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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 \underoverset {\underset{\mathcal{C}}{\longrightarrow}} {\overset{\mathcal{L}}{\longleftarrow}} {\bot} dgCoCAlg_k

between the category of dg-Lie algebrasand 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[1]\mathfrak{g}[1] and whose differential is given on the tensor product of two generators by the Lie bracket [,][-,-].

For (pointers to) the details, see at model structure on dg-Lie algebras – Relation to dg-coalgebras.

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 dg-Lie algebras

Throughout, let kk be of characteristic zero.

Definition

(Chevalley-Eilenberg dg-coalgebra)

Write

CE:dgLieAlg kdgCocAlg k CE \;\colon\; dgLieAlg_{k} \longrightarrow dgCocAlg_k

for the Chevalley-Eilenberg algebra functor. It sends a dg-Lie algebra (𝔤,,[,])(\mathfrak{g}, \partial, [-,-]) to the dg-coalgebra

CE(𝔤,,[,])( 𝔤[1],D=+[,]), CE(\mathfrak{g},\partial,[-,-]) \;\coloneqq\; \left( \vee^\bullet \mathfrak{g}[1] ,\; D = \partial + [-,-] \right) \,,

where on the right the extension of \partial and [,][-,-] to graded derivations is understood.

For dg-Lie algebras concentrated in degrees n1 \geq n \geq 1 this is due to (Quillen 69, appendix B, prop 6.2). For unbounded dg-algebras, this is due to (Hinich 98, 2.2.2).

Definition

For (X,D)dgCocalg k(X,D) \in dgCocalg_k write

(X,D)(F(X¯[1]),D+(Δ1idid1))dgLieAlg k \mathcal{L}(X,D) \coloneqq \left( F(\overline{X}[-1]),\; \partial \coloneqq D + (\Delta - 1 \otimes id - id \otimes 1) \right) \;\in dgLieAlg_k\;

where

  1. X¯ker(ϵ)\overline{X} \coloneqq ker(\epsilon) is the kernel of the counit, regarded as a chain complex;

  2. FF is the free Lie algebra functor (as graded Lie algebras);

  3. on the right we are extending (Δ1idid1):X¯X¯X¯(\Delta - 1 \otimes id - id \otimes 1) \colon \overline{X} \to \overline{X} \otimes \overline{X} as a Lie algebra derivation

For dg-Lie algebras concentrated in degrees n1 \geq n \geq 1 this is due to (Quillen 69, appendix B, prop 6.1). For unbounded dg-algebras, this is due to (Hinich 98, 2.2.1).

Proposition

The functors from def. 1 and def. 2 are adjoint to each other:

dgLieAlg kCEdgCocAlg k. dgLieAlg_k \underoverset {\underset{CE}{\longrightarrow}} {\overset{\mathcal{L}}{\longleftarrow}} {\bot} dgCocAlg_k \,.

Moreover, for XdgCocAlg kX \in dgCocAlg_k and 𝔤dgLieAlg k\mathfrak{g} \in dgLieAlg_k then the adjoint hom sets are naturally isomorphic

Hom((X),𝔤)Hom(X,CE(𝔤))MC(Hom(X¯,𝔤)) Hom(\mathcal{L}(X), \mathfrak{g}) \simeq Hom(X, CE(\mathfrak{g})) \simeq MC(Hom(\overline{X},\mathfrak{g}))

to the Maurer-Cartan elements in the Hom-dgLie algebra from X¯\overline{X} to 𝔤\mathfrak{g}.

For dg-Lie algebras concentrated in degrees n1 \geq n \geq 1 this is due to (Quillen 69, appendix B, somewhere). For unbounded dg-algebras, this is due to (Hinich 98, 2.2.5).

Proposition

The adjunction (CE)(\mathcal{L} \dashv CE) from prop. 2 is a Quillen adjunction between then projective model structure on dg-Lie algebras as the model structure on dg-coalgebras

(dgLieAlg k) projCE(dgCocAlg k) Quillen. (dgLieAlg_k)_{proj} \underoverset {\underset{CE}{\longrightarrow}} {\overset{\mathcal{L}}{\longleftarrow}} {\bot} (dgCocAlg_k)_{Quillen} \,.

(Hinich 98, lemma 5.2.2, lemma 5.2.3)

Moreover:

Proposition

In non-negatively graded dg-coalgebras, both Quillen functors (CE)(\mathcal{L} \dashv CE) from prop. 3 preserve all quasi-isomorphisms, and both the adjunction unit and the adjunction counit are quasi-isomorphisms.

For dg-algebras in degrees n1\geq n \geq 1 this is (Quillen 76, theorem 7.5). In unbounded degrees this is (Hinich 98, prop. 3.3.2)

Theorem

The Quillen adjunctin from prop. 3 is a Quillen equivalence:

(dgLieAlg k) proj qu QuCE(dgCocAlg k) Quillen. (dgLieAlg_k)_{proj} \underoverset {\underset{CE}{\longrightarrow}} {\overset{\mathcal{L}}{\longleftarrow}} {{}_{\phantom{qu}}\simeq_{Qu}} (dgCocAlg_k)_{Quillen} \,.

(Hinich 98, theorem 3.2) using (Quillen 76 II 1.4)

References

In characteristic zero and in positive degrees the model structure is due to

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

in non-negative degrees in

and in unbounded degrees in

See also

Review with discussion of homotopy limits and homotopy colimits is in

Revised on March 7, 2017 14:37:32 by Urs Schreiber (192.41.135.245)