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I am (slowly) working on bringing the entry differential cohomology in better shape.
For the moment I am adding a section on the Hopkins-Singer formulation.
I see Ulrich's influence :) Don't forget to learn something about eta invariants and determinant line bundles (in the business of zeta function renormalization and quantum anomalies), he has a number of papers on them (I do not know what was his motivation).
That’s a good suggestion, Zoran, but I am afraid there is no time for that. But I’ll see.
I added a (very brief so far) Properties-section on the Hopkins-Singer definition.
Then I wrote on my personal web in the section General differential cohomology the outline of an argument that is supposed to show that
for $\mathcal{A}$ a topological space
and $A = LConst Sing \mathcal{A}$ its incarnation as a constant oo-stack in $\mathbf{H} = Sh_{\infty}(CartSp)$,
the Hopkins-Singer differential cohomology with coefficients in $\mathcal{A}$ coincides with the differential cohomology with coefficients in $A$ in the (oo,1)-topos $\mathbf{H}$, as defined there.
Needs more attention, but I have to run now to get some breakfast…
I am finally getting to brushing-up the entry on differential cohomology (which has been – of all entries! – woefully neglected).
For the moment I have polished and considerably expanded the Idea section and brushed-up and commented the list of references.
In particular I am splitting off an entry differential cohomology hexagon.
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