Let $\mathcal C$ be a locally finitely presentable category, and let $\mathcal C_0 \subseteq \mathcal C$ be a dense generator of finitely-presentable objects. Then

Every object $C \in \mathcal C$ is a colimit of objects of $\mathcal C_0$, and

The closure $\overline{\mathcal C_0}$ of $\mathcal C_0$ under finite colimits comprises precisely the finitely-presentable objects of $\mathcal C$ [1].

I'm interested in cases where the closure process in (2) takes more than one step to form. So inductively define $\mathcal C_{n+1}$ to comprise the finite colimits of objects of $\mathcal C_n$. Then $\overline{\mathcal C_0} = \cup_{n \in \mathbb N} \mathcal C_n$.

**Questions:**

What is an example of a locally finitely presentable category $\mathcal C$, and a dense generator $\mathcal C_0 \subseteq \mathcal C$ of finitely-presentable objects, such that $\overline{\mathcal C_0} \neq \mathcal C_1$?

We might opt to treat retracts specially -- so what happens if we instead define $\mathcal C_{n+1}$ to comprise the retracts of finite colimits of objects of $\mathcal C_n$?

More generally, we can ask this for higher degrees of presentability (though the iterative construction of the closure of $\mathcal C_0$ under $\kappa$-small colimits may now in principle take transfinitely many steps). I'd be interested in such examples too.

**Notes:**

- This doesn't happen if $\mathcal C$ is a presheaf category and $\mathcal C_0$ is contained in the representables: any finitely-presentable preseheaf is a finite colimit of representables.

**EDIT:** The following examples are all at least potentially mistaken; see Jeremy Rickard's comments.

My favorite example of a finite-colimit-closure which takes several steps to form is the closure of $\{R\}$ under finite colimits in $Mod_R$, for appropriate rings $R$, e.g. $R = \mathbb Z$. But in this case, although $\{R\}$ is a strong generator of finitely-presentable objects, it is not a dense generator. And I think the finite-colimit closure of the dense generator $\{R \oplus R\}$ takes only one step to form.

Similarly, the finite colimit closure of $\{\mathbb Z\} \subseteq Grp$ takes at least two steps to form, but $\{\mathbb Z\}$ is not dense, and on the other hand, the finite colimit closure of the dense generator $\{F_2\}$ occurs in one step.

For another similar example, in the final paragraph of Section 5.9 of

*Basic Concepts of Enriched Category Theory*, Kelly claims that the walking idempotent is not a colimit (in $Cat$) of copies of the walking arrow. I don't follow his proof sketch, but perhaps if it could be understood, then the argument might show that the walking idempotent is also not the finite colimit of copies of the "composable pair" category $\bullet \to \bullet \to \bullet$, which is dense in $Cat$.

[1] This is not true $\infty$-categorically, where we need to additionally close under retracts (the indexing category for an idempotent not being finite in the $\infty$-categorical sense). For example, not every retract of a finite CW complex is homotopy equivalent to finite CW complex, by the Wall finiteness obstruction.

finitecolimit closure of $\{\mathbb{Z}\}$ is reached in one step. $\endgroup$1more comment