model structure on simplicial Lie algebras


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The model structure on simplicial algebras for simplicial Lie algebras.



For kk some field, write LieAg kLieAg_k for the category of Lie algebras over kk, and write

((LieAlg) k) Δ opsSetCat ((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) proj Δ op((LieAlg)_k)^{\Delta^{op}}_{proj} for the category equipped with these classes of morphism.


The category ((LieAlg) k) proj Δ op((LieAlg)_k)^{\Delta^{op}}_{proj} from def. 1 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).


Relation to dg-Lie algebras


Let (𝔤,[,](\mathfrak{g}, [-,-] be a simplicial Lie algebra. Then the normalized chains complex N𝔤N \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

[,] N𝔤:(N𝔤) k(N𝔤)N(𝔤 k𝔤)N([,])N(𝔤) [-,-]_{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:LieAlg k Δ opdgLieAlg k N \;\colon\; LieAlg_k^{\Delta^{op}} \longrightarrow dgLieAlg_k

from simplical Lie algebras to dg-Lie algebras.

(Quillen 69, (4.3))


The functor NN from simplicial Lie algebras to dg-Lie algebras from def. 2 has a left adjoint

(N *N):LieAlg k Δ opNN *dgLieAlg k. (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).


With respect to the projective model structure on simplicial Lie algebras from prop. 1 and the projective model structure on dg-Lie algebras (this prop.) the adjunction from prop. 2 is a Quillen adjunction

(N *N):(LieAlg k Δ op) projNN *(dgLieAlg k) proj. (N^* \dashv N) \;\colon\; (LieAlg_k^{\Delta^{op}})_{proj} \underoverset {\underset{N}{\longrightarrow}} {\overset{N^*}{\longleftarrow}} {\bot} (dgLieAlg_k)_{proj} \,.

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


The adjunction between homotopy categories of the derived functors of the Quillen adjunction in prop. 3

Ho(LieAlg k Δ op)N𝕃N *Ho(dgLieAlg k) 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 Δ op) 1N𝕃N *Ho(dgLieAlg k) 1. 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))


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

Revised on February 28, 2017 15:54:23 by Urs Schreiber (