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Urs Schreiber, On nonabelian differential cohomology, March 17, 2008, Slides.
Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Čech cocycles for differential characteristic classes, Advances in Theoretical and Mathematical Physics 16:1 (2012), 149–250, arXiv:1011.4735, doi:10.1007/BF02104916.
I have added pointer to Def. 9.3 (p. 161) in:
which gives the actual Hopkins-Singer-style definition of nonabelian differential cohomology, by using the “differential non-abelian character map” in place where Hopkins & Singer and followups, can, in the abelian case , more simply use smashing of spectra with .
(One can roughly see that the definition in “Čech cocycles for differential characteristic classes” wants to be a model of this homotopy pullback definition – but the actual relation remains open.)
Presumably this is Definition 4.38 in the arXiv version? (I am willing to bet that almost everybody will look at the arXiv version, not the actual book, so it seems to be that arXiv numbers are at least as useful, and probably much more useful.)
We had some debate on whether to update the published version on the arXiv. While this remains unresolved, for better or worse, the copy which I am maintaining (fixing typos etc.) is the pdf here.
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