Definition 29.1 (The standard simplex):

The standard $n-$simplex denoted by $\Delta_n$ is the convex hull of the $n+1$ the standard unit vectors in $\mathbb{R}^{n+1}$. Denoting the standard unit vectors by ${\bf e}_1, {\bf e}_2, \dots, {\bf e}_{n+1}$, their convex hull is the set

$$\Delta_n = \{(t_1, t_2, \dots, t_{n+1}),\;\; t_1\geq 0,\;t_2\geq 0,\dots,t_{n+1}\geq 0,\; t_1 + t_ 2 + \dots + t_{n+1} = 1\}.$$

We take the standard zero simplex $\Delta_0$ to be the point ${\bf e}_1$.

Thus $\Delta_2$ is the equilateral triangle with vertices $(1, 0, 0), (0, 1, 0)$ and $(0, 0, 1)$ and the one simplex $\Delta_1$ is the line segment in $\mathbb{R}^2$ joining the points $(1, 0)$ and $(0, 1)$.

Note that $\Delta_2$ contains three copies of $\Delta_1$ namely the sides of the equilateral triangle. Likewise $\Delta_3$ contains four copies of $\Delta_2$, the four faces of the regular tetrahedron. To formalize this idea, we introduce $(n+1)$ affine maps $\Delta_{n-1} \longrightarrow \Delta_{n}$ called the face maps. For $i = 1, 2, 3$, the $i-$th face of $\Delta_2$ is the face opposite to vertex ${\bf e}_i$ and consists of all points $(t_1, t_2, t_3)$ with non-negative entries and $t_1+t_2+t_3 = 1$ such that the $i-$th coordinate $t_i$ vanishes.

Now suppose that $(t_1, t_2, \dots, t_{n+1})$ denotes denotes a typical point on the last face of $\Delta_n$. Then since $t_{n+1}$ vanishes, we see that $(t_1, t_2, \dots, t_n)$ is a typical point on $\Delta_{n-1}$. Turning the argument around we define the map

$$\Delta_{n-1}\longrightarrow \Delta_n\phantom{XXXXX.}$$

$$(t_1, t_2, \dots, t_n)\mapsto (t_1, t_2, \dots, t_n, 0),$$

where the $t_i$ are all non-negtive and $\sum t_i = 1$, and call it the standard $n$-th face map. The $i-$th face map ( $0\leq i \leq n$) would be

$$\phantom{X.XXXXXXXXXXXXXXXX}\Phi_i^{n}:\Delta_{n-1} \longrightarrow \Delta_n$$

$$(t_1, t_2,\dots, t_n) \mapsto (t_1, t_2, \dots,t_{i-1},0, t_i,\dots, t_{n}),\phantom{XXXXXXX}{(29.1)}$$

We leave it to the reader to write down explicitly the maps $\Phi_j^{n}\circ \Phi_i^{n-1}:\Delta_{n-2}\longrightarrow \Delta_n$ and prove the following result: