model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
∞-Lie theory (higher geometry)
dg-Lie algebras may be thought of (see here) 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, theorem II) is one of the algebraic models for presenting rational homotopy theory (see there) of simply connected topological spaces.
There exists a model category structure $(dgLie_k)_{proj}$ on the category $dgLie_k$ of dg-Lie algebras over a commutative ring $k \supset \mathbb{Q}$ such that
weak equivalences the quasi-isomorphisms on the underlying chain complexes.
For dg-Lie algebras in degrees $\geq n \geq 1$, this is due to Quillen 69. For unbounded dg-Lie algebras this is due to (Hinich 97).
This becomes a simplicial category with simplicial mapping spaces given by
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.
(Hinich 97, 4.8.2, following Bousfield-Gugenheim 76tructure on dg-algebras#HomComplexes))
This enrichment satisfies together with the model structure some of the properties of a simplicial model category (Hinich 97, 4.8.3, 4.8.4), but not all of them.
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.
For more see at relation between L-∞ algebras and dg-Lie algebras.
Via the above relation to $L_\infty$-algebras, dg-Lie algebras are also connected by a composite adjunction to dg-coalgebras. We dicuss the direct adjunction.
Throughout, let $k$ be of characteristic zero.
(Chevalley-Eilenberg dg-coalgebra)
Write
for the Chevalley-Eilenberg algebra functor. It sends a dg-Lie algebra $(\mathfrak{g}, \partial, [-,-])$ to the dg-coalgebra
where on the right the extension of $\partial$ and $[-,-]$ to graded derivations is understood.
For dg-Lie algebras concentrated in degrees $\geq n \geq 1$ this is due to (Quillen 69, appendix B, prop 6.2). For unbounded dg-algebras, this is due to (Hinich 98, 2.2.2).
For $(X,D) \in dgCocalg_k$ write
where
$\overline{X} \coloneqq ker(\epsilon)$ is the kernel of the counit, regarded as a chain complex;
$F$ is the free Lie algebra functor (as graded Lie algebras);
on the right we are extending $(\Delta - 1 \otimes id - id \otimes 1) \colon \overline{X} \to \overline{X} \otimes \overline{X}$ as a Lie algebra derivation
For dg-Lie algebras concentrated in degrees $\geq n \geq 1$ this is due to (Quillen 69, appendix B, prop 6.1). For unbounded dg-algebras, this is due to (Hinich 98, 2.2.1).
The functors from def. and def. are adjoint to each other:
Moreover, for $X \in dgCocAlg_k$ and $\mathfrak{g} \in dgLieAlg_k$ then the adjoint hom sets are naturally isomorphic
to the Maurer-Cartan elements in the Hom-dgLie algebra from $\overline{X}$ to $\mathfrak{g}$.
For dg-Lie algebras concentrated in degrees $\geq n \geq 1$ this is due to (Quillen 69, appendix B, somewhere). For unbounded dg-algebras, this is due to (Hinich 98, 2.2.5).
The adjunction $(\mathcal{L} \dashv CE)$ from prop. is a Quillen adjunction between then projective model structure on dg-Lie algebras as the model structure on dg-coalgebras
(Hinich 98, lemma 5.2.2, lemma 5.2.3)
Moreover:
In non-negatively graded dg-coalgebras, both Quillen functors $(\mathcal{L} \dashv CE)$ from prop. preserve all quasi-isomorphisms, and both the adjunction unit and the adjunction counit are quasi-isomorphisms.
For dg-algebras in degrees $\geq n \geq 1$ this is (Quillen 76, theorem 7.5). In unbounded degrees this is (Hinich 98, prop. 3.3.2)
The Quillen adjunctin from prop. is a Quillen equivalence:
(Hinich 98, theorem 3.2) using (Quillen 76 II 1.4)
The normalized chains complex functor from simplicial Lie algebras constitutes a Quillen adjunction from the projective model structure on simplicial Lie algebras, see there.
The model structure on dg-Lie algebras in characteristic zero and in degrees $\geq n \geq 1$ goes back to
This is extended to a model structure on dg-Lie algebras in unbounded degrees in
and the corresponding Quillen adjunction to the model structure on dg-coalgebras in unbounded degrees is discussed in
See also section 2.1 of
Review with discussion of homotopy limits and homotopy colimits is in