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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 27th 2022
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI 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](https://arxiv.org/abs/1506.07557)_, 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 27th 2022
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   Author: Urs
   Format: MarkdownItexAs a step in the "Hodge filtration" of the Deligne complex this gadget appears as Prop. 1.3 in the talk notes [here](https://ncatlab.org/schreiber/files/DiffCoIsCoHo.pdf), which eventually evolved into Section 5.2.13.4 in dcct "v2", starting [p. 470 here](https://ncatlab.org/schreiber/files/dcct170811.pdf#page=497). In the example of the delooped canonical *multiplicative* bundle gerbe it appears on [p. 40 here](https://arxiv.org/pdf/1304.0236.pdf#page=40) and in an analogous form down in the degree you may be expecting on [p. 46](https://arxiv.org/pdf/1304.0236.pdf#page=46)).

   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).

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 27th 2022
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   Author: DavidRoberts
   Format: MarkdownItexHmm, 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!

   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!

4. • CommentRowNumber4.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMay 27th 2022
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   Author: Dmitri Pavlov
   Format: MarkdownItexThis 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?

   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?

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 27th 2022
   • (edited May 27th 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexas far as I remember, these truncated Deligne complexes appear on the nLab, so far, only [here](https://ncatlab.org/nlab/show/intermediate+Jacobian#CharacterizationAsHodgeTrivialDeligneCohomology) in the entry on [[intermediate Jacobians]]

   as far as I remember, these truncated Deligne complexes appear on the nLab, so far, only here in the entry on intermediate Jacobians