Does ${\bf CPO}$ have $\omega$-colimits? Does the category ${\bf CPO}$ have $\omega$-colimits? By ${\bf CPO}$ I mean the category that has as objects the $\omega$-complete pointed partial orders and as arrows $\omega$-continuous functions.
 A: No, it doesn't. Consider the $\omega$-diagram $D=(\{0\}\to \{-1,0\}\to \{-2,-1,0\}\to\cdots)$ in $\mathbf{CPO}.$ Suppose $X$ were the colimit of our diagram and let $Z=\{-\infty_1,-\infty_2,\ldots,-2,-1,0\},$ where $-\infty_1<-\infty_2$ and both $-\infty_i$ are strictly smaller than every integer in $Z.$ Then there should be a unique map $X\to Z$ consistent with the inclusions of the $\{-i,\ldots,0\}.$ This implies all the maps $\{-i,\ldots,0\}\to X$ are injective, since the corresponding maps to $Z$ are, and by the cocone property, $X$ has a sub-poset isomorphic to $\mathbb Z_{\le 0}=\{\ldots,-2,-1,0\}.$ Since $\mathbb Z_{\le 0}$ is not pointed (contains no $\perp$), $X$ must contain at least one further, distinct element $\perp$ below all of these.
Now any map $f:X\to Z$ consistent with the cocone maps must send $\perp$ to either $-\infty_1$ or $-\infty_2$ in order to be monotone. This produces a contradiction, since $Z$ has an endomorphism sending both $-\infty_i$ to $-\infty_2$ as well as one sending both $-\infty_i$ to $-\infty_1,$ composing with one of which changes $f$ while preserving its respect for the cocones.
Abstractly, the problem with $\mathbf{CPO}$ is the requirement that bottom elements exist, but not be preserved by the morphisms. If you looked at just $\omega$-complete posets, or required morphisms to preserve the bottom elements, then you would have categories that are cocomplete, and indeed, accessibly monadic over posets.
