# Visualizing Simplicial Objects

Definition. Let $[n]$ denote the set $\{0,1,\dots,n\}$. We define the category $\Delta\subseteq \mathcal{Set}$ (called the category of combinatorial simplices) as follows: The objects of $\Delta$ are the sets $[n]$. The morphisms in $\Delta$ are the order-preserving set functions.

Remark. For each $n$, the set $[n]$ is a category using the linear ordering on $n$. To be precise, the objects are the elements of $[n]$ and for $i,j\in [n]$,

$\hom(i,j)=\begin{cases} \leq:i\to j, &\mbox{if }i\leq j,\\ \emptyset, &\mbox{otherwise}. \end{cases}$

Definition. A simplicial object (resp. cosimplicial object) in a category $\mathcal{C}$ is a contravariant (resp. covariant) functor $K:\Delta^{\operatorname{op}}\to \mathcal{C}$ (resp. $K:\Delta\to \mathcal{C}$). In particular, if $\mathcal{C}=\mathcal{Set}$, simplicial objects are called simplicial sets and similarly for cosimplicial objects.

Put simply, a simplicial object $K$ is completely determined by the following data:

• An object $K_n:= K([n])$ called the $n$-skeleton of $K$,
• A $\mathcal{C}$-map $f^*:=K(f):K_n\to K_m$ for each $f\in \hom([m],[n])$ called the attachment maps of $K$.

For each $n\geq 0$, we have a natural simplicial set $\Delta^n$ called the $n$-simplex. This simplicial set is defined as follows:

• $\Delta^n_i :=\hom_\Delta ([i],[n])$.
• If $f:[i]\to [j]$, then $f^*:\hom_\Delta ([j],[n])\to \hom_\Delta ([i],[n])$ is given by $g\mapsto g\circ f$.

Some basic properties are:

• If $i>n$, then $\Delta^n_i=\emptyset$.
• If $m\leq n$, then $\Delta^m\subseteq \Delta^n$. (More precisely, there is an isomorphic copy of $\Delta^m$ in $\Delta^n$.)
• If $K$ is a simplicial set, $K_n\simeq \hom(\Delta^n, K)$ where morphisms are taken in the category of simplicial sets.

For all $n$, there is a canonical visualization of $\Delta^n$. For simplicity, we consider the $n=2$ case:

• For $\Delta^2_0$, label $0\mapsto 0$ by $(1,0,0)$, $0\mapsto 1$ by $(0,1,0)$, and $0\mapsto 2$ by $(0,0,1)$.
• For $\Delta^2_1$, label
$0\mapsto 0,1\mapsto 1$ by a path $\gamma_0:[0,1]\to \mathbb{R}^3$ taking $(1,0,0)$ to $(0,1,0)$,
$0\mapsto 0,1\mapsto 2$ by a path $\gamma_1:[0,1]\to \mathbb{R}^3$ taking $(1,0,0)$ to $(0,0,1)$,
$0\mapsto 1,1\mapsto 2$ by a path $\gamma_2:[0,1]\to \mathbb{R}^3$ taking $(0,1,0)$ to $(0,0,1)$.
• For $\Delta^2_2$, label the unique element by any continuous map $F:\mathbb{D}^2\to \mathbb{R}^3$ taking $\partial\mathbb{D}^2$ to $\cup_{i} \gamma_i$.

Using these identifications, $\Delta^2$ is noneother than the geometric $2$-simplex in $\mathbb{R}^3$.

As a final note, in motivation for Kan complexes, we define the notion of a horn.

Definition. Let $0\leq j\leq n$. The jth horn $\Lambda_j^n$ of a $n$-simplex $\Delta^n$ is defined by

$(\Delta_j^n)_i=\begin{cases}\Delta^n_i, &\mbox{if } i<n-1,\\ \Delta^n_i\setminus \{f\}, &\mbox{if }i=n-1, \\ \emptyset, &\mbox{if }i>n-1,\end{cases}$

where $f$ is the unique map with $f^{-1}(j)=\emptyset$.

Geometrically, the unique map $f$ corresponds to the $j$th face (i.e. the $j$th element in the $(n-1)$-skeleton of $\Delta^n$ using lexicographic ordering). Thus the $j$th horn is just the $n$-simplex with this face and the interior (the unique element in $\Delta_n^n$) removed.

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