model category

for ∞-groupoids

∞-Lie theory

# Contents

## Definition

The model category structure on the category $dgLie_k$ of dg-Lie algebras over a commutative ring $k \supset \mathbb{Q}$ has

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

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

where

• $\Omega^\bullet(\Delta^k)$ is the dg-algebra of polynomial differential forms on the $k$-simplex;

• $\Omega^\bullet(\Delta^k)\otimes \mathfrak{h}$ is the canonical dg-Lie algebra structure on the tensor product.

## Properties

### Rectification resolution for $L_\infty$-algebras

dg-Lie algebras with this model structure are a rectification of L-∞ algebras: for $Lie$ the Lie operad and $\widehat Lie$ its standard cofibrant resolution, algebras over an operad over $Lie$ in chain complexes are dg-Lie algebras and algebras over $\widehat Lie$ are L-∞ algebras and by the rectification result discussed at model structure on dg-algebras over an operad there is an induced Quillen equivalence

$Alg(\widehat Lie) \stackrel{\simeq}{\to} Alg(Lie)$

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/$L_\infty$-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

$(\mathcal{R} \dashv i) \;\colon\; dgLie \stackrel{\overset{\mathcal{R}}{\leftarrow}}{\underset{i}{\to}} dgCoCAlg$

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 $dgCoCAlg$ and the model structure on $dgLie$ (Hinich 98, theorem 3.2).

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

Review with discussion of homotopy limits and homotopy colimits is in

Revised on September 18, 2014 10:03:24 by Urs Schreiber (185.26.182.29)