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I have split off from differential form an entry integration of differential forms, without much ado. Maybe to be polished later…
I like very much the appendix on integration on manifolds from
which is a very careful axiomatic introduction to Lebesgue integration of differential forms on smooth manifolds.
Is it possible to sneak in something about integration without integration?
That’s discussed at Lie integration. For my present purpose I eventually want integration of differential forms have a decent detailed exposition of the standard story. This is supposed to be used eventually in an introductory course Geometry of Physics for students with no relevant background.
But for the moment the entry just sits there as is…
I put some more structure into the new entry.
Thanks! That’s looking better now.
I have added to integration of differential forms a section In cohesive homotopy-type theory, stating the result by Bunke-Nikolaus-Völkl that constructs the integration map in cohesive homotopy-type theory generally, proves Stokes generally, and shows how ordinary integration is recovered as a special case.
Am adding some related pointers also to Stokes theorem and maybe fiber integration.
Hm, that page should also link to cogerm differential form somewhere.
Please do, I haven’t been following the discussion on the cogerm forms much.
Also the entry differential form itself one day needs to be edited to something better.
Yeah, I’m developing a bad habit of commenting on forum threads saying “X should be done” rather than actually doing it… I think it’s an artifact of being short on time. Bare-bones links now added.
[ I’ll say something in reply later when I have a minute. ]
Oh,I removed it because the actual statement is better than my answer made it sound, and so as it stood it seemed to point to a problem where actually there is a beautiful story.
I realized this and wanted to fix it, but the gods of research are testing my faith these days, and didn’t allow me to find a quiet moment. So I just removed it for the time being.
But maybe to say it real quick: given an $\infty$-group $G$, then cohesion gives a canonical concept of flat (i.e. closed) $G$-valued differential forms, namely those things modulated by the object which I used to denote $\flat_{dR}\mathbf{B}G$ and which as of late I think I should better call $\flat_{dR} G$.
From this, non-closed differential forms do not appear intrinsically. They do however appear naturally when choosing a Hodge filtration on $\flat_{dR} G$. While then they are a natural choice, they are still a choice.
But if the group is even stably abelian, if it is a spectrum, then (at least then) also the object which I used to call $\Pi_{dR} \mathbf{B}G$ and which I think now I should better call $\Pi_{dR}G$ also has an interpretation in terms of differential forms, and this is now the non-flat $G$-valued forms but modulo the exact forms.
So stably, closed forms and “co-closed” forms are god-given. Fully general differential forms on the other hand appear naturally as stages in Hodge filtrations of either the closed or the co-closed forms.
Sorry if this sounds cryptic. One day I should write an exposition of this, maybe the day when I write the new version of “diff. coh. in a coh. topos” (hah!).
Meanwhile, most of what I am saying here is said, in less suggestive and more technical terms, at differential cohomology hexagon – Examples – ordinary differential cohomology.
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