nLab integration of differential forms



Integration theory

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from point-set topology to differentiable manifolds

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smooth space


The magic algebraic facts




infinitesimal cohesion

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graded differential cohesion

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id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


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Here we discuss the integration of a differential form (possibly twisted in some way) on a topological manifold (possibly with additional structure) over an appropriately structured submanifold (or formal linear combination thereof).


See at differential form for basic definitions.

Integration of top-dimension pseudoforms (pseudoforms to measures)

Let XX be an nn-dimensional topological manifold, and let ω\omega be a continuous nn-pseudoform on XX. Suppose that XX is paracompact and Hausdorff, so that we may find a locally finite cover of XX with a subordinate partition of unity and a continuous coordinate chart on each patch. (When XX is differentiable, or even smooth, then these may also be chosen to be differentiable or smooth, which may be convenient but is not necessary.) Then ω\omega defines a measure on XX as follows:

  • On each coordinate patch UU, fix the orientation given by the coordinates to turn ω\omega into an untwisted nn-form ω^\hat{\omega}; then write ω^\hat{\omega} in coordinates as

    ω^=ω Udx 1dx n. \hat{\omega} = \omega_U \wedge \mathrm{d}x^1 \wedge \cdots \wedge \mathrm{d}x^n .

    In this situation, it is convenient also to write

    ω=ω Udx 1dx n; \omega = \omega_U \mathrm{d}x^1 \cdots \mathrm{d}x^n ;

    in other words, we interpret dx 1dx n\mathrm{d}x^1 \cdots \mathrm{d}x^n as the absolute value of dx 1dx n\mathrm{d}x^1 \wedge \cdots \wedge \mathrm{d}x^n.

  • The coordinates on UU define a diffeomorphism between UU and an open subset of R n\mathbf{R}^n that we'll also call UU; so use the latter formula to interpret

    (1) Uω= Uω U(x 1,,x n)dx 1dx n, \int_U \omega = \int_U \omega_U(x^1,\ldots,x^n)\, \mathrm{d}x^1 \cdots \mathrm{d}x^n ,

    where the right-hand side is now interpreted in the usual way as an integral with respect to Lebesgue measure.

  • Using the partition of unity, write

    ω= Uw Uω U, \omega = \sum_U w_U \omega_U ,

    where w Uw_U is a weight function defined on UU and ω U\omega_U is the restriction of ω\omega to UU. Then we have

    Xω= U Uw U(x 1,,x n)ω U(x 1,,x n)dx 1dx n, \textstyle{\int}_X \omega = \sum_U \textstyle{\int}_U w_U(x^1,\ldots,x^n) \omega_U(x^1,\ldots,x^n)\, \mathrm{d}x^1 \cdots \mathrm{d}x^n ,

    or more generally,

    Eω= U Uχ E(x 1,,x n)w U(x 1,,x n)ω U(x 1,,x n)dx 1dx n \textstyle{\int}_E \omega = \sum_U \int_U \chi_E(x^1,\ldots,x^n) w_U(x^1,\ldots,x^n) \omega_U(x^1,\ldots,x^n)\, \mathrm{d}x^1 \cdots \mathrm{d}x^n

    for EE a measurable subset of XX and χ E\chi_E the characteristic function of EE. This makes ω\omega into a measure ω:E Eω\omega \colon E \mapsto \int_E \omega on XX.

A priori, this definition depends not only on the particular coordinate patches chosen but also on the partition of unity chosen to go with them. Furthermore, the definition could be done just as easily (perhaps even more easily) for something other than an nn-pseudoform. But the (perhaps surprising) fact that justifies it all is this:


When ω\omega is an nn-pseudoform, the definition of Eω\int_E \omega is independent of the coordinates and partition chosen. Furthermore, the map from nn-pseudoforms to measures is linear.

Note that, if ω\omega were an nn-form instead of a pseudoform, then the definition would depend on the orientation of the coordinates chosen. We could fix that by using the absolute value |ω U|{|\omega_U|} in place of ω U\omega_U in (1) and the following equations, but then the map from forms to measures would not be linear.

Measures to pseudoforms

It may also be enlightening to consider how to go back from a measure to an nn-pseudoform. If ω\omega is an absolutely continuous Radon measure on XX, then it defines an nn-pseudoform (which we may also call ω\omega) as follows:

  • Given a point aa, choose one of the two local orientations at aa.
  • Given nn linearly independent vectors (v 1,,v n)(v_1,\ldots,v_n) at aa, develop them into a coordinate system on a neighbourhood UU of aa.
  • For sufficiently large natural number kk, the coordinate cube C kC_k of points with coordinates in [0,1/k] n[0,1/k]^n exists (lies within UU).
  • Let LL be lim kk nω(C k)\lim_{k \to \infty} k^n \omega(C_k).
  • If the coordinate system on UU is positively oriented at aa, then let ω(v 1,,v n)\omega(v_1,\ldots,v_n) be LL; if the coordinate system on UU is negatively oriented at aa, then let ω(v 1,,v n)\omega(v_1,\ldots,v_n) be L-L.
  • Extend the definition to nn arbitrary vectors by continuity (which necessarily maps a linearly dependent tuple of vectors to zero).

Again, this definition is independent of the coordinate system chosen (as long as it extends the given vectors); or if that's not true, then we messed up and need to add further restrictions to the absolutely continuous Radon measure ω\omega. The definition is not independent of the orientation chosen, of course; thus we get a pseudoform rather than an untwisted form. You might try to ignore the orientation and take ω(v 1,,v n)\omega(v_1,\ldots,v_n) to be LL always, but that does not define an exterior form, as is most easily seen if two vectors are switched (which does not change LL). Instead, this would define an absolute differential form (which is equivalent to a pseudoform when, as here, the degree equals the dimension).

Integration of more general forms

One can integrate forms other than nn-pseudoforms, of course, but only over certain structures within the manifold XX. Specifically, if RR is a pp-dimensional submanifold of XX (that is a pp-dimensional manifold UU equipped with a map R:UXR\colon U \to X), then we would like to integrate pp-forms or pp-pseudoforms (defined on XX) over RR. Here is how we do this:

  • We may integrate a pp-form η\eta over RR if RR is oriented, that is if UU is oriented. We pull back η\eta from XX to UU, then use the orientation on UU to turn η\eta into a pp-pseudoform, which we can then integrate on the pp-dimensional manifold UU.

  • We may integrate a pp-pseudoform η\eta over RR if RR is pseudooriented, that is if it is equipped with a map that, for each point aa on UU, takes a local orientation of XX at R(a)R(a) to a local orientation of UU at aa, continuously in aa and taking opposite orientations to opposite orientations. Then locally, we turn η\eta into a pp-form on XX using a local orientation on XX, pull that back to UU, and use the corresponding local orientation on UU to turn that back into a pp-pseudoform, which we can then integrate on UU.

Thus, while integration of nn-pseudoforms is the most basic, integration of general pp-forms is actually a bit simpler than integration of general pp-pseudoforms. Integration of other twisted or vector-valued forms can also be done, again given appropriate structure on RR.

Note that, if XX is thought of a submanifold of itself, then it has a natural pseudoorientation that takes each local orientation to itself, and so we recover the original definition of integration of nn-pseudoforms on XX. Also, if XX is an oriented manifold, then it may be viewed as an oriented submanifold of itself, giving a definition of integration of nn-forms on XX.

To integrate on unoriented submanifolds of arbitrary dimension, use absolute differential forms. To integrate nonlinear differential forms, use cogerm differential forms.

Integration on other domains

One often sees the definition of integration given for parametrised submanifolds, that is submanifolds where UU is an open subset of R p\mathbf{R}^p. This amounts to a combination of the concepts above, with the two uses of UU (as a coordinate patch in XX or as the source of a submanifold of XX) identified. The theorem that the integral of an nn-pseudoform on XX is independent of the coordinates chosen now becomes a theorem that the integral of a parametrised submanifold is independent of the parametrisation (up to some details about orientation), which in the end returns the result that one can integrate forms over arbitrary submanifolds (given an orientation or pseudoorientation as above).

We can also integrate on a formal linear combination of submanifolds, which is handy since these are the chains of de Rham homology?. This is straightforward:

iα iU iω iα i U iω. \int_{\sum_i \alpha_i U_i} \omega \coloneqq \sum_i \alpha_i \int_{U_i} \omega .

In cohesive homotopy-type theory

We discuss here a general abstract formulation of differential forms, their integration and the Stokes theorem in cohesive (∞,1)-toposes, hence the axiomatics of cohesive homotopy type theory (following Bunke-Nikolaus-Völkl 13, theorem 3.2).

General definition

Let H\mathbf{H} be a cohesive (∞,1)-topos. We write (Π)(\Pi \dashv \flat \dashv \sharp) for the defining triple of adjoint modalities (shape modality \dashv flat modality \dashv sharp modality).

As usual (see at structures in a cohesive ∞-topos – de Rham cohomology) we deduce from this the secondary cohesive modalities Π dR\Pi_{dR} and dR\flat_{dR}: for a pointed object with loop space object GG (for instance the delooping BG\mathbf{B}G) dRG\flat_{dR}G is defined by the long homotopy fiber sequence

Gθ G dRGBGBG G \stackrel{\theta_G}{\longrightarrow} \flat_{dR} G \stackrel{}{\longrightarrow} \flat \mathbf{B}G \longrightarrow \mathbf{B}G

where the morphism on the right is the \flat-counit. Here θ G\theta_G is the Maurer-Cartan form on GG.

(Beware that elsewhere we write dRBG\flat_{dR}\mathbf{B}G for what here we write dRG\flat_{dR}G.)

Dually Π dR\Pi_{dR} is defined.

Write now

Stab(H)=T *HTHH Stab(\mathbf{H}) = T_\ast \mathbf{H} \hookrightarrow T \mathbf{H} \to \mathbf{H}

for tangent cohesive (∞,1)-topos of H\mathbf{H}, extending it by its stabilization Stab(H)Stab(\mathbf{H}) given by spectrum objects in H\mathbf{H}.

We assume that there is an interval object

**(i 0,i 1)Δ 1 \ast \cup \ast \stackrel{(i_0, i_1)}{\longrightarrow} \Delta^1

“exhibiting the cohesion” (see at continuum) in that there is a (chosen) equivalence between the shape modality Π\Pi and the localization L Δ 1L_{\Delta^1} at the the projection maps out of Cartesian products with this line Δ 1×()()\Delta^1\times (-) \to (-)

ΠL Δ 1. \Pi \simeq L_{\Delta^1} \,.

This is the case for instance for the “standard continuum”, the real line in H=\mathbf{H} = Smooth∞Grpd.

It follows in particular that there is a chosen equivalence of (∞,1)-categories

(H)L Δ 1H \flat(\mathbf{H})\simeq L_{\Delta^1}\mathbf{H}

between the flat modal homotopy-types and the Δ 1\Delta^1-homotopy invariant homotopy-types.

Given a stable homotopy type E^Stab(H)TH\hat E \in Stab(\mathbf{H})\hookrightarrow T \mathbf{H} cohesion provides two objects

Π dRE^, dRE^Stab(H) \Pi_{dR} \hat E \,,\;\; \flat_{dR}\hat E \;\; \in Stab(\mathbf{H})

which may be interpreted as de Rham complexes with coefficients in Π( dRΣE^)\Pi(\flat_{dR} \Sigma \hat E), the first one restricted to negative degree, the second to non-negative degree. Moreover, there is a canonical map

Π dRE^ d dRE^ ι θ E^ E^ \array{ \Pi_{dR}\hat E && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{dR}\hat E \\ & {}_{\mathllap{\iota}}\searrow && \nearrow_{\mathrlap{\theta_{\hat E}}} \\ && \hat E }

which interprets as the de Rham differential d\mathbf{d}. See at differential cohomology diagram for details.

Throughout in the following we leave the “inclusion” ι\iota of “differential forms regarded as E^\hat E-connections on trivial EE-bundles” implicit.


Integration of differential forms is the morphism

Δ 1:[Δ 1, dRE^]Π dRE^ \int_{\Delta^1} \;\colon\; [\Delta^1, \flat_{dR}\hat E] \longrightarrow \Pi_{dR}\hat E

which is induced via the homotopy cofiber property of dRΩE^\flat_{dR}\Omega \hat E from the counit naturality square of the flat modality on [(**(i 0,i 1)Δ 1),][(\ast \coprod \ast \stackrel{(i_0, i_1)}{\to} \Delta ^1 ), -], using that this square exhibits a null homotopy due to the Δ 1\Delta^1-homotopy invariance of E^\flat \hat E.

(Bunke-Nikolaus-Völkl 13, (9), (10)


The Stokes theorem holds:

0 1di 1 *i 0 *. \int_0^1 \circ \mathbf{d} \simeq i_1^\ast - i_0^\ast \,.

(Bunke-Nikolaus-Völkl 13, theorem 3.2

Recovering traditional integration of differential forms


In H=\mathbf{H} = Smooth∞Grpd let E^Stab(H)\hat E \in Stab(\mathbf{H}) be given, under the stable Dold-Kan correspondence, by the traditional truncated de Rham complex E^Ω n\hat E \coloneqq \Omega^{\bullet \geq n }.

Then on this object the general cohesive integration map of def. reduces on the n-nth homotopy group to the tradition fiber integration of differential forms as above:

(Δ 1×X)/X:Ω(Δ 1×X) cl nΩ n1(X)/im(d). \int_{(\Delta^1 \times X)/X} \;\colon\; \Omega(\Delta^1 \times X)_{cl}^n \longrightarrow \Omega^{n-1}(X)/im(\mathbf{d}) \,.

(Bunke-Nikolaus-Völkl 13, lemma 4.2 (7)



Exposition with an eye towards applications in physics

Discussion in the abstract context of cohesion and differential cohomology is in

Last revised on June 9, 2023 at 17:32:52. See the history of this page for a list of all contributions to it.