Let $\mathcal{C}$ be a cocomplete category and $\mathcal{S} \subseteq \mathcal{C}$ be a full subcategory. The *colimit completion* $\mathrm{Colim}^\mathcal{C}(\mathcal{S})$of $\mathcal{S}$ in $\mathcal{C}$ is the smallest full subcategory $\mathcal{S} \subseteq \mathrm{Colim}^\mathcal{C}(\mathcal{S}) \subseteq \mathcal{C}$ which is closed under the formation of small colimits in $\mathcal{C}$. Naturally, it can be attained via a transifinite induction: set $\mathrm{Colim}^\mathcal{C}_0(\mathcal{S}) = \mathcal{S}$; for successor ordinals define $\mathrm{Colim}^\mathcal{C}_{\alpha+1}(\mathcal{S})$ to be the full subcategory of $\mathcal{C}$ consisting of objects which are the colimits of small diagrams in $\mathrm{Colim}^\mathcal{C}_\alpha(\mathcal{S})$; and for limit ordinals take unions. If we iterate through *all* ordinals, we get $\mathrm{Colim}^\mathcal{C}(\mathcal{S})$ at the end.

In category theory, one is often interested in cases where this process stabilizes after one step. For instance if $\mathcal{S}$ is a dense generator in $\mathcal{C}$ (or even just colimit-dense), then the process stabilizes after one step. If $\mathcal{S}$ is a regular generator in $\mathcal{C}$, then the process stabilizes after at most two steps. And the canonical example of a category lacking a strong generator is $\mathsf{Top}$ where one can argue as for $k$-spaces that the process always stops after one step. One is also interested in cases where we close under not all colimits, but just those of certain shapes; if the class of diagrams is saturated or almost so, then in the free case where $\mathcal{C}$ is a presheaf category and $\mathcal{S}$ is the representables, the process stabilizes in one step.

But I'd like to see some examples where things go bad.

**Question:**

- What's an example of $\mathcal{S} \subseteq \mathcal{C}$ where $\mathrm{Colim}^\mathcal{C}(\mathcal{S})$ takes more than two steps to attain?
- Infinitely many steps?
- A proper class of steps?

I'm really interested in seeing a proper class of steps, but I'd be happy to at least see something easier. And I'm not at all averse to taking advantage of duality / doing everything with limits instead of colimits. And on the flip side,

- If $\mathcal{S}$ is small and $\mathcal{C}$ is locally presentable, must the process stop after a small number of steps?