∞-Lie differentiation and integration
infinitesimal path ∞-groupoid?
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
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 $R \in Diff \hookrightarrow C$ 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?
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
(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 at 12:10:30. See the history of this page for a list of all contributions to it.