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\begin{document}

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\section*{simplex category}

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - 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{ordinal}{Monoidal structure}\dotfill \pageref*{ordinal} \linebreak
\noindent\hyperlink{As2Categories}{$\Delta$ and $\Delta_a$ as 2-categories}\dotfill \pageref*{As2Categories} \linebreak
\noindent\hyperlink{universal_properties}{Universal properties}\dotfill \pageref*{universal_properties} \linebreak
\noindent\hyperlink{DualityWithIntervals}{Duality with intervals}\dotfill \pageref*{DualityWithIntervals} \linebreak
\noindent\hyperlink{combinatorics}{Combinatorics}\dotfill \pageref*{combinatorics} \linebreak
\noindent\hyperlink{related_constructions}{Related constructions}\dotfill \pageref*{related_constructions} \linebreak
\noindent\hyperlink{SimplicialSets}{Simplicial sets}\dotfill \pageref*{SimplicialSets} \linebreak
\noindent\hyperlink{RealizationAndNerve}{Realization and nerve}\dotfill \pageref*{RealizationAndNerve} \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 \textbf{simplex category} $\Delta$ encodes one of the main [[geometric shapes for higher structures]]. Its objects are the standard cellular $n$-[[simplices]]. It is also called the \emph{[[simplicial category]]}, but that term is ambiguous.

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

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{augmented simplex category} $\Delta_a$ is the [[full subcategory]] of [[Cat]] on the [[free categories]] of [[finite set|finite]] linear [[directed graphs]]

\begin{displaymath}
\{c_0 \to c_1 \to \cdots \to c_n\}
  \,.
\end{displaymath}
Equivalently this is the category whose [[objects]] are finite [[total order|totally ordered sets]], or finite ordinals, and whose [[morphisms]] are order-preserving functions between them.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
The \textbf{simplex category} $\Delta$ is the [[full subcategory]] of $\Delta_a$ (and hence of $Cat$) consisting of the free categories on finite and \textbf{[[inhabited set|inhabited]]} linear directed graphs, hence of \emph{non-empty} finite linear orders or \emph{non-zero} ordinals.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
It is common, convenient and without risk to use a [[skeleton]] of $\Delta$ or $\Delta_a$, where we pick a fixed representative in each [[isomorphism class]] of objects. Since isomorphisms of finite linearly ordered sets are \emph{unique} this step is so trivial that it is often not even mentioned explicitly.

With this the objects of $\Delta$ are in bijection with [[natural numbers]] $n \in \mathbb{N}$ and one usually writes

\begin{displaymath}
[n] = \{0 \to 1 \to \cdots \to n\}
\end{displaymath}
for the object of $\Delta$ given by the category with $(n+1)$ [[object]]s. Geometrically one may think of this as the [[spine]] of the standard cellular $n$-[[simplex]], see the discussion of \hyperlink{SimplicialSets}{simplicial sets} below. In this context one also writes $\Delta[n]$ or $\Delta^n$ for the [[simplicial set]] [[representable functor|represented]] by the object $[n]$: the simplicial $n$-[[simplex]]. By the [[Yoneda lemma]] one may identify the subcategory of simplicial sets on the $\Delta[n]$ with $\Delta$.

With this convention the first few objects of $\Delta$ are

\begin{displaymath}
[0] = \{0\}
\end{displaymath}
\begin{displaymath}
[1] = \{0 \to 1\}
\end{displaymath}
\begin{displaymath}
[2] = \{0 \to 1 \to 2\}
\end{displaymath}
etc.

The category $\Delta_a$ contains one more object, corresponding to the empty category $\emptyset$. When sticking to the above standard notation for the objects of $\Delta$, that extra object is naturally often denoted

\begin{displaymath}
[-1] = \emptyset
  \,.
\end{displaymath}
However, in contexts where only $\Delta_a$ and not $\Delta$ plays a role, some authors prefer to start counting with 0 instead of with $-1$. Then for instance the notation

\begin{displaymath}
\mathbf{0} = \emptyset
\end{displaymath}
\begin{displaymath}
\mathbf{1} = [0] =  \{0\}
\end{displaymath}
\begin{displaymath}
\mathbf{2} = [1]  = \{0 \to 1\}
\end{displaymath}
and generally

\begin{displaymath}
\mathbf{n} = [n-1]
\end{displaymath}
may be used.

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
The [[skeleton|skeletal]] version of the \textbf{augmented simplex category} $\Delta_a$ can be presented as follows:

\begin{itemize}%
\item objects are the finite totally ordered sets $\mathbf{n} \coloneqq \{0 \lt 1 \lt  \cdots \lt n-1\}$ for all $n \in \mathbb{N}$;


\item morphisms \emph{generated} by (are all expressible as finite compositions of) the following two elementary kinds of maps

\begin{enumerate}%
\item \textbf{face maps}: $\;\delta_i^n :\: \mathbf{n-1} \hookrightarrow \mathbf{n}\;$ is the injection whose image leaves out $i \in [n]\quad$ ($n \gt 0$ and $0 \leq i \lt n$);


\item \textbf{degeneracy maps}: $\;\sigma_i^n :\: \mathbf{n+1} \to  \mathbf{n}\;$ is the surjection such that $\sigma_i(i) = \sigma_i(i+1) = i\quad$ ($n \gt 0$ and $0 \leq i \lt n$);



\end{enumerate}


\end{itemize}
subject to the following relations, called the \textbf{simplicial relations} or \textbf{[[simplicial identities]]}:

\begin{displaymath}
\itexarray{
     \delta_j^{n+1} \circ \delta_i^n
     =
     \delta_i^{n+1}\circ \delta_{j-1}^n
     &
     \qquad 0 \leq i \lt j \leq n
     \\
     \sigma_j^n \circ \sigma_i^{n+1}
     =
     \sigma_i^n \circ \sigma_{j+1}^{n+1}
     &
     \qquad 0 \leq i \leq j \lt n
  }
\end{displaymath}
\begin{displaymath}
\sigma_j^n \circ \delta_i^{n+1}
  =
  \left\lbrace
    \itexarray{
      \delta_i^n \circ \sigma_{j-1}^{n-1}
      &
      \qquad 0 \leq i \lt j \lt n
      \\
      Id_n
      &
      \quad 0 \leq j \lt n \quad and\quad i = j \;or\; i = j+1
      \\
      \delta^n_{i-1} \circ \sigma_{j}^{n-1}
      &
      \qquad 0 \leq j \quad and\quad j +1 \lt i \leq n
    }
  \right.
\end{displaymath}
\end{prop}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{ordinal}{}\subsubsection*{{Monoidal structure}}\label{ordinal}

The addition of [[natural number]]s extends to a [[functor]] $\oplus : \Delta_a \times \Delta_a \to \Delta_a$ and $\oplus : \Delta \times \Delta \to \Delta$, by taking $\mathbf{m} \oplus \mathbf{n}$ to be the [[disjoint union]] of the underlying sets of $\mathbf{m}$ and $\mathbf{n}$, with the linear order that extends those on $\mathbf{m}$ and $\mathbf{n}$ by putting every element of $\mathbf{m}$ below every element of $\mathbf{n}$. This is called the \emph{[[ordinal sum]]} functor. If we visualise $\mathbf{n}$ as a totally ordered set $\{0 \lt 1 \lt \cdots \lt n-1\}$, and similarly for $\mathbf{m}$, then $\mathbf{m} \oplus \mathbf{n}$ looks like

\begin{displaymath}
\mathbf{m} \oplus \mathbf{n} = \{0 \lt 1 \lt  \cdots \lt m-1 \lt 0^*\lt 1^* \lt \cdots \lt (n-1)^*\}
\end{displaymath}
where $k^*$ denotes $k$ considered as an element of $\mathbf{n}$.

Clearly $\oplus : \Delta_a \times \Delta_a \to \Delta_a$ acts on objects as

\begin{displaymath}
\mathbf{n} \oplus \mathbf{m} = \mathbf{n+m},
\end{displaymath}
On morphisms, given $f : \mathbf{m} \to \mathbf{m}'$ and $g : \mathbf{n} \to \mathbf{n}'$, we have

\begin{displaymath}
(f\oplus g)(i) = \left\lbrace
     \itexarray{
       f(i) & if \; 0 \leq i  \leq m - 1
       \\
       m' + g(i-m) & if \; m \leq i \leq (m+n-1)
     }
  \right.
  \,.
\end{displaymath}
so that $f \oplus g$ can be visualised as $f$ and $g$ placed side by side.

It is easy to see now that $(\Delta_a,\oplus,\mathbf{0})$ is a [[strict monoidal category]].

It is important to note that this tensor does \emph{not} give a monoidal structure to $\Delta$, as that category does not contain the unit $\mathbf{0} = [-1] = \emptyset$.

Also note that this [[monoidal structure]] is \emph{not} [[symmetric monoidal category|symmetric]]! One does have isomorphisms $\mathbf{m} \oplus \mathbf{n} \simeq \mathbf{n+m} \simeq \mathbf{n} \oplus \mathbf{m}$ for all $m,n$, but it is easy to see that they are not [[bifunctorial]].

Under [[Day convolution]] this monoidal structure induces the [[join of simplicial sets]].

\hypertarget{As2Categories}{}\subsubsection*{{$\Delta$ and $\Delta_a$ as 2-categories}}\label{As2Categories}

Being full subcategories of the [[2-category]] $Cat$, $\Delta$ and $\Delta_a$ are themselves 2-categories: their [[2-cell|2-cells]] $f \Rightarrow g$ are given by the pointwise order on monotone functions. Equivalently, they are generated under (vertical and [[horizontal composition|horizontal]]) composition by the inequalities

\begin{displaymath}
\delta^n_{i+1} \leq \delta^n_i \qquad \qquad \sigma^n_i \leq \sigma^n_{i+1} \, .
\end{displaymath}
Of course, the ordinal sum functor $\oplus$ extends to a [[2-functor]] in the obvious way.

For each $n$ there is \href{https://ncatlab.org/nlab/show/adjoint+string}{a string of adjunctions}

\begin{displaymath}
\delta^n_{n-1} \dashv \sigma^n_{n-2} \dashv \delta^n_{n-2} \dashv
  \cdots \dashv \delta^n_1 \dashv \sigma^n_0 \dashv \delta^n_0
\end{displaymath}
where the counit of $\sigma_i \dashv \delta_i$ and the unit of $\delta_{i+1} \dashv \sigma_i$ are identities.

For each $n \geq 2$, the object $\mathbf{n+1}$ is given by the [[pushout]]

\begin{displaymath}
\itexarray{
\mathbf{n-1} & \overset{\delta_0}{\to} & \mathbf{n} \\
\mathllap{\scriptsize{\delta_{n-1}}} \downarrow & & \downarrow \mathrlap{\scriptsize{\delta_n}} \\
\mathbf{n} & \underset{\delta_0}{\to} & \mathbf{n+1}
}
\end{displaymath}
This means that $\Delta_a$ is generated as a 2-category by these pushouts and by taking adjoints of morphisms. Its monoidal structure is also determined in this way: for each $n$, write $\bot_n = \delta_{n-1}\cdots\delta_2\delta_1$ for the (morphism $\mathbf{1} \to \mathbf{n}$ corresponding to the) least element $0$ of $\mathbf{n}$, and $\top_n = \delta_0\cdots\delta_0\delta_0$ for the greatest. Then there are [[cospans]] $\mathbf{1} \to \mathbf{n} \leftarrow \mathbf{1}$ given by $\top_n$ and $\bot_n$, and each such is equivalent to the $(n-1)$ fold cospan composite (i.e. pushout) of $\mathbf{1} \to \mathbf{2} \leftarrow \mathbf{1}$ with itself. The ordinal sum $\mathbf{n} \oplus \mathbf{m}$ is given by the composite

\begin{displaymath}
\itexarray{
  & & & & \mathbf{n} \oplus \mathbf{m} & & & & \\
  & & & \nearrow & & \nwarrow & & & \\
  & & \mathbf{n} & & & & \mathbf{m} & & \\
  & \nearrow & & \nwarrow & & \nearrow & & \nwarrow & \\
  \mathbf{1} & & & & \mathbf{1} & & & & \mathbf{1}
}
\end{displaymath}
The universal property of pushouts, together with those of the initial and terminal objects $\mathbf{0},\mathbf{1}$, then suffices to define $\oplus$ as a 2-functor.

\hypertarget{universal_properties}{}\subsubsection*{{Universal properties}}\label{universal_properties}

The morphisms $\mathbf{0} \overset{\delta_0}{\to} \mathbf{1} \overset{\sigma_0}{\leftarrow} \mathbf{2}$ in $\Delta_a$ make $\mathbf{1}$ into a [[monoid object]]. Indeed, it is easy to see that

\begin{displaymath}
\begin{aligned}
\delta^n_i & = \mathbf{i} \oplus \delta^0_0 \oplus \mathbf{n-i} \\
\sigma^n_i & = \mathbf{i} \oplus \sigma^1_0 \oplus \mathbf{n-i-1}
\end{aligned}
\end{displaymath}
so that the morphisms of $\Delta_a$ are generated under $\circ$ and $\oplus$ by $\delta^0_0$ and $\sigma^1_0$, together with exactly the equations needed to make them the structure maps of the monoid $[1]$. The objects of $\Delta_a$ are the elements of the free monoid generated by $\mathbf{1}$ and $\oplus$.

$\Delta_a$ thus becomes the universal category-equipped-with-a-monoid, in the sense that for any strict monoidal category $B$, there is a bijection between monoids $(M,m,e)$ in $B$ and strict [[monoidal functors]] $\Delta_a \to B$ such that $\mathbf{1} \mapsto M$, $\sigma_0 \mapsto m$ and $\delta_0 \mapsto e$.

In particular, for $K$ a 2-category, [[monads]] in $K$ correspond to 2-functors $\mathbf{B}\Delta_a \to K$, where $\mathbf{B}\Delta_a$ is $\Delta_a$ considered as a one-object 2-category. Because monads in $K$ are also the same as [[lax functors]] $1 \to K$, this correspondence exhibits $\mathbf{B}\Delta_a$ as the [[lax morphism classifier]] for the terminal category $1$.

When $\Delta_a$ is considered as a 2-category, a similar argument to the above shows that the one-object [[3-category]] $\mathbf{B}\Delta_a$ classifies [[lax-idempotent monads]]: given a 3-category $M$ and a lax-idempotent monad $t$ therein, there is a unique 3-functor $\mathbf{B}\Delta_a \to M$ sending $[1]$ to $t$, essentially because $\sigma^1_0 \dashv \delta^1_0 = \delta^0_0 \oplus \mathbf{1}$ with identity counit.

\hypertarget{DualityWithIntervals}{}\subsubsection*{{Duality with intervals}}\label{DualityWithIntervals}

Recall that an [[interval]] is a [[linearly ordered set]] with a top and bottom element; interval maps are [[monotone functions]] which preserve top and bottom.

Parallel to the categories $\Delta$ and $\Delta_a$, let $\nabla$ denote the category of finite intervals where the top and bottom elements are \emph{distinct}, and let $\nabla_a$ denote the category of all finite intervals, including the terminal one where top and bottom coincide. Then we have [[duality|concrete dualities]], or equivalences of the form

\begin{displaymath}
\Delta_a^{op} \simeq \nabla_a; \qquad \Delta^{op} \simeq \nabla,
\end{displaymath}
both induced by the [[duality|ambimorphic object]] $\mathbf{2}$, seen as both an ordinal and an interval. In other words, we have in each case an adjoint equivalence

\begin{displaymath}
Int(-, \mathbf{2})^{op} \dashv Ord(-, \mathbf{2})
\end{displaymath}
inducing the first equivalence $Ord(-, \mathbf{2}): \Delta_a^{op} \to \nabla_a$, and the second equivalence by restriction.

This fact is mentioned in (\hyperlink{Joyal}{Joyal}), to help give some intuition for his [[Theta category|category]] $\Theta$ as dual to a category of disks. See also \emph{\href{interval#RelationToSimplices}{Interval -- Relation to simplices}}, and the section on dualities in (\hyperlink{Wraith}{Wraith}).

\hypertarget{combinatorics}{}\subsubsection*{{Combinatorics}}\label{combinatorics}

The homogenous symmetric function $h_m(x_1,x_2,{\dots},x_n)$ is the generating function for the set of all morphisms from $[m-1]$ to $[n-1]$.

As an order-preserving function between finite ordinals, any morphism $f : \mathbf{m} \to \mathbf{n}$ in $\Delta_a$ is completely specified by fixing $k$ elements of $\mathbf{n}$ as the [[image]] of $f$, together with a [[partition\#of\_numbers|composition]] of $\mathbf{m}$ into $k$ parts, each part denoting a non-empty, contiguous subset of elements of $\mathbf{m}$ sharing their value of $f$. That is, each such composition is given by a collection of $k$ interval parts $[0,i_1], [i_1 + 1, i_2], \ldots, [i_{k-1}+1, m-1]$, determined by a $(k-1)$-element subset $\{i_1, \ldots, i_{k-1}\}$ of an $(m-1)$-element set $\{0, \ldots, m-2\}$. Hence, there are a total of

\begin{displaymath}
\sum_k \binom{n}{k} \binom{m-1}{k-1} = \sum_k \binom{n}{k} \binom{m-1}{m-k} = \binom{n+m-1}{m}
\end{displaymath}
different morphisms of type $\mathbf{m} \to \mathbf{n}$ in $\Delta_a$, where we obtain the expression on the right by applying the [[Chu–Vandermonde identity]]. For example, there are

\begin{displaymath}
\binom{2}{1}\binom{2}{0} + \binom{2}{2}\binom{2}{1} = 2+2 = 4 = \binom{4}{3}
\end{displaymath}
different morphisms $\mathbf{3} \to \mathbf{2}$, corresponding to the four functions

\begin{enumerate}%
\item $f(0) = f(1) = 0, f(2) = 1$
\item $f(0) = 0, f(1) = f(2) = 1$
\item $f(0) = f(1) = f(2) = 0$
\item $f(0) = f(1) = f(2) = 1$

\end{enumerate}
A more direct [[bijective proof]] of the identity $|\Delta_a(\mathbf{m},\mathbf{n})| = \binom{n+m-1}{m}$ is also possible: see \href{http://nforum.ncatlab.org/discussion/3636/number-of-morphisms-in-the-simplex-category/?Focus=29835#Comment_29835}{this comment} on the nForum.

As some interesting special cases, taking $m=n$ gives the number of monotone endofunctions on $\mathbf{n}$ (OEIS sequence \href{https://oeis.org/A088218}{A088218}, or \href{https://oeis.org/A001700}{A001700} if we consider endomorphisms $[n] \to [n] \in \Delta$), while taking $m=2$ gives the triangular numbers (OEIS sequence \href{https://oeis.org/A000217}{A000217}).

\hypertarget{related_constructions}{}\subsection*{{Related constructions}}\label{related_constructions}

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

[[presheaf|Presheaves]] on $\Delta$ are [[simplicial set|simplicial sets]]. Presheaves on $\Delta_a$ are \textbf{[[augmented simplicial sets]].}

Under the [[Yoneda embedding]] $Y : \Delta \to$ [[SSet]] the object $[n]$ induces the standard simplicial $n$-[[simplex]] $Y([n]) =: \Delta^n$. So 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.

The face and degeneracy maps and the relation they satisfy are geometrically best understood in terms of the [[full and faithful functor|full and faithful]] image under $Y$ in [[SSet]]:

\begin{itemize}%
\item the face map $Y(\delta_i) : \Delta^{n-1} \to \Delta^{n}$ injects the standard simplicial $(n-1)$-simplex as the $i$th face into the standard simplicial $n$-simplex;


\item the degeneracy map $Y(\sigma_i) : \Delta^{n+1} \to \Delta^{n}$ projects the standard simplicial $(n+1)$-simplex onto the standard simplicial $n$-simplex by collapsing its vertex number $i$ onto the face opposite to it.



\end{itemize}
\hypertarget{RealizationAndNerve}{}\subsubsection*{{Realization and nerve}}\label{RealizationAndNerve}

There are important standard functors from $\Delta$ to other categories which \emph{[[nerve and realization|realize]]} $[n]$ as a concrete model of the standard $n$-[[simplex]].

\begin{itemize}%
\item The functor $\Delta[-] : \Delta \to$ [[sSet]] (the [[Yoneda embedding]]) realizes $[n]$ as a [[simplicial set]].


\item The functor $|\cdot| : \Delta \to$ [[Top]]

sends $[n]$ to the standard topological $n$-simplex $[n] \mapsto \{x_0 \leq x_1 \leq \cdots \leq x_n \leq 1\}\subset \mathbb{R}^{n}$. This functor induced [[geometric realization]] of [[simplicial sets]].


\item The functor $O : \Delta \to Str\omega Cat$ sends $[n]$ to the $n$th [[oriental]]. This induces simplicial [[nerves]] of [[omega-category|omega-categories]].

Under the functor $Str \omega Cat \to Cat$ which discards all higher morphisms and identifies all 1-morphisms that are connected by a 2-morphisms, this becomes again the identification of $\Delta$ with the full subbcategory of $Cat$ on linear [[quivers]] that we started the above definition with

\begin{displaymath}
[n] \mapsto \{0 \to 1 \to \cdots \to n\}
  \,.
\end{displaymath}


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

\begin{itemize}%
\item [[geometric shapes for higher structures]]

\begin{itemize}%
\item \textbf{simplex category}


\item [[cube category]]


\item [[globe category]]


\item [[cell category]]


\item [[Segal's category]]



\end{itemize}

\item [[simplicial set]]


\item [[augmented simplicial set]]


\item [[join of simplicial sets]]



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

See the references at [[simplicial set]].

See also:

\begin{itemize}%
\item Section VII.5 of [[Categories Work|Categories for the Working Mathematician]]


\item Section II.2 of P. Gabriel and M. Zisman, \emph{Calculus of Fractions and Homotopy Theory}. Springer, 1967


\item Section 2, Doctrines, of [[Ross Street]], \href{http://www.numdam.org/item?id=CTGDC_1980__21_2_111_0}{\emph{Fibrations in Bicategories}}, 1980



\end{itemize}
The relation to [[intervals]] and the generalization to the [[cell category]] is due to

\begin{itemize}%
\item [[André Joyal]], \emph{Disks, duality and $\Theta$-categories}, preprint, 1997 (\href{http://ncatlab.org/nlab/files/JoyalThetaCategories.pdf}{pdf})

\end{itemize}
A discussion of the opposite categories of $\Delta, \Delta_a$ and related categories can be found here:

\begin{itemize}%
\item [[Gavin C. Wraith]], \emph{Using the generic interval}, Cah. Top. G\'e{}om. Diff. Cat. \textbf{XXXIV} 4 (1993) pp.259-266. (\href{http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1993__34_4/CTGDC_1993__34_4_259_0/CTGDC_1993__34_4_259_0.pdf}{pdf})

\end{itemize}
[[!redirects augmented simplex category]]

category: category



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