<p>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.</p>
<p>CPOs (respectively CPOs admitting bottom elements) are categories <a href="https://ncatlab.org/nlab/show/enriched+category" rel="nofollow noreferrer">enriched</a> in the <a href="https://ncatlab.org/nlab/show/interval+category" rel="nofollow noreferrer">interval category</a>, which admit a certain class of colimits (namely, of <span class="math-container">$\omega$</span>-cochains). The theory of Kelly–Lack's <a href="http://www.tac.mta.ca/tac/volumes/7/n7/7-07abs.html" rel="nofollow noreferrer">On the monadicity of categories with chosen colimits</a> therefore applies, so that <span class="math-container">$\mathbf{CPO}$</span> is (2-)monadic over <span class="math-container">$\mathbf{Cat}$</span>. Since the colimits are bounded by <span class="math-container">$\omega$</span>, this monad is finitary, and so <span class="math-container">$\mathbf{CPO}$</span> <a href="https://ncatlab.org/nlab/show/colimits+in+categories+of+algebras#for_locally_presentable_categories" rel="nofollow noreferrer">inherits colimits</a> from <span class="math-container">$\mathbf{Cat}$</span>, including in particular colimits of <span class="math-container">$\omega$</span>-cochains.</p>

CC BY-SA 4.0
2022-12-02T20:10:32.820Z
Thanks! Is there a standard reference for this? [Smyth and Plotkin](https://epubs.siam.org/doi/10.1137/0211062) mention that **CPO** has all $\omega^{op}$-limits but, strangely, don't mention anything about colimits.
LaR
CC BY-SA 4.0
2022-12-02T20:50:32.003Z
@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.
varkor
CC BY-SA 4.0
2022-12-02T18:51:22.793Z
2022-12-02T20:50:59.630Z
2022-12-02T20:50:59.630Z
varkor
<p>Does the category <span class="math-container">${\bf CPO}$</span> have <span class="math-container">$\omega$</span>-colimits? By <span class="math-container">${\bf CPO}$</span> I mean the category that has as objects the <span class="math-container">$\omega$</span>-complete pointed partial orders and as arrows <span class="math-container">$\omega$</span>-continuous functions.</p>

<ct.category-theory><semantics>
Does ${\bf CPO}$ have $\omega$-colimits?
LaR
# Does ${\bf CPO}$ have $\omega$-colimits?

(Asked by: username_0 on 2022-12-02)

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.

**Tags:** ct.category-theory, semantics


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## 1 answers

### Answer by username_1 on 2022-12-02. Score: 5

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](https://ncatlab.org/nlab/show/enriched+category) in the [interval category](https://ncatlab.org/nlab/show/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](http://www.tac.mta.ca/tac/volumes/7/n7/7-07abs.html) 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](https://ncatlab.org/nlab/show/colimits+in+categories+of+algebras#for_locally_presentable_categories) from $\mathbf{Cat}$, including in particular colimits of $\omega$-cochains.

> **Comments:**
> **username_0**: Thanks! Is there a standard reference for this? [Smyth and Plotkin](https://epubs.siam.org/doi/10.1137/0211062) mention that **CPO** has all $\omega^{op}$-limits but, strangely, don't mention anything about colimits.  
> **username_1**: @username_0: 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.  

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