on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
∞-Lie theory (higher geometry)
The model structure on simplicial algebras for simplicial Lie algebras.
For $k$ some field, write $LieAg_k$ for the category of Lie algebras over $k$, and write
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.
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).
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
where the first morphism is the Eilenberg-Zilber map.
This construction extends to a functor
from simplical Lie algebras to dg-Lie algebras.
The functor $N$ from simplicial Lie algebras to dg-Lie algebras from def. has a left adjoint
This is (Quillen 69, prop. 4.4).
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
By the standard Dold-Kan correspondence $N$ 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.
restricts on connected simplicial algebras and connected dg-Lie algebras (meaning: having the trivial Lie algebra in degree 0) to an equivalence of categories:
(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
and specifically for reduced simplicial Lie algebras due to
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
Last revised on February 28, 2017 at 15:54:23. See the history of this page for a list of all contributions to it.