My question is: If I get rid of the assumption on n, but still have an abelian group object, etc., then what goes wrong with this classification? That is: In the special case that X corresponds to a space, and A corresponds to an abelian group, what is the difference between an (EM) 0-gerbe banded by A and an A-torsor? Similarly, what is the difference between an (EM) 1-gerbe banded by A and an A-banded gerbe (in the classical setting)?

I tried to parse the nonabelian classification given on the nLab involving a slightly different notion of gerbe in order to apply it to this question, but couldn't manage it.

I suppose another answer to my question would be a clear definition (for all n) of some objects that H^n+1(X, A) does classify. The end of the infty-gerbe article promises this in section 2.3 of something that Urs wrote, but I couldn't find a section 2.3 in the linked page... and clicking on "2." in the description of sections didn't seem to do anything... Maybe something's wrong with my browser. ]]>