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following public demand, I created an entry ordinary differential cohomology.
added brief discussion of two of the exact sequences characterizing ordinary differential cohomology
It may be there is a small terminological inconsistency between Idea sections in differential cohomology and ordinary differential cohomology.
Differential cohomology is a refinement of ordinary cohomology such that a differential cocycle is to its underlying ordinary cocycle as a bundle with connection is to its underlying bundle.
…
The best known version of differential cohomology is a differential refinement of generalized cohomology in the sense of the generalized Eilenberg–Steenrod axioms.
Word ordinary above seems to mean generalized cohomology in the sense of Eilenberg-Steenrod ?
ordinary differential cohomology says
Ordinary differential cohomology is the differential cohomology-refinement of ordinary cohomology.
So here ordinary means really ordinary/classical.
Phrase differential refinement may mean a construction, or an a priori definition. It is used in ordinary differential cohomology in a phrase that it “exists” so one probably assumes the a priori definition for the case of generalized cohomology theory like in Bunke-Schick. Right ? In that case, Urs, should we make an entry, link or redirect for differential refinement ? I mean, one thing is to go into the whole theory of differential cohomology and how it is constructed and another is just the statement what it a priori means. The entry ordinary differential cohomology seems to be built on the notion of differential refinement, which has to be searched for.
Okay, right, “ordinary” means different things here.
I have changed the first “ordinary” to “plain”, such as to read
Differential cohomology is a refinement of plain cohomology such that a differential cocycle is to its underlying ordinary cocycle as a bundle with connection is to its underlying bundle.
The “ordinary” at ordinary differential cohomology I have accompanied with the last words of
Ordinary differential cohomology is the differential cohomology-refinement of ordinary cohomology, for instance realized as singular cohomology.
Notice that the paragraph below that is all about the distinction between ordinary and generalized cohomology.
Phrase differential refinement may mean a construction, or an a priori definition. It is used in ordinary differential cohomology in a phrase that it “exists” so one probably assumes the a priori definition for the case of generalized cohomology theory like in Bunke-Schick. Right ?
Yes. One reason why the entry on differential cohomology is still in such a preliminary state “under construction” is that I am still thinking about how to present it most systematically. Of course I have my own ideas about how to make things systematic in terms of cohesive $\infty$-toposes, but ignoring that, the state of the literature is slightly inconclusive:
Hopkins-Singer gave an explicit construction of cocycles and cohomology for differential cohomology. Bunke-Schick rather listed a bunch of axioms they want to have satisfied, and then constructed various models for these. These axioms do not uniquely fix the models, there are slight differences in the various differential K-theories out there.
It’s good that (if) you are getting interested in this. Somebody should push me to polish up that entry on differential cohomology.
Maybe next semester I’ll be teaching a master course on this. At the very least by then I should produce decent writeups.
Of course I have my own ideas about how to make things systematic in terms of cohesive $\infty$-toposes, but ignoring that, the state of the literature is slightly inconclusive
I am glad somebody spelled this out. When in a cohesive $\infty$-topos the things are more canonical I had in back of my mind the slight non-uniqueness of the differential refinements of K-theory (if I understood correctly) in the work of Bunke-Schick. On the other hand, I like that they have axiomatics in classical terms, as one indeeds refines classical cohomology.
Yes, Jim, Urs is well aware of this situation, I am sure. The fact that the Bunke-Schick axioms do not fix uniquely the differential refinement for all generalized cohomology theories but do fix for some, can be understood in two ways. One that indeed more theories are interesting; and another that we still have to search for axioms in classical setup. Of course, Urs said, the cohesive topos tells you the things canonically, but maybe the classical language can fix it as well. So, the questions is in some sense genuinely open.
see also a paper by Simons and Sullivan
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