# Schreiber Sullivan differential forms

∞-Lie theory

## Idea

In rational homotopy theory (as described there) one central tool is the definition of a dg-algebra of differential forms on a simplicial set, and hence on a topological space.

This may be understood as a special case of the following general construction in ∞-Lie theory:

In a smooth (∞,1)-topos $H=\left(\mathrm{SPSh}\left(C{\right)}^{\mathrm{loc}}{\right)}^{\circ }$ with underlying lined topos $\left(𝒯,R\right)$ there is canonically the cosimplicial object

${\Delta }_{R}:\Delta \to \mathrm{SPSh}\left(C\right)$\Delta_R : \Delta \to SPSh(C)

modeled on the interval object $R$ (as discussed there).

This induces the smooth realization functor

$\mid -{\mid }_{R}:\mathrm{SSet}\to \mathrm{SPSh}\left(C\right)$|-|_R : SSet \to SPSh(C)

that sends a simplicial set to an ∞-Lie groupoid given by

$\mid {S}_{•}\mid ={\int }^{\left[n\right]\in \Delta }{S}_{n}\cdot {\Delta }_{R}^{n}\phantom{\rule{thinmathspace}{0ex}}.$|S_\bullet| = \int^{[n] \in \Delta} S_n \cdot \Delta_R^n \,.

Notably when $R\in \mathrm{Diff}↪C$ models the standard real line, ${\Delta }_{R}^{n}$ is the standard $n$-simplex regarded as a smooth manifold (though typically collared, see interval object) and $\mid S{\mid }_{R}$ is the piecewise smooth manifold obtained by gluing together one copy of ${\Delta }_{C}^{n}$ for each $n$-simplex in $S$.

While $S$ itself had no sensible smooth structure, the smooth realization $\mid S{\mid }_{R}$ does, being an object of $\mathrm{SPSh}\left(C\right)$, and we may form its infinitesimal path ∞-groupoid?

$\begin{array}{rl}{\Pi }^{\mathrm{inf}}\left(\mid S\mid \right)& ={\Pi }^{\mathrm{inf}}\left({\int }^{\left[n\right]\in \Delta }{S}_{n}\cdot {\Delta }_{C}^{n}\right)\\ & ={\int }^{n}{S}_{n}\cdot {\Pi }^{\mathrm{inf}}\left({\Delta }_{C}^{n}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}},$\begin{aligned} \Pi^{inf}(|S|) & = \Pi^{inf}(\int^{[n] \in \Delta} S_n \cdot \Delta^n_C) \\ & = \int^n S_n \cdot \Pi^{inf}(\Delta^n_C) \end{aligned} \,,

where we used that ${\Pi }^{\mathrm{inf}}$, being a left adjoint, preserves coends and colimits.

This is manifestly an ∞-Lie algebroid. To recognize it, we form its Chevalley-Eilenberg algebra by applying the left adjoint $\mathrm{CE}\left(-\right):\mathrm{SPSh}\left(C\right)\to {\mathrm{dgAlg}}^{\mathrm{op}}$ to get

$\mathrm{CE}\left({\Pi }^{\mathrm{inf}}\left(\mid S{\mid }_{R}\right)\right)\simeq {\int }^{n\in \Delta }{S}_{n}\cdot {\Omega }^{•}\left({\Delta }_{C}^{n}\right)\phantom{\rule{thinmathspace}{0ex}}.$CE(\Pi^{inf}(|S|_R)) \simeq \int^{n \in \Delta} S_n \cdot \Omega^\bullet(\Delta^n_C) \,.

(Notice that the coend and the tensor in the integrand is taken in ${\mathrm{dgAlg}}^{\mathrm{op}}$).

If instead of smooth differential forms here we took polynomial forms with rational coefficients, this would be Sullivan’s construction of different forms on a simplicial set as known in rational homotopy theory.

Created on January 13, 2010 12:13:38 by Urs Schreiber (82.113.106.228)