Before doing any serious work on representability, must read carefully the article of Deglise and Mazzari on syntomic cohomology.
In the Sardinia talk I had a diagram of CTs: All, ordinary, orientable, Q-coeffs, BO, Beilinson, Weil, each corresponding to a certain subcat or version of SH, where in particular SH_{BO} should be related to DM via some adjunction or something. This should be one way of understanding properties of CTs.
1015 : Remark on rigidity over several fields by Serge Yagunov
It is shown that T-spectrum representable cohomology theories on smooth algebraic varieties satisfy normalization condition over nonreal fields. As a consequence, one can see that the rigidity property holds for all representable theories over considered fields.
Pull out the stuff not specific to Deligne cohomology, and put in a separate section. This could also be done for comparison between Gillet char classes and the CD map, with sufficient hyps on the target cohom (including maybe something about the equivariance).
Check Levine MM article for Bloch-Ogue theories. I think his axioms implies Chern classes. Is there any reason to think that they imply Galois or Nisnevich descent??
See paper note from Cisinski conversation for notes on the G-model structure for G being the class or of representable sheaves.
Deglise said in a conversation that to prove that some complex satisfies Nis descent, it is enough to prove that any distinguished square gives rise to a long exact sequence in cohomology.
Deglise also said that Cech covers and hypercoverings defs are not equivalent in the case of Cech descent, see paper notes from conversation.
Read the Arakelov arxiv preprint and pull out the essential/abstract part of the equivariance argument for the Deligne spectrum.
Here we could put a slightly original explanation of the difference between satisfying Kunneth for Gm and for P1.
Tony mentioned (May 2011) something he called closed Mayer-Vietoris, I think in relation to understanding K-theory or K’-theory of a curve with singularities “analytically like the crossing of coordinate axes”. What is this closed MV? Does it have an interpretation in terms of descent or the six functors?
Jakob regarding switching between covariant and contravariant motives: You replace * by ! throughout, e.g. in my old paper, the motive is f_* f^* \one, but in our paper it is f_! f^! \one. The motive with comp. supp was f_! f^* \one, in our paper it would be (but we don’t seem to need it) f_* f^! \one. Andreas Holmstrom: and how is the intermediate extension defined in the C and D world? Jakob Scholbach: just replace * by ! and reverse the direction of all arrows Jakob Scholbach: so in this case im (j_! –> j_) doesn’t change its shape. END
nLab page on Representable CTs temp