model category

for ∞-groupoids

# Contents

## Idea

A model category structure on the category of dg-coalgebras.

## Definition

Let $k$ be a field of characteristic 0.

###### Proposition

There is a pair of adjoint functors

$(\mathcal{L} \dashv \mathcal{C}) \;\colon\; dgLieAlg_k \stackrel{\overset{\mathcal{L}}{\leftarrow}}{\underset{\mathcal{C}}{\to}} dgCoCAlg_k$

between the category of dg-Lie algebras (on unbounded chain complexes) and that of dg cocommutative coalgebras, where the right adjoint sends a dg-Lie algebra $(\mathfrak{g}_\bullet, [-,-])$ to its “Chevalley-Eilenberg coalgebra”, whose underlying coalgebra is the free graded co-commutative coalgebra on $\mathfrak{g}_\bullet$ and whose differential is given on the tensor product of two generators by the Lie bracket $[-,-]$.

###### Theorem

There exists a model category structure on $dgCoCAlg_k$ for which

• the cofibrations are the (degreewise) injections;

• the weak equivalences are those morphisms that become quasi-isomorphisms under the functor $\mathcal{L}$ from prop. 1.

Moreover, this is naturally a simplicial model category structure.

This is (Hinich98, theorem, 3.1). More details on this are in the relevant sections at model structure for L-infinity algebras.

## Properties

### Relation to the model structure on dg-Lie algebras

###### Proposition

The functor $\mathcal{C}$ from prop. 1 is a right Quillen functor.

Hence $(\mathcal{L} \dashv \mathcal{C})$ is a Quillen adjunction to the model structure on dg-algebras.

###### Proposition

The adjunction of prop. 1 constitutes a Quillen equivalence to the model structure on dg-Lie algebras.

This is (Hinich98, theore, 3.2).

## References

• Dan Quillen, Rational homotopy theory , Annals of Math., 90(1969), 205–295. (see Appendix B)

• Ezra Getzler, Paul Goerss, A model category structure for differential graded coalgebras (ps)

• Vladimir Hinich, Homological algebra of homotopy algebras , Comm. in algebra, 25(10)(1997), 3291–3323.

Revised on March 29, 2013 03:16:52 by Urs Schreiber (89.204.139.104)