From P100, http://angg.twu.net/MINICATS/awodey__category_theory.pdf
I am a bit confused about a part of the discussion on $\omega$-CPO. To briefly describe the problem, definition goes following
$\omega = (N, \le)$, $\omega_n = \{ k \le n : k \in \omega \}$.
The last paragraph says that
$\omega_0 \to \omega_1 \to \omega_2 \to ...$
hence all the $\omega_n$ has a colimit which is $\omega$. This makes sense, because all those $\omega_n$ has inclusion map to $\omega$. Then it says $\omega$ itself has no colimit, which more or less makes sense, because the set of natural numbers has no supremum. But the last paragraph really confuses me:
So the colimit of the $\omega_n$ in the category of $\omega$-CPOs, if it exists, must be something else. In fact it is $\omega + 1$.
$0 \le 1 \le 2 \le ... \le \omega$
For then any bounded sequence has a colimit in the bounded part and any unbounded one has $\omega$ as colimit.
I don't quite understand this part. I thought the colimit of $\omega_n$ should be the supremum of it, therefore it needs to be $n$. The book says it's $\omega+1$. What is that? The sequence ends with $\omega$, but $\omega$ is defined to be a set. I am very confused here. I don't fully know order theory and never heard of $\omega$-CPO before this. Any idea about what the book's implying?
I found Awodey's explanation confusing as well. Here is what I think is going on:
In the category $\textbf {Pos},$ an $\omega-$CPO is a poset $D$ for which every diagram $d:\omega\to D$ has a colimit $d_{\omega}\in D;\ $ i.e. $D$ is cocomplete.
In Awodey's example $D=\omega$ and $d:\omega\to \omega$ is the diagram $\omega_0\to \omega_1\to \cdots\to $ where the arrows are inclusions. The colimit of this diagram is $\omega$, which in the language of ordinals, is $\bigcup \omega.$
On the other hand, since each $\omega_n$ is a (finite) CPO, we may, as Awodey does, consider $d$ to be a diagram in the category of $\omega-$CPO's, so that the colimit, if it exists, must be a CPO. But, $\omega$ is not cocomplete (because $d$ has no colimit $in\ \omega$) and so is not a CPO, which means the colimit cannot be $\omega.$