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.
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If you relax the assumption that the CPOs have bottom elements (i.e. initial objects), or require that the morphisms also preserve the bottom element, then the answer is yes.
CPOs (respectively CPOs admitting bottom elements) are categories enriched in the interval category, which admit a certain class of colimits (namely, of $\omega$-cochains). The theory of Kelly–Lack's On the monadicity of categories with chosen colimits therefore applies, so that $\mathbf{CPO}$ is (2-)monadic over $\mathbf{Cat}$. Since the colimits are bounded by $\omega$, this monad is finitary, and so $\mathbf{CPO}$ inherits colimits from $\mathbf{Cat}$, including in particular colimits of $\omega$-cochains.
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$\begingroup$ Thanks! Is there a standard reference for this? Smyth and Plotkin mention that CPO has all $\omega^{op}$-limits but, strangely, don't mention anything about colimits. $\endgroup$– LaRCommented Dec 2, 2022 at 20:10
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1$\begingroup$ @LaR: looking at the paper, I realise I missed that your morphisms don't preserve the initial objects (which is denoted in that paper by $\mathbf{CPO}_\bot$), so this does not answer your question. Not requiring preservation of one of the colimits is unnatural from a categorical point of view, so I would need to think some more about this. $\endgroup$– varkorCommented Dec 2, 2022 at 20:50