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
∞-Lie theory (higher geometry)
The model category structure on the category $dgLie_k$ of dg-Lie algebras over a commutative ring $k \supset \mathbb{Q}$ has
weak equivalences the quasi-isomorphisms on the underlying chain complexes.
This is a simplicial model category with respect to the sSet-hom functor
where
$\Omega^\bullet(\Delta^k)$ is the dg-algebra of polynomial differential forms on the $k$-simplex;
$\Omega^\bullet(\Delta^k)\otimes \mathfrak{h}$ is the canonical dg-Lie algebra structure on the tensor product.
dg-Lie algebras with this model structure are a rectification of L-∞ algebras: for $Lie$ the Lie operad and $\widehat Lie$ its standard cofibrant resolution, algebras over an operad over $Lie$ in chain complexes are dg-Lie algebras and algebras over $\widehat Lie$ are L-∞ algebras and by the rectification result discussed at model structure on dg-algebras over an operad there is an induced Quillen equivalence
between the model structure for L-∞ algebras which is transferred from the model structure on chain complexes (unbounded propjective) to the above model structure on chain complexes.
There is also a Quillen equivalence from the model structure on dg-Lie algebras to the model structure on dg-coalgebras. This is part of a web of Quillen equivalences that identifies dg-Lie algebra/$L_\infty$-algebras with infinitesimal derived ∞-stacks (“formal moduli problems”). More on this is at model structure for L-∞ algebras.
Specifically, there is (Quillen 69) an adjunction
between dg-coalgebras and dg-Lie algebras, where the right adjoint is the (non-full) inclusion that regards a dg-Lie algebra as a differential graded coalgebra with co-binary differential, and where the left adjoint $\mathcal{R}$ (“rectification”) sends a dg-coalgebra to a dg-Lie algebra whose underlying graded Lie algebra is the free Lie algebra on the underlying chain complex. Over a field of characteristic 0, this adjunction is a Quillen equivalence between the model structure for L-∞ algebras on $dgCoCAlg$ and the model structure on $dgLie$ (Hinich 98, theorem 3.2).
In particular, therefore the composite $i \circ \mathcal{R}$ is a resolution functor for $L_\infty$-algebras.
The model structure on dg-Lie algebras goes back to appendix B of
For more discussion see
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)
and section 2.1 of
Review with discussion of homotopy limits and homotopy colimits is in