nLab de Rham differential

The de Rham differential

The de Rham differential


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

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.

Last revised on December 28, 2020 at 10:49:40. See the history of this page for a list of all contributions to it.