Schreiber Sullivan differential forms

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 $\mathbf{H} = (SPSh(C)^{loc})^\circ$ with underlying lined topos $(\mathcal{T},R)$ there is canonically the cosimplicial object

$$\Delta_R : \Delta \to SPSh(C)$$

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

This induces the smooth realization functor

$$|-|_R : SSet \to SPSh(C)$$

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

$$|S_\bullet| = \int^{[n] \in \Delta} S_n \cdot \Delta_R^n \,.$$

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

While $S$ itself had no sensible smooth structure, the smooth realization $|S|_R$ does, being an object of $SPSh(C)$, and we may form its infinitesimal path ∞-groupoid?

$$\begin{aligned} \Pi^{inf}(|S|) & = \Pi^{inf}(\int^{[n] \in \Delta} S_n \cdot \Delta^n_C) \\ & = \int^n S_n \cdot \Pi^{inf}(\Delta^n_C) \end{aligned} \,,$$

where we used that $\Pi^{inf}$, 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 $CE(-) : SPSh(C) \to dgAlg^{op}$ to get

$$CE(\Pi^{inf}(|S|_R)) \simeq \int^{n \in \Delta} S_n \cdot \Omega^\bullet(\Delta^n_C) \,.$$

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

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.