TITLE: What's an example of a subcategory whose closure under colimits takes a lot of steps to form by iteration?
QUESTION [17 upvotes]: 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?

REPLY [3 votes]: I wish I could give a reference but in 1965 John Isbell claimed that the limit completion of the 2 element group (considered as a one object category) requires a proper class of steps, but I didn't understand his argument and I don't believe he ever published it.