**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?



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




---


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).


