nLab 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 $\mathbf{H}$ 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.

Properties

• The de Rham differential acts as a graded derivation on the de Rham algebra.