Induced map on simplices from order-preserving maps between finite ordinal numbers Let $\Delta$ be the category of finite ordinal numbers with order-preserving maps, i.e., $\Delta$ consists of objects strings $[n]: 0→1→2→⋯→n$.
A morphism $f:[n]→[m]$ is an order-preserving function (a functor) and we can think of the morphism like diagrams where arrows don't cross.
For an arrow $[n] \overset{\Theta}{\to} [m]$ we get an induced map $| \Delta^n | \overset{\Theta_*}{\to} | \Delta^m |$ given by $\Theta_* ( t_0 , \ldots , t_n ) = ( s_0 , \ldots , s_n )$ where $$s_i = \begin{cases}0, &\text{if }\Theta^{-1}(i) = \emptyset \\ \sum_{j \in \Theta^{-1}(i)} t_i, &\text{if }\Theta^{-1}(i) \neq \emptyset.\end{cases}$$
Here $| \Delta^n |$ is the topological $n$-simplex defined as a space: $$| \Delta^n | := \{ (t_0 , \ldots , t_n) \in \mathbb{R}^{n+1} : {\textstyle \sum_{i=0}^n} t_i = 1, t_i \in [0,1] \}.$$
I don't understand why I get that induced map, and why it's defined that way. I'm having trouble understanding the $s_i$. Can someone explain what's happening? 
 A: As always to understand a general definition one needs to look at concrete examples. Let's consider the cases when $n=1$ and $m=2$ and when $n=2$ and $m=1$. 
The standard $1$-simplex is a line from $(1, 0)$ to $(0, 1)$ in $\mathbb{R}^2$ like so.  
And the standard $2$-simplex is the triangle with vertices $(1, 0, 0)$, $(0,1, 0)$ and $(0, 0, 1)$ in $\mathbb{R}^3$ like so
Let us see what order-preserving functions there are from $[1]$ to $[2]$. there are three constant functions $c_0$, $c_1$ and $c_2$ plus three injective ones, $f_0$, $f_1$ and $f_2$ defined as
\begin{align*} f_0(0) = 0 \text{ and } f_0(1) = 1\\ f_1(0) = 0 \text{ and }  f_1(1) = 2\\
f_2(0) = 1 \text{ and }  f_2(1) = 2
\end{align*}
Let's pick $f_0$ and see what $(f_0)_*$ is. We easily compute
$$ (f_0)_*(t_0, t_1) = (t_0, t_1, 0). $$
What this means, then, is that $(f_0)_*$ embeds the $1$-simplex as the bottom yellow line in the $2$-simplex (if we interpret the third coordinate as the vertical axis). Similarly the other two injective functions will embed the $1$-simplex as green and red lines in the $2$-simplex.
In general, an injective function will embed the $n$-simplex as an $n$-face of the $m$-simplex.
How about functions that are non-injective? Let $\Theta : [2] \to [1]$ be the function \begin{align*} \Theta(0) &= 0 \\ \Theta(1) &= 1\\ \Theta(2) &= 1.\end{align*}
Then we compute $$\Theta_*(t_0, t_1, t_2) = (t_0, t_1 + t_2).$$ Consider the line in the $2$-simplex where the first coordinate is, say, $\frac{1}{2}$. This is the red line depicted below. 
What the map $\Theta_*$ does is map that whole line to the point $(\frac{1}{2}, \frac{1}{2})$ in the standard $1$-simplex. Similarly it will map the line where the first coordinate is $\frac{1}{3}$ to the point $(\frac{1}{3}, \frac{2}{3})$ in the standard $1$-simplex.
I hope this helps in understanding why the definition is the way it is.
