nLab de Rham differential

The de Rham differential

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

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 }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

The de Rham differential

Traditional

In differential geometry the de Rham differential is the differential in the de Rham complex, “exterior derivative” acting on differential forms.

In cohesive homotopy theory

Let H\mathbf{H} be a cohesive (∞,1)-topos and write THT \mathbf{H} for its tangent cohesive (∞,1)-topos.

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

Π dRΩE^, dRΣE^Stab(H) \Pi_{dR} \Omega \hat E \,,\;\; \flat_{dR}\Sigma \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 dRΩE^ d dRΣE^ ι θ E^ E^ \array{ \Pi_{dR}\Omega \hat E && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{dR}\Sigma \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.

Properties

Last revised on May 3, 2023 at 05:00:03. See the history of this page for a list of all contributions to it.