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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeSep 26th 2012

I have split off from differential form an entry integration of differential forms, without much ado. Maybe to be polished later…

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeSep 26th 2012
• (edited Sep 26th 2012)

I like very much the appendix on integration on manifolds from

• Shlomo Sternberg, Lectures on differential geometry

which is a very careful axiomatic introduction to Lebesgue integration of differential forms on smooth manifolds.

• CommentRowNumber3.
• CommentAuthorEric
• CommentTimeSep 26th 2012

Is it possible to sneak in something about integration without integration?

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeSep 26th 2012

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…

• CommentRowNumber5.
• CommentAuthorTobyBartels
• CommentTimeSep 27th 2012

I put some more structure into the new entry.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeSep 28th 2012

Thanks! That’s looking better now.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeApr 29th 2014
• (edited Apr 29th 2014)

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.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeApr 29th 2014

Hm, that page should also link to cogerm differential form somewhere.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeApr 29th 2014

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.

• CommentRowNumber10.
• CommentAuthorMike Shulman
• CommentTimeApr 29th 2014

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.

• CommentRowNumber11.
• CommentAuthorColin Tan
• CommentTimeJun 30th 2014
Reading Urs' habilitationsschrift, I wonder : is there a universal property which characterizes the universal differential 1-form, say, or the smooth space which classifies differential 1-form? I mean, such a universal property would help us understnad what makes differential forms so special in a cohesive oo-topos which makes us want to consider them when studying, say, differential cohomology. Or, a weaker problem, to find a property which the differential 1-form classifier satisfies of which maps into such classifier give meaningful quantities of smooth spaces. If the last question is not precise, then a specific analogous situation in homotopy theory was cohomology classes, the speicific Eilenberg-ManLane spaces as classifiers and therafter the concept of ring spectrum of which the Eilenberg-MacLane spectrum is one such.

A concrete particular of a classifier which has a universal property is the subobject classifier of a 1-topos.
• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJun 30th 2014
• (edited Jun 30th 2014)

[ I’ll say something in reply later when I have a minute. ]

• CommentRowNumber13.
• CommentAuthorColin Tan
• CommentTimeJul 16th 2014
Urs, I enjoyed your original answer at #12. Did you remove it intentionally?
• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeJul 16th 2014
• (edited Jul 16th 2014)

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.