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 with underlying lined topos there is canonically the cosimplicial object
modeled on the interval object (as discussed there).
This induces the smooth realization functor
that sends a simplicial set to an ∞-Lie groupoid given by
Notably when models the standard real line, is the standard -simplex regarded as a smooth manifold (though typically collared, see interval object) and is the piecewise smooth manifold obtained by gluing together one copy of for each -simplex in .
While itself had no sensible smooth structure, the smooth realization does, being an object of , and we may form its infinitesimal path ∞-groupoid?
where we used that , 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 to get
(Notice that the coend and the tensor in the integrand is taken in ).
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.