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I have a vague memory that someone in the higher geometry crew once mentioned this in a more or less formal setting (a paper or talk, but maybe only here). Namely the geometric objects classified by $\mathbf{B}(\mathbf{B}^2U(1)_{conn})$. Unfortunately the only explicit mention I can find is (2.2) in The WZW term of the M5-brane and differential cohomotopy, and it’s not what I was thinking of.
This should be something like bundle 2-gerbes with connection and curving, but where the trivialisation on triple overlaps is as a bundle gerbe with connection. This is stronger than is usually specified in the literature, and I needed to point to a place where this is discussed, if at all. If such a discussion doesn’t exist, then that’s totally ok, I just wanted to check my memory wasn’t tricking me.
As a step in the “Hodge filtration” of the Deligne complex this gadget appears as Prop. 1.3 in the talk notes here, which eventually evolved into Section 5.2.13.4 in dcct “v2”, starting p. 470 here.
In the example of the delooped canonical multiplicative bundle gerbe it appears on p. 40 here and in an analogous form down in the degree you may be expecting on p. 46).
Hmm, Ok. I’m going to do a little dig through Konrad’s papers, if he doesn’t offer an answer here, but I might have just invented something or confused what I’d read with something else. Thanks, though!
This type of object is where the off-diagonal characteristic classes live. See Chapter 16 in Amabel–Debray–Haine.
By the way, does the nLab has anything about off-diagonal characteristic classes in differential cohomology?
as far as I remember, these truncated Deligne complexes appear on the nLab, so far, only here in the entry on intermediate Jacobians
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