I have some confusion regarding the following lecture note by Gereon Quick here. The complete set of notes can be found here.
It is written that
The standard simplices are related by face maps for $0 \le i \le n$ which can described as $\phi_i^n(t_0, \dots ,t_{n-1}) = (t_0, \dots, t_{i-1}, 0, t_i, \dots, t_{n-1})$ with the $0$ inserted at the $i^{\text{th}}$ coordinate ($t_0$ is the $0^{\text{th}}$ coordinate).
My thinking : I think $0$ is not inserted at the $i^{\text{th}}$ coordinate, it's inserted between the $(i-1)^{\text{st}}$ and the $i^{\text{th}}$ coordinates. I know that $(t_0,\dots,t_{i-1},0,t_i,\dots,t_{n-1}) \in \Delta^{n-1}$.
My question :I'm not getting why $0$ is inserted at the $i^{\text{th}}$ coordinate. Why isn't $1$ inserted at the $i^{\text{th}}$ coordinate instead?
First of all, $0$ is inserted in the $i^{\text{th}}$ coordinate of the point in $\mathbb{R}^{n+1}$ (numbering from $0$). The fact that $t_{i-1}$ is the previous coordinate, and $t_i$ is the next coordinate is irrelevant: the subscripts refer to the coordinates of the point in $\Delta^{n-1}$, not $\Delta^n$. Said another way $$(t_0, \dots, t_{i-1}, 0, t_i, \dots, t_{n-1}) = (x_0, \dots, x_{i-1}, x_i, x_{i+1}, \dots, x_n)$$ with $x_0 = t_0, \dots, x_{i-1} = t_{i-1}, x_i = 0, x_{i+1} = t_i, \dots, x_n = t_{n-1}$, so $0$ is in the $i^{\text{th}}$ coordinate (because $x_i = 0$).
As $(t_0, \dots, t_{n-1}) \in \Delta^{n-1}$, we have $t_0 + \dots + t_{n-1} = 1$, so $(t_0, \dots, t_{i-1}, 1, t_i, \dots, t_{n-1}) \not\in \Delta^n$ because $t_0 + \dots + t_{i-1} + 1 + t_i + \dots + t_{n-1} = 2 \neq 1$. On the other hand, $(t_0, \dots, t_{i-1}, 0, t_i, \dots, t_{n-1}) \in \Delta^n$. Moreover, the set of such points actually constitutes a face of $\Delta^n$. For example, when $n = 2$, the regular $2$-simplex $\Delta^2$ is the intersection of the affine plane $x + y + z = 1$ in $\mathbb{R}^3$ with the octant $x, y, z \geq 0$. In particular, $\Delta^2$ is a triangle (image taken from here).
$\hspace{66mm}$
The three sides of the triangle correspond to the images of the three face maps $\phi^2_0$, $\phi^2_1$, and $\phi^2_2$. Namely, the image of $\phi^2_0$ is the edge in the $yz$-plane, the image of $\phi^2_1$ is the edge in the $xz$-plane, and the image of $\phi^2_2$ is the edge in the $xy$-plane.