model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
and
rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)
Examples of Sullivan models in rational homotopy theory:
(Quillen adjunction between simplicial sets and connective dgc-algebras)
The PL de Rham complex-construction is the left adjoint in a Quillen adjunction between
That the PL de Rham complex functor preserves cofibrations, hence sends injections of simplicial sets to surjections of dgc-algebras, is immediate from its construction.
That its right adjoint preserves fibrations, hence sends cofibrations of dgc-algebras to Kan fibrations, is the statement of Bousfield-Gugenheim 76, Lemma 8.2.
fundamental theorem of dg-algebraic rational homotopy theory
Quillen adjunction between equivariant simplicial sets and equivariant connective dgc-algebras