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model structure on dg-Lie algebras

Definition

The model category structure on the category ${\mathrm{dgLie}}_k$ of dg-Lie algebras over a commutative ring $k\supset \mathbb{Q}$ has

• fibrations the surjective maps

• weak equivalences the quasi-isomorphisms on the underlying chain complexes.

This is a simplicial model category with respect to the sSet-hom functor

$$\mathrm{dgLie}(𝔤,𝔥):=([k]\mapsto {\mathrm{Hom}}_{\mathrm{dgLie}}(𝔤,{\Omega }^{•}({\Delta }^k)\otimes 𝔥)),$$dgLie(\mathfrak{g}, \mathfrak{h}) := ([k] \mapsto Hom_{dgLie}(\mathfrak{g} , \Omega^\bullet(\Delta^k) \otimes\mathfrak{h})) \,,

where

• ${\Omega }^{•}({\Delta }^k)$ is the dg-algebra of polynomial differential forms on the $k$-simplex;

• ${\Omega }^{•}({\Delta }^k)\otimes 𝔥$ is the canonical dg-Lie algebra structure on the tensor product.

Properties

Rectification resolution for $L_{\mathrm{\infty }}$-algebras

dg-Lie algebras with this model structure are a rectification of L-∞ algebras: for $\mathrm{Lie}$ the Lie operad and $\widehat{\mathrm{Lie}}$ its standard cofibrant resolution, algebras over an operad over $\mathrm{Lie}$ in chain complexes are dg-Lie algebras and algebras over $\widehat{\mathrm{Lie}}$ are L-∞ algebras and by the rectification result discussed at model structure on dg-algebras over an operad there is an induced Quillen equivalence

$$\mathrm{Alg}(\widehat{\mathrm{Lie}})\stackrel{\simeq }{\to }\mathrm{Alg}(\mathrm{Lie})$$Alg(\widehat Lie) \stackrel{\simeq}{\to} Alg(Lie)

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/$L_{\mathrm{\infty }}$-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

$$(ℛ⊣i):\mathrm{dgLie}\stackrel{\stackrel{ℛ}{\leftarrow }}{\underset{i}{\to }}\mathrm{dgCoCAlg}$$(\mathcal{R} \dashv i) \;\colon\; dgLie \stackrel{\overset{\mathcal{R}}{\leftarrow}}{\underset{i}{\to}} dgCoCAlg

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 $ℛ$ (“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 $\mathrm{dgCoCAlg}$ and the model structure on $\mathrm{dgLie}$ (Hinich 98, theorem 3.2).

In particular, therefore the composite $i\circ ℛ$ is a resolution functor for $L_{\mathrm{\infty }}$-algebras.

Related concepts

• model structure on dg-coalgebras

• model structure on simplicial Lie algebras

• model structure for L-∞ algebras

References

The model structure on dg-Lie algebras goes back to appendix B of

• Dan Quillen, Rational homotopy theory , Annals of Math., 90(1969), 205–295.

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

• Jacob Lurie, Formal moduli problems