nLab model structure on dg-coalgebras



Model category theory

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Rational homotopy theory



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


Let kk be a field of characteristic 0.


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.


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. .

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.


Relation to dg-Lie algebras

Throughout, let kk be of characteristic zero.


(Chevalley-Eilenberg dg-coalgebra)


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).


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\;


  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).


The functors from def. and def. 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).


The adjunction (CE)(\mathcal{L} \dashv CE) from prop. 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)



In non-negatively graded dg-coalgebras, both Quillen functors (CE)(\mathcal{L} \dashv CE) from prop. 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)


The Quillen adjunctin from prop. 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)


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

Last revised on March 22, 2023 at 11:59:41. See the history of this page for a list of all contributions to it.