I think one of the main goals here should be the framework of Flach. Some of the things that have to be modified: Integral coeffs Deligne cohomology: This might be a problem at several stages: first to get the actual complexes of sheaves, with product, and then if we want to take a resolution. For the resolution there might be at least two ways: Either use Thom-Sullivan in Levine’s book, or use infinity-categories: more precisely take the nerve of our category, formally invert the unit object or some other already invertible guy according to the recipe of Smith, in order to get an infinity category which satisfies Lurie’s criteria for rigidification construction(s). One could also hope to avoid the resolution issue, by looking at some Hodge complexes in the spirit of Huber.
More about Flach: Talk about the h-topology method suggested by Cisinski. A general philosophical point is that Flach constructs CTs by topos theory, and constructs for example the Weil-etale topos. We construct a CT via motivic stable homotopy theory. To construct a combined version of the two, maybe one has to use higher topos theory, or some stable infty-cat related to the Weil-etale topos.
For the regulator: Need to work in (I think) the category of sheaves of spectra with transfers, and use for motivic cohomology the h-sheafification of the spectrum associated to the motivic complexes of Voevodsky or maybe some other version like Suslin-Friedlander or Chow complexes sheafified or maybe C-D or motivic functors or something else. Then Deligne cohomology should also be a commutative monoid object in there and the motivic one should be the initial commutative monoid. This should give a map in the homotopy category I imagine, but that might be enough, or we might be able to lift it by some fibrant-cofibrant argument or by arguments similar to the original C-D theorem, or by something from Riou’s thesis.
Another point which seems crucial is one should work with etale motivic cohomology (also pointed out by Huber), or maybe even Weil-etale in some sense, or something even more refined. In the etale setting, things might be known by the next article of C and D, which are not known in the standard setting, for example Pic-hat comparison maybe? Could ask Deglise about the latter.
Maybe email Flach and just point out that Goncharov’s groups might inherit some good properties from comparison with Feliu, this could be useful for him. Should also discuss with him what he thinks about pairing with homology, is it ok, is it better? What is homology of a topos???
I don’t know if it’s relevant at all, but Soule has some ideas on what he calls adelic intersection theory in some paper I don’t have, maybe in a PSPUM volume.
Hornbostel proves that motivic homotopy theory becomes a HA(G) context, OSLT. Does this have any effect on some future research projects, or some speculation, here or in the Peter article?
Mention generalized schemes with a formalism of SH (and DM?), maybe everything should be “circular” in a suitable sense, i.e. the same objects/categories everywhere, at all stages.
Check what’s in Gillet’s various slides, see if our stuff can be used for arithmetic intersection theory on stacks.
nLab page on Old Arakelov III preview