category theory

# Contents

## 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 free categories of finite linear directed graphs

$\left\{{c}_{0}\to {c}_{1}\to \cdots \to {c}_{n}\right\}\phantom{\rule{thinmathspace}{0ex}}.$\{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.

###### Definition

The simplex category $\Delta$ is the full subcategory of ${\Delta }_{a}$ (and hence of $\mathrm{Cat}$) consisting of the free categories on finite and inhabited directed graphs, hence of 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 $n\in ℕ$ and one usually writes

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

for the object of $\Delta$ given by the category with $\left(n+1\right)$ 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 $\Delta \left[n\right]$ or ${\Delta }^{n}$ for the simplicial set represented by the object $\left[n\right]$: the simplicial $n$-simplex. By the Yoneda lemma one may identify the subcategory of simplicial sets on the $\Delta \left[n\right]$ with $\Delta$.

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

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

etc.

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

$\left[-1\right]=\varnothing \phantom{\rule{thinmathspace}{0ex}}.$[-1] = \emptyset \,.

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

$0=\varnothing$\mathbf{0} = \emptyset
$1=\left[0\right]=\left\{0\right\}$\mathbf{1} = [0] = \{0\}
$2=\left[1\right]=\left\{0\to 1\right\}$\mathbf{2} = [1] = \{0 \to 1\}

and generally

$n=\left[n-1\right]$\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:=\left\{0<1<\cdots for all $n\in ℕ$;

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

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

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

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}& \phantom{\rule{thickmathspace}{0ex}}i\array{ \delta_j^{n+1} \circ \delta_i^n = \delta_i^{n+1}\circ \delta_{j-1}^n & \; i \lt j \\ \sigma_j^n \circ \sigma_i^{n+1} = \sigma_i^{n-1} \circ \sigma_{j+1}^n & \; i \leq j }
${\sigma }_{j}^{n}\circ {\delta }_{i}^{n+1}=\left\{\begin{array}{cc}{\delta }_{i}^{n}\circ {\sigma }_{j-1}^{n-1}& ij+1\end{array}$\sigma_j^n \circ \delta_i^{n+1} = \left\lbrace \array{ \delta_i^n \circ \sigma_{j-1}^{n-1} & i \lt j \\ Id_n & \; i = j \;or\; i = j+1 \\ \delta^n_{i-1} \circ \sigma_{j}^{n-1} & i \gt j +1 } \right.

## Properties

### Monoidal structure

The addition of natural numbers extends to a functor $\oplus :{\Delta }_{a}×{\Delta }_{a}\to {\Delta }_{a}$ and $\oplus :\Delta ×\Delta \to \Delta$, 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 $\left\{0<1<\cdots , and similarly for $m$, then $m\oplus n$ looks like

$m\oplus n=\left\{0<1<\cdots \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 $\oplus :{\Delta }_{a}×{\Delta }_{a}\to {\Delta }_{a}$ 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

$\left(f\oplus g\right)\left(i\right)=\left\{\begin{array}{cc}f\left(i\right)& \mathrm{if}\phantom{\rule{thickmathspace}{0ex}}0\le i\le m-1\\ m\prime +g\left(i-m\right)& \mathrm{if}\phantom{\rule{thickmathspace}{0ex}}m\le i\le \left(m+n-1\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$(f\oplus g)(i) = \left\lbrace \array{ f(i) & if \; 0 \leq i \leq m - 1 \\ m' + g(i-m) & if \; m \leq 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 $\left({\Delta }_{a},\oplus ,\left[0\right]\right)$ 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 $0=\left[-1\right]=\varnothing$.

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 $f⇒g$ are given by the pointwise order on monotone functions. Equivalently, they are generated under (vertical and horizontal) composition by the inequalities

${\delta }_{i+1}^{n}\le {\delta }_{i}^{n}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}{\sigma }_{i}^{n}\le {\sigma }_{i+1}^{n}\phantom{\rule{thinmathspace}{0ex}}.$\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

$\begin{array}{ccc}n-1& \stackrel{{\delta }_{0}}{\to }& n\\ {\delta }_{n-1}↓& & ↓{\delta }_{0}\\ n& \underset{{\delta }_{0}}{\to }& n+1\end{array}$\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 $1\to n←1$ given by ${\top }_{n}$ and ${\perp }_{n}$, and each such is equivalent to the $\left(n-1\right)$ fold cospan composite (i.e. pushout) of $1\to 2←1$ with itself. The ordinal sum $n\oplus m$ is given by the composite

$\begin{array}{ccccccccc}& & & & n\oplus m& & & & \\ & & & ↗& & ↖& & & \\ & & n& & & & m& & \\ & ↗& & ↖& & ↗& & ↖& \\ 1& & & & 1& & & & 1\end{array}$\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 $0\stackrel{{\delta }_{0}}{\to }1\stackrel{{\sigma }_{0}}{←}2$ 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 $\left[1\right]$. 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 $\left(M,m,e\right)$ in $B$ and strict monoidal functors ${\Delta }_{a}\to B$ such that $1↦M$, ${\sigma }_{0}↦m$ and ${\delta }_{0}↦e$.

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 $\left[1\right]$ to $t$, essentially because ${\sigma }_{0}^{1}⊣{\delta }_{0}^{1}={\delta }_{0}^{0}\oplus 1$ with identity counit.

### Duality with intervals

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 distinct, and let ${\nabla }_{a}$ denote the category of all finite intervals, including the terminal one where top and bottom coincide. Then we have concrete dualities, or equivalences of the form

${\Delta }_{a}^{\mathrm{op}}\simeq {\nabla }_{a};\phantom{\rule{2em}{0ex}}{\Delta }^{\mathrm{op}}\simeq \nabla ,$\Delta_a^{op} \simeq \nabla_a; \qquad \Delta^{op} \simeq \nabla,

both induced by the ambimorphic object $2$, seen as both an ordinal and an interval. In other words, we have in each case an adjoint equivalence

$\mathrm{Int}\left(-,2{\right)}^{\mathrm{op}}⊣\mathrm{Ord}\left(-,2\right)$Int(-, \mathbf{2})^{op} \dashv Ord(-, \mathbf{2})

inducing the first equivalence $\mathrm{Ord}\left(-,2\right):{\Delta }_{a}^{\mathrm{op}}\to {\nabla }_{a}$, and the second equivalence by restriction.

This fact is mentioned in (Joyal), to help give some intuition for his category $\Theta$ as dual to a category of disks. See also at Interval – Relation to simplices.

### 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 $\left[n\right]$ induces the standard simplicial $n$-simplex $Y\left(\left[n\right]\right)=:{\Delta }^{n}$. So in particular we have $\left({\Delta }^{n}\right)\left[m\right]={\mathrm{Hom}}_{\Delta }\left(\left[m\right],\left[n\right]\right)$ and hence ${\Delta }^{n}\left[m\right]$ is a finite set with $\left(\genfrac{}{}{0}{}{n+m+1}{n}\right)$ elements.

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 $Y\left({\delta }_{i}\right):{\Delta }^{n-1}\to {\Delta }^{n}$ injects the standard simplicial $\left(n-1\right)$-simplex as the $i$th face into the standard simplicial $n$-simplex;

• the degeneracy map $Y\left({\sigma }_{i}\right):{\Delta }^{n+1}\to {\Delta }^{n}$ projects the standard simplicial $\left(n+1\right)$-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 $\left[n\right]$ as a concrete model of the standard $n$-simplex.

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

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

sends $\left[n\right]$ to the standard topological $n$-simplex $\left[n\right]↦\left\{{x}_{0}\le {x}_{1}\le \cdots \le {x}_{n}\le 1\right\}\subset {ℝ}^{n}$. This functor induced geometric realization of simplicial sets.

• The functor $O:\Delta \to \mathrm{Str}\omega \mathrm{Cat}$ sends $\left[n\right]$ 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

$\left[n\right]↦\left\{0\to 1\to \cdots \to n\right\}\phantom{\rule{thinmathspace}{0ex}}.$[n] \mapsto \{0 \to 1 \to \cdots \to n\} \,.

## References

See the references at simplicial set.

• André Joyal, Disks, duality and $\Theta$-categories, preprint, (1997)