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

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

This induces the smooth realization functor

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

Notably when $R\in \mathrm{Diff}\hookrightarrow C$ 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 $\mathrm{SPSh}(C)$, and we may form its infinitesimal path ∞-groupoid?

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 $\mathrm{CE}(-):\mathrm{SPSh}(C)\to {\mathrm{dgAlg}}^{\mathrm{op}}$ to get

(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.