on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
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.
Let $k$ be a commutative ring. Write $Ch_\bullet(k)$ for the category of unbounded chain complexes of $k$-modules.
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 (…)
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).
Let $P$ be a $\Sigma$-split operad, def. 1, 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 $Ch_\bullet(k)$.
is therefore a Quillen adjunction (see at transferred model structure).
This appears as (Hinich, theorem 4.1.1).
If $P_1 \to P_2$ is a quasi-isomorphism of $\Sigma$-split operads compatible with splittings, then there is an induced Quillen equivalence
between the corresponding model structures on their algebras, as above.
This is (Hinich, theorem 4.7.4).
Theorem 2 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.
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:
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
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
be the morphism of dg-algebras given on generators by
This yields a simplicial commutative dg-algebra
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
given by
where on the right be have the coend over the copowering of $cdgAlg_k^{op}$ over Set.
For $P$ a dg-operad as above, define sSet-hom-objects between objects $A,B \in Alg(P)$ by
These simplicial hom-objects satisfy the dual of the pushout-product axiom (see enriched model category).
This is (Hinich, lemma 4.8.4).
For $S$ a degreewise finite simplicial set, we have a natural isomorphism
This is (Hinich, lemma 4.8.3).
The homotopy category of $Alg_P$ is given by
where $Q A$ is a cofibrant resolution of $A$.
This appears as (Hinich, section 4.8.10).
model structure on dg-algebras over an operad
The model structure on dg-algebras over an operad is discussed in
based on results in