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I considerably expanded and reorganized the discussion at Chern-Simons 2-gerbe.
There is now a decvent Idea-section;
I created three subsections for three different kinds of constructions of this beast.
The first subsection contains now a detailed account of the consztruction by Brylinski-McLaughline of an explicit Cech-Deligne cocycle. I end this with saying that once the construction is there, proving that it is correct is easy. The mystery is maybe how one comes up with the construction in the first place!
So then in the next subsection I rederive this construction as a special case of the general methods described at infinity-Chern-Weil theory. So I show that from Lie integration of the underlying Lie algebra cocycle one gets a canonical lift to pseudo-connections with values in the Lie algebra, and turning the crank, out drops the Brylinski-McLaughlin construction. I’ll later see if I can streamline this discussion a bit more.
Then there is a third subsection which is supposed to deal with the construction of bundle 2-gerbe representatives. But here I am being lazy and just give the references so far. Even though the construction is actually simple.
I spent some time on trying to make the section on the derivation of the Deligne-cocycle from “oo-Chern-Weil theory” more readable, See here.
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