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model structure on dg-Lie algebras

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Model category theory

model category

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Model structures

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\infty-Lie theory

∞-Lie theory

Background

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Contents

Definition

The model category structure on the category dgLie kdgLie_k of dg-Lie algebras over a commutative ring kk \supset \mathbb{Q} has

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

dgLie(𝔤,𝔥):=([k]Hom dgLie(𝔤,Ω (Δ k)𝔥)), dgLie(\mathfrak{g}, \mathfrak{h}) := ([k] \mapsto Hom_{dgLie}(\mathfrak{g} , \Omega^\bullet(\Delta^k) \otimes\mathfrak{h})) \,,

where

Properties

Rectification resolution for L L_\infty-algebras

dg-Lie algebras with this model structure are a rectification of L-∞ algebras: for LieLie the Lie operad and Lie^\widehat Lie its standard cofibrant resolution, algebras over an operad over LieLie in chain complexes are dg-Lie algebras and algebras over Lie^\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

Alg(Lie^)Alg(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 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

(i):dgLieidgCoCAlg (\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 \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 dgCoCAlgdgCoCAlg and the model structure on dgLiedgLie (Hinich 98, theorem 3.2).

In particular, therefore the composite ii \circ \mathcal{R} is a resolution functor for L L_\infty-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

and section 2.1 of

Revised on May 12, 2013 08:22:32 by Urs Schreiber (89.204.153.90)