nLab
simplex category

Idea

The 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 simplicial category, but that term is ambiguous.

Definition

Definition

• The augmented simplex category ${\Delta }_a$ is the full subcategory of Cat on the finite linear quivers

$$\{c_0\to c_1\to \cdots \to c_n\}.$$\{c_0 \to c_1 \to \cdots \to c_n\} \,.

Equivalently this is the category whose objects are finite totally ordered sets, or finite ordinals, and whose morphisms are order-preserving functions between them.

• The simplex category $\Delta$ is the full subcategory of ${\Delta }_a$ (and hence of $\mathrm{Cat}$) consisting of the finite inhabited linear quivers, non-empty linear orders or non-zero ordinals.

Remark

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 totally ordered sets are 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

$$[n]=\{0\to 1\to \cdots \to n\}$$[n] = \{0 \to 1 \to \cdots \to n\}

for the object of $\Delta$ given by the category with $(n+1)$ objects. Geometrically one may think of this as the spine of the standard cellular $n$-simplex, see the discussion of simplicial sets below. In this context one also writes $\Delta [n]$ or ${\Delta }^n$ for the simplicial set 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

$$[0]=\{0\}$$[0] = \{0\}

$$[1]=\{0\to 1\}$$[1] = \{0 \to 1\}

$$[2]=\{0\to 1\to 2\}$$[2] = \{0 \to 1 \to 2\}

etc.

The category ${\Delta }_a$ contains one more object, correspondig to the empty category $\varnothing$. When sticking to the above standard notation for the objects of $\Delta$, that extra object is naturally often denoted

$$[-1]=\varnothing .$$[-1] = \emptyset \,.

However, in contexts where only ${\Delta }_a$ and not $\Delta$ plays a role, one rathers starts counting with 0 instead of with $-1$. Then for instance the notation

$$0=\varnothing$$\mathbf{0} = \emptyset

$$1=[-1]=\{0\}$$\mathbf{1} = [-1] = \{0\}

$$2=[1]=\{0\to 1\}$$\mathbf{2} = [1] = \{0 \to 1\}

and generally

$$n=[n-1]$$\mathbf{n} = [n-1]

may be used.

Proposition

The skeletal version of the augmented simplex category ${\Delta }_a$ can be presented as follows:

• objects are the finite totally ordered sets $n:=\{0<1<\cdots <n-1\}$ for all $n\in \mathbb{N}$;

• morphisms generated by (are all expressible as finite compositions of) the following two elementary kinds of maps

  1. face maps: ${\delta }_i:={\delta }_i^n:n-1\hookrightarrow n$ is the injection whose image leaves out $i\in [n]$;

  2. degeneracy maps: ${\sigma }_i:={\sigma }_i^n:n+1\to n$ is the surjection such that ${\sigma }_i(i)={\sigma }_i(i+1)=i$;

subject to the following relations, called the simplicial relations or simplicial identities:

$$\begin{array}{cc}{\delta }_j^{n+1}\circ {\delta }_i^n={\delta }_i^{n+1}\circ {\delta }_{j-1}^n& \mathrm{for}i<j\\ {\sigma }_j^n\circ {\sigma }_i^{n+1}={\sigma }_i^{n-1}\circ {\sigma }_{j+1}^n& \mathrm{for}i\le j\end{array}$$\array{ \delta_j^{n+1} \circ \delta_i^n = \delta_i^{n+1}\circ \delta_{j-1}^n & for i \lt j \\ \sigma_j^n \circ \sigma_i^{n+1} = \sigma_i^{n-1} \circ \sigma_{j+1}^n & for i \leq j }

$${\sigma }_j^n\circ {\delta }_i^{n+1}=\{\begin{array}{cc}{\delta }_i^n\circ {\sigma }_{j-1}^{n-1}& \mathrm{if}i<j\\ {\mathrm{Id}}_n& \mathrm{if}i=j\mathrm{or}i=j+1\\ {\delta }_{i-1}^n\circ {\sigma }_j^{n-1}& \mathrm{if}i>j+1\end{array}$$\sigma_j^n \circ \delta_i^{n+1} = \left\lbrace \array{ \delta_i^n \circ \sigma_{j-1}^{n-1} & if i \lt j \\ Id_n & if i = j or i = j+1 \\ \delta^n_{i-1} \circ \sigma_{j}^{n-1} & if i \gt j +1 } \right.

Properties

Monoidal structure

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

$$m\oplus n=\{0<1<\cdots <m-1<0^{*}<1^{*}<\cdots <(n-1{)}^{*}\}$$\mathbf{m} \oplus \mathbf{n} = \{0 \lt 1 \lt \cdots \lt m-1 \lt 0^*\lt 1^* \lt \cdots \lt (n-1)^*\}

where $k^{*}$ denotes $k$ considered as an element of $n$.

Clearly $\oplus :{\Delta }_a\times {\Delta }_a\to {\Delta }_a$ acts on objects as

$$n\oplus m=n+m,$$\mathbf{n} \oplus \mathbf{m} = \mathbf{n+m},

On morphisms, given $f:m\to m\prime$ and $g:n\to n\prime$, we have

$$(f\oplus g)(i)=\{\begin{array}{cc}f(i)& \mathrm{if}0\le i\le m-1\\ m\prime +g(i-m)& \mathrm{if}m<i\le (m+n-1)\end{array}.$$(f\oplus g)(i) = \left\lbrace \array{ f(i) & if 0 \leq i \leq m - 1 \\ m' + g(i-m) & if m \lt i \leq (m+n-1) } \right. \,.

so that $f\oplus g$ can be visualised as $f$ and $g$ placed side by side.

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

It is important to note that this tensor does not give a monoidal structure to $\Delta$, as that does not have the unit $0=[-1]=\varnothing$.

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

$\Delta$ and ${\Delta }_a$ as 2-categories

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

$${\delta }_{i+1}^n\le {\delta }_i^n{\sigma }_i^n\le {\sigma }_{i+1}^n.$$\delta^n_{i+1} \leq \delta^n_i \qquad \qquad \sigma^n_i \leq \sigma^n_{i+1} \, .

Of course, the ordinal sum functor $\oplus$ extends to a 2-functor in the obvious way.

For each $n$ there is a string of adjunctions

$${\delta }_{n-1}^n⊣{\sigma }_{n-2}^n⊣{\delta }_{n-2}^n⊣\cdots ⊣{\delta }_1^n⊣{\sigma }_0^n⊣{\delta }_0^n$$\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

where the counit of ${\sigma }_i⊣{\delta }_i$ and the unit of ${\delta }_{i+1}⊣{\sigma }_i$ are identities.

For each $n\ge 2$, the object $n+1$ is given by the pushout

$$\begin{array}{ccc}n-1& \stackrel{{\delta }_0}{\to }& n\\ {\delta }_{n-1}\downarrow & & \downarrow {\delta }_0\\ n& \underset{{\delta }_0}{\to }& n+1\end{array}$$\array{ \mathbf{n-1} & \overset{\delta_0}{\to} & \mathbf{n} \\ \mathllap{\scriptsize{\delta_{n-1}}} \downarrow & & \downarrow \mathrlap{\scriptsize{\delta_0}} \\ \mathbf{n} & \underset{\delta_0}{\to} & \mathbf{n+1} }

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 ${\perp }_n={\delta }_{n-1}\cdots {\delta }_2{\delta }_1$ for the (morphism $1\to n$ corresponding to the) least element $0$ of $n$, and ${\top }_n={\delta }_0\cdots {\delta }_0{\delta }_0$ for the greatest. Then there are cospans $1\to n\leftarrow 1$ given by ${\top }_n$ and ${\perp }_n$, and each such is equivalent to the $(n-1)$ fold cospan composite (i.e. pushout) of $1\to 2\leftarrow 1$ with itself. The ordinal sum $n\oplus m$ is given by the composite

$$\begin{array}{ccccccccc}& & & & n\oplus m& & & & \\ & & & \nearrow & & \nwarrow & & & \\ & & n& & & & m& & \\ & \nearrow & & \nwarrow & & \nearrow & & \nwarrow & \\ 1& & & & 1& & & & 1\end{array}$$\array{ & & & & \mathbf{n} \oplus \mathbf{m} & & & & \\ & & & \nearrow & & \nwarrow & & & \\ & & \mathbf{n} & & & & \mathbf{m} & & \\ & \nearrow & & \nwarrow & & \nearrow & & \nwarrow & \\ \mathbf{1} & & & & \mathbf{1} & & & & \mathbf{1} }

The universal property of pushouts, together with those of the initial and terminal objects $0,1$, then suffices to define $\oplus$ as a 2-functor.

Universal properties

The morphisms $0\stackrel{{\delta }_0}{\to }1\stackrel{{\sigma }_0}{\leftarrow }2$ in ${\Delta }_a$ make $1$ into a monoid object. Indeed, it is easy to see that

$$\begin{array}{rl}{\delta }_i^n& =i\oplus {\delta }_0^0\oplus n-i\\ {\sigma }_i^n& =i\oplus {\sigma }_0^1\oplus n-i-1\end{array}$$\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}

so that the morphisms of ${\Delta }_a$ are generated under $\circ$ and $\oplus$ by ${\delta }_0^0$ and ${\sigma }_0^1$, 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 $1$ 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 $1\mapsto M$, ${\sigma }_0\mapsto m$ and ${\delta }_0\mapsto e$.

In particular, for $K$ a 2-category, monads in $K$ correspond to 2-functors $B{\Delta }_a\to K$, where $B{\Delta }_a$ 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 $B{\Delta }_a$ 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 $B{\Delta }_a$ classifies lax-idempotent monads: given a 3-category $M$ and a lax-idempotent monad $t$ therein, there is a unique 3-functor $B{\Delta }_a\to M$ sending $[1]$ to $t$, essentially because ${\sigma }_0^1⊣{\delta }_0^1={\delta }_0^0\oplus 1$ with identity counit.

Related constructions

Simplicial sets

Presheaves on $\Delta$ are simplicial sets. Presheaves on ${\Delta }_a$ are augmented simplicial sets..

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

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

• 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;

• 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.

Realization and nerve

There are important standard functors from $\Delta$ to other categories which realize $[n]$ as a concrete model of the standard $n$-simplex.

• The functor $\Delta [-]:\Delta \to$ sSet (the Yoneda embedding) realizes $[n]$ as a simplicial set.

• The functor $\mid \cdot \mid :\Delta \to$ Top

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

• The functor $O:\Delta \to \mathrm{Str}\omega \mathrm{Cat}$ sends $[n]$ to the $n$th oriental. This induces simplicial nerves of omega-categories.

  Under the functor $\mathrm{Str}\omega \mathrm{Cat}\to \mathrm{Cat}$ 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 $\mathrm{Cat}$ on linear quivers that we started the above definition with

  $$[n]\mapsto \{0\to 1\to \cdots \to n\}.$$[n] \mapsto \{0 \to 1 \to \cdots \to n\} \,.

Related concepts

• geometric shapes for higher structures

  • simplex category

  • cube category

  • globe category

• simplicial set

• augmented simplicial set

• join of simplicial sets

References

See the references at simplicial set.

category: category