Let $\Delta$ be the category of finite ordinal numbers with order-preserving maps, i.e., $\Delta$ consists of objects strings $$ [n]: 0 \to 1 \to 2 \to \dots \to n. $$ A morphism $f:[n] \to [m]$ is an order-preserving function (a functor) and we can think of the morphisims like diagrams where arrows don't cross.
My questions are:
- Why can't the arrows cross?
- What does it mean if an arrow crosses?
I hope I explained myself :) Thank you!
A morphism in this category is an order preserving map. Look at a morphism $f$ from $[m]$ to $[n]$, and draw it in the following way:
In one line, write the numbers $1$ to $m$ in order.
In another line, write the numbers $1$ to $n$ in order.
Connect each $x$ in the first line to $f(x)$ in the second line, using a straight arrow.
The arrows don't intersect ("cross") since the morphism is order-preserving. Indeed, if two arrows $x \to f(x)$ and $y \to f(y)$ crossed, say $x < y$ and $f(x) > f(y)$, then $f$ wouldn't be order-preserving. The converse (if the arrows don't cross then $f$ is order-preserving) is true as well.
What happens in the arrows cross? Then the corresponding function doesn't appear as a morphism in our category, since this is how we defined our category.