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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 $G$.
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 $G$.