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

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

$\Pi_{dR} \Omega \hat E \,,\;\; \flat_{dR}\Sigma \hat E \;\; \in Stab(\mathbf{H})$

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

$\array{ \Pi_{dR}\Omega \hat E && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{dR}\Sigma \hat E \\ & {}_{\mathllap{\iota}}\searrow && \nearrow_{\mathrlap{\theta_{\hat E}}} \\ && \hat E }$

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

Last revised on December 28, 2020 at 05:49:40. See the history of this page for a list of all contributions to it.