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model structure on dg-Lie algebras

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Model category theory

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Contents

Idea

dg-Lie algebras may be thought of as the “strict” strong homotopy Lie algebras. As such they support a homotopy theory. The model category structure on dg-Lie algebras is one way to present this homotopy theory. This is used for instance in deformation theory, see at formal moduli problems.

For dg-Lie algebras in positive degree and over the rational numbers this model structure, due to (Quillen 69) is one of the algebraic models for presenting rational homotopy theory (see there) of simply connected topological spaces.

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

  • Dan Quillen, section II.5 and appendix B of Rational homotopy theory, Annals of Math., 90(1969), 205–295 (JSTOR, pdf)

For more discussion see

and section 2.1 of

Review with discussion of homotopy limits and homotopy colimits is in

Revised on February 21, 2017 03:16:03 by Urs Schreiber (94.220.74.41)