This question tells us that in general colimits do not exist in the category of manifolds.

However, this negative answer is not very satisfying. A manifold can be considered as a colimit of its altas. In this sense, it seems that constructing manifolds via atlas is a means of completing the model category of open subsets of $R^n$ (or more precisely a comma category based on this), so it should be reasonable that at least certain kinds of colimits should hold in the category of manifolds.

Considering the altas as a "good" system, do colimits of systems which are

1) of same-dimensional manifolds

2) have only open injections as morphisms,

3) is countable

and

4) (add your additional condition here),

exist?

PS the counter example in the above referred question fails conditions 1 and 2.

EDIT: After our good discussion below, perhaps let me restate my original question. I am motivated by the fact that a manifold is a colimit of its altas. What I'm looking for is a generalization of this fact. Can we extract out some general properties of an atlas that makes its colimit exist?

For instance do colimits of countable inverse limits of open smooth injections of (neccessarily) same-dimensional manifolds exist? (remove as many quantifiers as is uncessary).

I really do mean inverse system and not direct system. Observe that the altas (if we take all finite intersections) is filtered in the direction away from the colimit.

filteredcolimits of open embeddings would exist. I would imagine coequalizers are problematic. $\endgroup$generalnotion of manifold based on the notion of pseudogroup of homeomorphisms, which covers many special cases: topological, smooth, analytic, complex, foliated, elliptic, hyperbolic, etc. (reference: Thurston's Three-Dimensional Geometry and Topology). It seems to me a really good answer to your question ought to work in this generality. $\endgroup$