Look up Giraud's "Cohomologie Non-abelienne", it should answer most of your questions. And, for the record, for $i=1$, by an analagous argument, you get a cohomology set (only a group if abelian) which is the moduli of vector bundles of rank $n$, or more generally principal $G$-bundles, if you replace $GL(N,\mathbb{C})$ with another group $G$.
Edit: "answer most of your questions" is somewhat stronger than what I actually meant. It's a place to start looking, and to see why the answer is far from trivial. I've heard vaguely that Lurie has some answers to this question (perhaps in Higher Topos Theory?) but I haven't had a chance to go through in detail and look at it.
