Which abelian groups are $\aleph_1$-filtered colimits of finitely-generated abelian groups?

Question

Observation: Every $\aleph_1$-directed colimit $\varinjlim_i X_i$ of finite sets is finite.

Proof: Because the $X_i$'s are finite, the Mittag-Leffler condition holds, so by passing to the diagram of essential images, we may assume that the transition maps are injective. Therefore the cardinalities of the $X_i$ must be bounded (here is where we use $\aleph_1$-directednes), and by passing to a cofinal sequence we may assume that the cardinalities are constant. By the pigeonhole principle, all the transition maps are bijections. So the colimit is given by evaluation at any of its terms, and is finite.

Question: Is every $\aleph_1$-directed colimit of finitely-generated abelian groups finitely-generated? How about not-necessarily-abelian groups?

More generally, let $\mathcal C$ be a locally finitely-presentable category. Is every $\aleph_1$-directed colimit of finitely-presentable objects finitely-presentable?

Answer 1

Yes. This can be rephrased as: let $G$ be an abelian group [resp. group] with a chain of f.g. subgroups $G_\alpha$ for $\alpha<\omega_1$. Is $G$ f.g.?

The answer is yes for abelian groups:

The answer is clearly yes if $G_\alpha=G$ for large $\alpha$. Otherwise, up to extract we can suppose that all inclusions are proper. Then $\bigcup_n G_n$ is an infinitely generated group. But then it is contained in $G_\omega$, which is therefore not f.g.

The answer is no for groups. Indeed, every countable group embeds into a f.g. group, so it is easy to define by induction such a directed limit of non-surjective injections, embedding the colimit into a f.g. group at each countable ordinal.

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By the way, there exist familiar groups that are directed limit over a $\omega_1$-complete nets (nets in which countable subsets are bounded) of f.g. groups. An example is the group of permutations of any infinite set (Galvin 1995).

Answer 2

Ycor answered the question, so I shall add a couple of remarks providing some insight about why these results should be true at all.

Ind-completions are nested $$\text{Ind}(C) \supset \text{Ind}_{\aleph_1}(C) \supset \text{Ind}_{\aleph_2}(C) \supset \dots \supset \bigcap_{\lambda \in \text{RegCard}} \text{Ind}_{\lambda}(C).$$

Via the classical representation theorem $\text{Ind}_{\lambda}(C) \simeq \text{Flat}_\lambda(C^\circ,\text{Set}) \simeq \text{Cocont}\lambda\text{cont}(\text{Psh}(C^\circ),\text{Set})$, we see that their intersection actually coincides with the category of tiny objects in the presheaf category, that is it coincides with the Cauchy completetion $\hat{C}$ of $C$.

So, it is indeed true that, in a sense, $\text{Ind}_\lambda(C)$ converges to $\hat{C}$ when $\lambda$ grows. Moreover, for a small category $C$, there exists a $\lambda$ such that this matrioska stabilizes. I think one can show that such $\lambda$ is $\kappa^{++}$, where $\kappa$ is the cardinality of $C$.

As a result, for every small category, there must be a regular cardinal $\lambda$ such that $\text{Ind}_\lambda(C) \simeq \hat{C}$. Of course, if $C$ is already cauchy complete, we get $\text{Ind}_\lambda(C) \simeq \hat{C} \simeq C$.

I do not have a precise reference for this, but I think some shadow of these ideas sit at the very core of a recent work by Tendas: On continuity of accessible functors. Also 2.6 in LPAC actually relies on this kind of ideas.