on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
A model category structure on the category of dg-coalgebras.
Let $k$ be a field of characteristic 0.
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 $(\mathfrak{g}_\bullet, [-,-])$ to its “Chevalley-Eilenberg coalgebra”, whose underlying coalgebra is the free graded co-commutative coalgebra on $\mathfrak{g}_\bullet$ and whose differential is given on the tensor product of two generators by the Lie bracket $[-,-]$.
There exists a model category structure on $dgCoCAlg_k$ for which
the cofibrations are the (degreewise) injections;
the weak equivalences are those morphisms that become quasi-isomorphisms under the functor $\mathcal{L}$ from prop. 1.
Moreover, this is naturally a simplicial model category structure.
This is (Hinich98, theorem, 3.1). More details on this are in the relevant sections at model structure for L-infinity algebras.
The functor $\mathcal{C}$ from prop. 1 is a right Quillen functor.
Hence $(\mathcal{L} \dashv \mathcal{C})$ is a Quillen adjunction to the model structure on dg-algebras.
The adjunction of prop. 1 constitutes a Quillen equivalence to the model structure on dg-Lie algebras.
This is (Hinich98, theore, 3.2).
Dan Quillen, Rational homotopy theory , Annals of Math., 90(1969), 205–295. (see Appendix B)
Ezra Getzler, Paul Goerss, A model category structure for differential graded coalgebras (ps)
Vladimir Hinich, Homological algebra of homotopy algebras , Comm. in algebra, 25(10)(1997), 3291–3323.
Vladimir Hinich, DG coalgebras as formal stacks (arXiv:math/9812034)
Jonathan Pridham, Unifying derived deformation theories, Adv. Math. 224 (2010), no.3, 772-826 (arXiv:0705.0344)
Review with discussion of homotopy limits and homotopy colimits is in