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[[!redirects cosimplicial set]] [[!redirects cosimplicial sets]]

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category : simplicial object



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