Sullivan differential forms

**∞-Lie theory**
## ∞-Lie groupoids and -algebroids ##
* ∞-Lie groupoid
* ∞-Lie algebroid
* Chevalley-Eilenberg algebra
* Weil algebra
* ∞-Lie differentiation and integration
* path ∞-groupoid
* infinitesimal path ∞-groupoid?
* integration of ∞-Lie algebroid valued differential forms
## ∞-Chern-Weil theory ##
* invariant polynomial
* differential cohomology in an (∞,1)-topos
* ∞-Lie algebroid valued differential forms
* curvature?
* integration
* Cartan-Ehresmann ∞-connection
## symplectic ∞-geometry ##
* symplectic ∞-Lie algebroid

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.

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