By way of background to this question, I am interested in the properties of direct limits. They are usually defined in terms that assume there is an underlying directed poset, but according to category theory, direct limits do in fact exist for general diagrams that are not directed.
I am interested in getting an intuitive feel for what direct limits which are not based on directed sets "look like". In other words, what do we lose, in not specifying the directed nature of the poset? Clearly we don't lose existence, since category theory tells us that the limits exist. So what is it?
Every group G is a direct limit (in the conventional sense) of its finitely generated subgroups, because any two such subgroups generate another f.g. subgroup, so there is a natural directed partial order on the set of f.g. subgroups. But the poset of cyclic subgroups is not directed, yet the direct limit must exist, and must contain G.
Can anyone give an example where the direct limits of the cyclic subgroups is not G itself, and in that case what is it?
For the broader question of why directedness is usually specified in defining direct limits (except in books on category theory), I suspect but cannot prove to myself that it is connected with the idea that the concept of a direct limit is in some way topological. Topologies introduce directed sets in a natural way, in that the basis of neighbourhoods of a point is directed downwards (the intersection of two neighbourhoods is another one contained on both). Can anyone put this vague intuition on a firmer basis (no pun intended)?