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

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.

Created on April 29, 2014 03:53:30
by Urs Schreiber
(89.204.155.67)