nLab model structure on dg-coalgebras

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

$$(\mathcal{L} \dashv \mathcal{C}) \;\colon\; dgLieAlg_k \underoverset {\underset{\mathcal{C}}{\longrightarrow}} {\overset{\mathcal{L}}{\longleftarrow}} {\bot} dgCoCAlg_k$$

between the category of dg-Lie algebrasand that of dg cocommutative coalgebras, where the right adjoint sends a dg-Lie algebra $(\mathfrak{g}_\bullet, [-,-])$ to its Chevalley-Eilenberg coalgebra, whose underlying coalgebra is the free graded co-commutative coalgebra on $\mathfrak{g}[1]$ 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 $dgCoCAlg_k$ 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.

Properties

Relation to dg-Lie algebras

Throughout, let $k$ be of characteristic zero.

Definition

(Chevalley-Eilenberg dg-coalgebra)

Write

$$CE \;\colon\; dgLieAlg_{k} \longrightarrow dgCoCAlg_k$$

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

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

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

For dg-Lie algebras concentrated in degrees $\geq n \geq 1$ 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 $(X,D) \in dgCoCAlg_k$ write

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

where

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

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

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

For dg-Lie algebras concentrated in degrees $\geq n \geq 1$ 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. and def. are adjoint to each other:

$$dgLieAlg_k \underoverset {\underset{CE}{\longrightarrow}} {\overset{\mathcal{L}}{\longleftarrow}} {\bot} dgCoCAlg_k \,.$$

Moreover, for $X \in dgCoCAlg_k$ and $\mathfrak{g} \in dgLieAlg_k$ then the adjoint hom sets are naturally isomorphic

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

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

For dg-Lie algebras concentrated in degrees $\geq n \geq 1$ 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 $(\mathcal{L} \dashv CE)$ from prop. is a Quillen adjunction between then projective model structure on dg-Lie algebras as the model structure on dg-coalgebras

$$(dgLieAlg_k)_{proj} \underoverset {\underset{CE}{\longrightarrow}} {\overset{\mathcal{L}}{\longleftarrow}} {\bot} (dgCoCAlg_k)_{Quillen} \,.$$

(Hinich 98, lemma 5.2.2, lemma 5.2.3)

Moreover:

Proposition

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

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

Theorem

The Quillen adjunctin from prop. is a Quillen equivalence:

$$(dgLieAlg_k)_{proj} \underoverset {\underset{CE}{\longrightarrow}} {\overset{\mathcal{L}}{\longleftarrow}} {{}_{\phantom{qu}}\simeq_{Qu}} (dgCoCAlg_k)_{Quillen} \,.$$

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

Related concepts

• model structure on dg-comodules

• model structure for L-∞ algebras

• model structure on dg-Lie algebras, model structure on simplicial Lie algebras

• model structure on coalgebras over a comonad

• rational homotopy theory

• relation between L-∞ algebras and dg-Lie algebras

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

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

and in unbounded degrees in

• Vladimir Hinich, DG coalgebras as formal stacks, Journal of Pure and Applied Algebra Volume 162, Issues 2–3, 24 August 2001, Pages 209–250 [arXiv:math/9812034, doi:10.1016/S0022-4049(00)00121-3

• Jonathan Pridham, Unifying derived deformation theories, Adv. Math. 224 (2010), no.3, 772-826 (arXiv:0705.0344)

See also

• Jacob Lurie, Formal Moduli Problems

Review with discussion of homotopy limits and homotopy colimits is in

• Ben Walter, section 5 of Rational Homotopy Calculus of Functors (arXiv:math/0603336)