I once heard a rumour that various nice categories of stacks were co-complete. Gepner and Henriques, working from the groupoids point of view, give a construction [link] of 2-colimits of *topological* groupoids, but I haven't seen any definitive statement on whether there is a construction that works for, eg., differential or holomorphic stacks (let alone a proof). Personally I don't expect it, but I just want a definitive answer, so that I know that certain constructions I'm working on aren't a waste of time.

So what I'm asking is: are differential stacks co-complete? How about holomorphic stacks? If so, are there easy constructions? (Groupoid point of view preferred, but I'll take what I can get.) If the answer's no, are there any counterexamples? If none of these questions can be answered, can an expert come on here and say "it's not known"?

groupoidsdoes not mean that there are 2-colimits of topologicalstacks. The Yoneda embedding generally fails to preserve colimits! $\endgroup$ – David Carchedi May 30 '13 at 11:26