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

Δ R:ΔSPSh(C) \Delta_R : \Delta \to SPSh(C)

modeled on the interval object RR (as discussed there).

This induces the smooth realization functor

R:SSetSPSh(C) |-|_R : SSet \to SPSh(C)

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

S = [n]ΔS nΔ R n. |S_\bullet| = \int^{[n] \in \Delta} S_n \cdot \Delta_R^n \,.

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

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

Π inf(S) =Π inf( [n]ΔS nΔ C n) = nS nΠ inf(Δ C n), \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 Π inf\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)dgAlg opCE(-) : SPSh(C) \to dgAlg^{op} to get

CE(Π inf(S R)) nΔS nΩ (Δ C n). 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 opdgAlg^{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)