group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Stokes theorem asserts that the integration of differential forms of the de Rham differential of a differential form over a domain equals the integral of the form itself over the boundary of the domain.
Let
be the cosimplicial object of standard $k$-simplices in SmoothMfd: in degree $k$ this is the standard $k$-simplex $\Delta^k_{Diff} \subset \mathbb{R}^k$ regarded as a smooth manifold with boundary and corners. This may be parameterized as
In this parameterization the coface maps of $\Delta_{Diff}$ are
For $X$ any smooth manifold a smooth $k$-simplex in $X$ is a smooth function
The boundary of this simplex in $X$ is the chain (formal linear combination of smooth $(k-1)$-simplices)
Let $\omega \in \Omega^{k-1}(X)$ be a degree $(k-1)$-differential form on $X$.
(Stokes theorem)
The integral of $\omega$ over the boundary of the simplex equals the integral of its de Rham differential over the simplex itself
It follows that for $C$ any $k$-chain in $X$ and $\partial C$ its boundary $(k-1)$-chain, we have
We discuss here a general abstract formulation of differential forms, their integration and Stokes theorem in the axiomatics of cohesive homotopy type theory (following Bunke-Nikolaus-Völkl 13, theorem 3.2).
Let $\mathbf{H}$ be a cohesive (∞,1)-topos and write $T \mathbf{H}$ for its tangent cohesive (∞,1)-topos.
Assume that there is an interval object
“exhibiting the cohesion” (see at continuum) in that there is a (chosen) equivalence between the shape modality $\Pi$ and the localization $L_{\Delta^1}$ at the the projection maps out of Cartesian products with this line $\Delta^1\times (-) \to (-)$
This is the case for instance for the “standard continuum”, the real line in $\mathbf{H} =$ Smooth∞Grpd.
It follows in particular that there is a chosen equivalence of (∞,1)-categories
between the flat modal homotopy-types and the $\Delta^1$-homotopy invariant homotopy-types.
Given a stable homotopy type $\hat E \in Stab(\mathbf{H})\hookrightarrow T \mathbf{H}$ cohesion provides two objects
which may be interpreted as de Rham complexes with coefficients in $\Pi(\flat_{dR} \Sigma \hat E)$, the first one restricted to negative degree, the second to non-negative degree. Moreover, there is a canonical map
which interprets as the de Rham differential $\mathbf{d}$. See at differential cohomology diagram for details.
Throughout in the following we leave the “inclusion” $\iota$ of “differential forms regarded as $\hat E$-connections on trivial $E$-bundles” implicit.
Integration of differential forms is the map
which is induced via the homotopy cofiber property of $\flat_{dR}\Omega \hat E$ from the counit naturality square of the flat modality on $[(\ast \coprod \ast \stackrel{(i_0, i_1)}{\to} \Delta ^1 ), -]$, using that this square exhibits a null homotopy due to the $\Delta^1$-homotopy invariance of $\flat \hat E$.
Stokes’ theorem holds:
(Bunke-Nikolaus-Völkl 13, theorem 3.2)
Stokes theorem
a special case is Cauchy's integral theorem
A standard account is for instance in
Discussion of Stokes theorem on manifolds with corners is in
Discussion for manifolds with more general singularities on the boundary is in
Discussion in cohesive homotopy type theory is in