nLab model structure on simplicial Lie algebras

Idea

The model structure on simplicial algebras for simplicial Lie algebras.

Details

Definition

For $k$ some field, write $LieAlg_k$ for the category of Lie algebras over $k$, and write

$$((LieAlg)_k)^{\Delta^{op}} \in sSetCat$$

for the category of simplicial objects in that category, naturally regarded as a simplicial category. Say a morphism in that category is a weak equivalence or fibration, if its underlying morphism of simplicial sets is a weak equivalence or fibration, respectively, in the classical model structure on simplicial sets. Write $((LieAlg)_k)^{\Delta^{op}}_{proj}$ for the category equipped with these classes of morphism.

Proposition

The category $((LieAlg)_k)^{\Delta^{op}}_{proj}$ from def. is a simplicial model category.

Since Lie algebras are algebras over a Lawvere theory, this is a special case of (Reedy 74, theorem I) discussed at model structure on simplicial algebras. Without the simplicial enrichment and on reduced simplicial Lie algebras this model structure is a special case of (Quillen 67, II.4 theorem 4) and also left as an exercise for the reader in (Quillen 69, below theorem 4.7).

Properties

Relation to dg-Lie algebras

Definition

Let $(\mathfrak{g}, [-,-]$ be a simplicial Lie algebra. Then the normalized chains complex $N \mathfrak{g}$ of the underlying simplicial abelian group becomes a dg-Lie algebra by equipping it with the Lie bracket given by the following composite morphisms

$$[-,-]_{N \mathfrak{g}} \;\colon\; (N \mathfrak{g}) \otimes_k (N \mathfrak{g}) \overset{\nabla}{\longrightarrow} N (\mathfrak{g} \otimes_k \mathfrak{g}) \overset{N([-,-])}{\longrightarrow} N (\mathfrak{g})$$

where the first morphism is the Eilenberg-Zilber map.

This construction extends to a functor

$$N \;\colon\; LieAlg_k^{\Delta^{op}} \longrightarrow dgLieAlg_k$$

from simplical Lie algebras to dg-Lie algebras.

(Quillen 69, (4.3))

Proposition

The functor $N$ from simplicial Lie algebras to dg-Lie algebras from def. has a left adjoint

$$(N^* \dashv N) \;\colon\; LieAlg_k^{\Delta^{op}} \underoverset {\underset{N}{\longrightarrow}} {\overset{N^*}{\longleftarrow}} {\bot} dgLieAlg_k \,.$$

This is (Quillen 69, prop. 4.4).

Proposition

With respect to the projective model structure on simplicial Lie algebras from prop. and the projective model structure on dg-Lie algebras (this prop.) the adjunction from prop. is a Quillen adjunction

$$(N^* \dashv N) \;\colon\; (LieAlg_k^{\Delta^{op}})_{proj} \underoverset {\underset{N}{\longrightarrow}} {\overset{N^*}{\longleftarrow}} {\bot} (dgLieAlg_k)_{proj} \,.$$

Proof

By the standard Dold-Kan correspondence $N$ preserves fibrations and weak equivalences (this prop., Schwede-Shipley 02, section 4.1, p.17).

Proposition

The adjunction between homotopy categories of the derived functors of the Quillen adjunction in prop.

$$Ho(LieAlg_k^{\Delta^{op}}) \underoverset {\underset{\mathbb{R}N}{\longrightarrow}} {\overset{\mathbb{L}N^*}{\longleftarrow}} {\bot} Ho(dgLieAlg_k)$$

restricts on connected simplicial algebras and connected dg-Lie algebras (meaning: having the trivial Lie algebra in degree 0) to an equivalence of categories:

$$Ho(LieAlg_k^{\Delta^{op}})_{\geq 1} \underoverset {\underset{\mathbb{R}N}{\longrightarrow}} {\overset{\mathbb{L}N^*}{\longleftarrow}} {\simeq} Ho(dgLieAlg_k)_{\geq 1} \,.$$

(Quillen 76, section II.4, see p. 224 (20 of 92), see also figure 2 on p. 211 (8 of 92))

Related concepts

• model structure on dg-Lie algebras

• model structure for L-infinity algebras

• model structure on dg-coalgebras

References

The model structure for for simplicial %T%Palgebras is originally due to

• Dan Quillen, Homotopical Algebra, Lectures Notes in Mathematics 43, Springer Verlag, Berlin, (1967)

and specifically for reduced simplicial Lie algebras due to

• Dan Quillen, Rational homotopy theory, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (JSTOR, pdf)

On the homotopy theory of simplicial Lie algebras see also

• Stewart Priddy, On the homotopy theory of simplicial Lie algebras, (pdf)

• Graham Ellis, Homotopical aspects of Lie algebras Austral. Math. Soc. (Series A) 54 (1993), 393-419 (web)

See also

• Stefan Schwede, Brooke Shipley, Equivalences of monoidal model categories, Algebr. Geom. Topol. 3 (2003), 287–334 (arXiv:math.AT/0209342)