model structure on dg-Lie algebras
Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
∞-Lie theory (higher geometry)
Formal Lie groupoids
The model category structure on the category of dg-Lie algebras over a commutative ring has
This is a simplicial model category with respect to the sSet-hom functor
Rectification resolution for -algebras
dg-Lie algebras with this model structure are a rectification of L-∞ algebras: for the Lie operad and its standard cofibrant resolution, algebras over an operad over in chain complexes are dg-Lie algebras and algebras over are L-∞ algebras and by the rectification result discussed at model structure on dg-algebras over an operad there is an induced Quillen equivalence
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/-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
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 (“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 and the model structure on (Hinich 98, theorem 3.2).
In particular, therefore the composite is a resolution functor for -algebras.
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
Review with discussion of homotopy limits and homotopy colimits is in
Revised on September 18, 2014 10:03:24
by Urs Schreiber