related by the Dold-Kan correspondence
There is a pair of adjoint functors
between the category of dg-Lie algebras (on unbounded chain complexes) and that of dg cocommutative coalgebras, where the right adjoint sends a dg-Lie algebra to its “Chevalley-Eilenberg coalgebra”, whose underlying coalgebra is the free graded co-commutative coalgebra on and whose differential is given on the tensor product of two generators by the Lie bracket .
There exists a model category structure on for which
the cofibrations are the (degreewise) injections;
Moreover, this is naturally a simplicial model category structure.
This is (Hinich98, theore, 3.2).
Dan Quillen, Rational homotopy theory , Annals of Math., 90(1969), 205–295. (see Appendix B)
Vladimir Hinich, Homological algebra of homotopy algebras , Comm. in algebra, 25(10)(1997), 3291–3323.