nLab model structure on dg-algebras over an operad

Idea

This entry discusses model category structures on categories of algebras over an operad in the category of chain complexes (unbounded).

This is a special case of the general discussion at model structure on algebras over an operad, but the category of (unbounded) chain complexes warrents some special attention.

For the operad of ordinary (commutative) algebras see also model structure on dg-algebras.

Definition

Let $k$ be a commutative ring. Write $Ch_\bullet(k)$ for the category of unbounded chain complexes of $k$-modules.

Definition

An operad $P$ over $Ch_\bullet(k)$ is called $\Sigma$-split if (…)

A quasi-isomorphism between such operads $P_1 \to P_2$ is said to be compatible with $\Sigma$-splitting if (…)

Proposition

If $k$ contains the ring of rational numbers, $\mathbb{Q} \hookrightarrow k$, then every $Ch_\bullet(k)$-operad is $\Sigma$-split and every quasi-isomorphism of operads is compatible with $\Sigma$-splitting.

The associative operad $Assoc_k$ is $\Sigma$-split for all $k$.

This is (Hinich, example 4.2.5).

Theorem

Let $P$ be a $\Sigma$-split operad, def. , in $Ch_\bullet(k)$. Then the category $Alg_{Ch_\bullet(k)}(P)$ of algebras over the operad admits a model category structure whose

• weak equivalences are the underlying quasi-isomorphisms

• whose fibrations are the degreewise surjections

in $Ch_\bullet(k)$.

The free-forgetful adjunction

$$Alg_{Ch_\bullet(k)}(P) \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} Ch_\bullet(k)$$

is therefore a Quillen adjunction (see at transferred model structure).

This appears as (Hinich, theorem 4.1.1).

Properties

Invariance under equivalence and rectification

Theorem

If $P_1 \to P_2$ is a quasi-isomorphism of $\Sigma$-split operads compatible with splittings, then there is an induced Quillen equivalence

$$Alg(P_1) \stackrel{\overset{}{\leftarrow}}{\underset{}{\rightarrow}} Alg(P_2)$$

between the corresponding model structures on their algebras, as above.

This is (Hinich, theorem 4.7.4).

Remark

Theorem in particular provides rectification results for homotopy algebras: if $P$ is some operad and $\tilde P \stackrel{\simeq}{\to} P$ a cofibrant resolution in the suitable model structure on operads, then the theorem says that $P$-homotopy algebras have the same homotopy theory as the plain $P$-algebras.

Famous examples include the Quillen equivalence between the model structure on dg-Lie algebras and the model structure for L-infinity algebras.

Simplicial enrichment

We discuss how the above model structure on $Alg_{Ch_\bullet(k)}(P)$ is almost enhanced to a simplicial model category structure.

First we recall the standard definition of polynomial differential forms on simplices:

Definition

For $n \in \mathbb{N}$ let $\Omega_{poly}^\bullet(\Delta^n)$ be the commutative dg-algebra of polynomial differential forms on the $n$-simplex:

as a graded algebra it is

$$\Omega_{poly}^{\bullet}(\Delta^n) := k[t_0, \cdots, t_n, d t_0, \cdots, d t_n]/(\sum t_i -1, \sum d t_i)$$

with the differential the usual de Rham differential under the embedding $\Omega^\bullet_{poly}(\Delta^n) \hookrightarrow \Omega^\bullet(\Delta^n)$.

For $f : [k] \to [l]$ a morphism in the simplex category let

$$\Omega^\bullet_{poly}(f) : \Omega^\bullet_{poly}(\Delta^l) \to \Omega^\bullet_{poly}(\Delta^k)$$

be the morphism of dg-algebras given on generators by

$$\Omega^\bullet_{poly}(f) : t_i \mapsto \sum_{f(j) = i} t_j \,.$$

This yields a simplicial commutative dg-algebra

$$\Omega^\bullet_{poly}(\Delta^{(-)}) : \Delta^{op} \to cdgAlg_k$$

or equivalently a cosimplicial object in the opposite category $cdgAlg_k^{op}$.

By the general definition of differential forms on presheaves this extends by left Kan extension to a functor

$$\Omega^\bullet_{poly} : sSet \to cdgAlg_k^{op}$$

given by

$$\Omega^\bullet_{poly}(S) = \int^{[k]\in \Delta} S_k \cdot \Omega^\bullet_{poly}(\Delta^k) \,,$$

where on the right be have the coend over the copowering of $cdgAlg_k^{op}$ over Set.

Definition

For $P$ a dg-operad as above, define sSet-hom-objects between objects $A,B \in Alg(P)$ by

$$Alg_P(A,B) := ([n] \mapsto Hom_{Alg_P}(A, B \otimes \Omega^\bullet_{poly}(\Delta^n)) \in sSet \,.$$

Proposition

These simplicial hom-objects satisfy the dual of the pushout-product axiom (see enriched model category).

This is (Hinich, lemma 4.8.4).

Proposition

For $S$ a degreewise finite simplicial set, we have a natural isomorphism

$$Hom_{Alg_P}(A, B \otimes \Omega^\bullet_{poly}(S)) \simeq Hom_{sSet}(S, Alg_P(A,B)) \,.$$

This is (Hinich, lemma 4.8.3).

Proposition

The homotopy category of $Alg_P$ is given by

$$Ho(Alg_P)(A,B) \simeq \pi_0 Alg_P(Q A,B) \,,$$

where $Q A$ is a cofibrant resolution of $A$.

This appears as (Hinich, section 4.8.10).

Examples

• model structure for L-∞ algebras

Related concepts

• model structure on dg-operads

• model structure on dg-algebras over an operad

  • model structure on dg-algebras

  • model structure on dg-categories

References

The model structure on dg-algebras over an operad is discussed in

• Vladimir Hinich, Homological algebra of homotopy algebras Communications in algebra, 25(10). 3291-3323 (1997)(arXiv:q-alg/9702015, Erratum (arXiv:math/0309453))

based on results in

• A.K. Bousfield, V.K.A.M. Gugenheim, On PL de Rham theory and rational homotopy type , Memoirs AMS, t.8, 179(1976)