Dense generator whose closure under finite colimits takes several steps to form?

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

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

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

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:

1. 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 $$\overline{\mathcal C_0} \neq \mathcal C_1$$?

2. 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 $$\mathcal C_n$$?

3. 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:

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 $$Cat$$.

 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.

• You’ve linked to a proof that not every abelian group is a colimit of copies of $\mathbb{Z}$. But every finitely generated abelian group is a finite colimit of copies of $\mathbb{Z}$, so the finite colimit closure of $\{\mathbb{Z}\}$ is reached in one step. Jun 28 '20 at 8:47
• Isn't it always true that every finitely presentable object is a retract of a finite colimit of the generators ? I haven't thought very much about it, but I'm under the impression that looking at the 'finite subdiagram' of the canonical diagram gives that every every object is filtered colimit of finite colimits of the generators ? Is there something going wrong with this ? Jun 28 '20 at 12:53
• I think the "walking idempotent" is a finite colimit of copies of the "composable pair" category. Take the diagram with two copies of $1\to 2\to 3$, with three maps $E,F,G$ from the first copy to the second copy, where $E(1)=1$, $E(2)=E(3)=2$, $F(1)=2$, $F(2)=F(3)=3$, and $G(1)=1$, $G(2)=G(3)=3$. Jun 28 '20 at 13:09
• @SimonHenry I think you're suggesting to look at the subdiagram $\mathcal C_1 / C \subseteq \mathcal C / C$ -- but there's no reason to think this diagram is filtered unless $\mathcal C_1$ is known to be closed under finite colimits! Jun 28 '20 at 15:20
• I'm probably wrong, but what I had in mind was given $C_0/X$ the indexing category of the canonical diagram whose colimits is $X$, look at the class of all finitely generated categories $F \to C_0 /X$. This is a filtered diagram, and each colimits along finitely generated categories can be rewritten as a finite colimit. Jun 28 '20 at 15:54

I think Simon Henry's comment works to show that there are no examples of (2). That is

Theorem: Let $$\mathcal C$$ be a locally finitely-presentable category, let $$\mathcal C_0 \subseteq \mathcal C$$ be a dense generator of finitely-presentable objects. Let $$X \in \mathcal C$$ be finitely-presentable. Then $$X$$ is a retract of a finite colimit of objects of $$\mathcal C_0$$.

So the only question is whether the retract can be eliminated, both in the finitary and the infinitary case.

Remark: I've always been a little hazy on when a colimit can be decomposed using a colimit decomposition of the indexing diagram. But let $$K$$ be a simplicial set, and let $$\{K_I \mid I \in J\}$$ be a directed sub-poset of the collection of simplicial subsets of $$K$$ with $$\cup_{I \in J} K_I = K$$. Then, according to HTT Rmk 4.2.3.9, for any diagram $$F: K \to \mathcal C$$ in a cocomplete quasicategory, we have $$\varinjlim_{k \in K} F(k) = \varinjlim_{I \in J} \varinjlim_{k \in K_I} F(k)$$.

Proof of Thm: Using the Remark, with $$K = \mathcal C_0 / X$$ and $$J$$ the (directed) collection of finitely-generated subcategories of $$K$$, we obtain a $$J$$-indexed diagram with colimit $$X$$. Since $$X$$ is finitely-presentable, we get that $$X$$ is a retract of the colimit of a finitely-generated subdiagram of $$\mathcal C_0 / X$$. According to a theorem of Pare, any finitely-generated category admits a final functor from a finite category. So $$X$$ is equally a retract of a finite colimit of objects of $$\mathcal C_0$$.

• Thanks for clarifying this ! Jun 28 '20 at 18:40
• Note to self: it's actually possible to decompose a colimit in terms of a colimit decomposition of the indexing category in complete generality. Sep 3 '20 at 23:36
• Note to self: Still have to understand Reid Barton's argument to see if it generalizes to more general dense generating sets. Feb 4 '21 at 15:07