# 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.

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.
• 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.
• @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. Dec 2, 2022 at 20:50