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)
dg-Lie algebras may be thought of as the “strict” strong homotopy Lie algebras. As such they support a homotopy theory. The model category structure on dg-Lie algebras is one way to present this homotopy theory. This is used for instance in deformation theory, see at formal moduli problems.
For dg-Lie algebras in positive degree and over the rational numbers this model structure, due to (Quillen 69) is one of the algebraic models for presenting rational homotopy theory (see there) of simply connected topological spaces.
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
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