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.

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

]]>A concrete particular of a classifier which has a universal property is the subobject classifier of a 1-topos. ]]>

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.

]]>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.

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

]]>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*.

Thanks! That’s looking better now.

]]>I put some more structure into the new entry.

]]>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…

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

]]>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.

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