nLab
model structure on dg-coalgebras

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Contents

Idea

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

Definition

Let kk be a field of characteristic 0.

Proposition

There is a pair of adjoint functors

(𝒞):dgLieAlg k𝒞dgCoCAlg k (\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 (𝔤 ,[,])(\mathfrak{g}_\bullet, [-,-]) to its “Chevalley-Eilenberg coalgebra”, whose underlying coalgebra is the free graded co-commutative coalgebra on 𝔤 \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 kdgCoCAlg_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 𝒞\mathcal{C} from prop. 1 is a right Quillen functor.

(Hinich98, lemma 5.3.2)

Hence (𝒞)(\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

Review with discussion of homotopy limits and homotopy colimits is in

Revised on September 18, 2014 10:00:05 by Urs Schreiber (185.26.182.29)