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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 ${\mathrm{Ch}}_{•}(k)$ for the category of unbounded chain complexes of $k$-modules.

Definition

An operad $P$ over ${\mathrm{Ch}}_{•}(k)$ is called $\Sigma$-split if (…)

Proposition

• The associative operad ${\mathrm{Assoc}}_k$ is $\Sigma$-split for all $k$.

• If $k$ contains the ring of rational numbers, $\mathbb{Q}\hookrightarrow k$, then every ${\mathrm{Ch}}_{•}(k)$-operad is $\Sigma$-split.

This is (Hinich, example 4.2.5).

Proposition

Let $P$ be a $\Sigma$-split operad in ${\mathrm{Ch}}_{•}(k)$. Then the category ${\mathrm{Alg}}_{{\mathrm{Ch}}_{•}(k)}(P)$ of algebras over the operad admits a model category structure whose weak equivalences are the quasi-isomorphisms and whose fibrations are the degreewise surjections in ${\mathrm{Ch}}_{bulle}(k)$.

This appears as (Hinich, theorem 4.1.1).

Properties

Simplicial enrichment

We discuss how the above model structure on ${\mathrm{Alg}}_{{\mathrm{Ch}}_{•}(k)}(P)$ is almost enhanced to a simplicial model category structure.

we have the standard definition of polynomial differential forms on simplices.

Definition

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

as a graded algebra it is

$${\Omega }_{\mathrm{poly}}^{•}({\Delta }^n):=k[t_0,\cdots ,t_n,d{t}_0,\cdots ,d{t}_n]/(\sum t_i-1,\sum d{t}_i)$$\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 ${\Omega }_{\mathrm{poly}}^{•}({\Delta }^n)\hookrightarrow {\Omega }^{•}({\Delta }^n)$.

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

$${\Omega }_{\mathrm{poly}}^{•}(f):{\Omega }_{\mathrm{poly}}^{•}({\Delta }^l)\to {\Omega }_{\mathrm{poly}}^{•}({\Delta }^k)$$\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

$${\Omega }_{\mathrm{poly}}^{•}(f):t_i\mapsto \sum _{f(j)=i}{t}_j.$$\Omega^\bullet_{poly}(f) : t_i \mapsto \sum_{f(j) = i} t_j \,.

This yields a simplicial commutative dg-algebra

$${\Omega }_{\mathrm{poly}}^{•}({\Delta }^{(-)}):{\Delta }^{\mathrm{op}}\to {\mathrm{cdgAlg}}_k$$\Omega^\bullet_{poly}(\Delta^{(-)}) : \Delta^{op} \to cdgAlg_k

or equivalently a cosimplicial object in the opposite category ${\mathrm{cdgAlg}}_k^{\mathrm{op}}$.

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

$${\Omega }_{\mathrm{poly}}^{•}:\mathrm{sSet}\to {\mathrm{cdgAlg}}_k^{\mathrm{op}}$$\Omega^\bullet_{poly} : sSet \to cdgAlg_k^{op}

given by

$${\Omega }_{\mathrm{poly}}^{•}(S)={\int }^{[k]\in \Delta }{S}_k\cdot {\Omega }_{\mathrm{poly}}^{•}({\Delta }^k),$$\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 ${\mathrm{cdgAlg}}_k^{\mathrm{op}}$ over Set.

Definition

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

$${\mathrm{Alg}}_P(A,B):=([n]\mapsto {\mathrm{Hom}}_{{\mathrm{Alg}}_P}(A,B\otimes {\Omega }_{\mathrm{poly}}^{•}({\Delta }^n))\in \mathrm{sSet}.$$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

$${\mathrm{Hom}}_{{\mathrm{Alg}}_P}(A,B\otimes {\Omega }_{\mathrm{poly}}^{•}(S))\simeq {\mathrm{Hom}}_{\mathrm{sSet}}(S,{\mathrm{Alg}}_P(A,B)).$$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 ${\mathrm{Alg}}_P$ is given by

$$\mathrm{Ho}({\mathrm{Alg}}_P)(A,B)\simeq {\pi }_0{\mathrm{Alg}}_P(QA,B),$$Ho(Alg_P)(A,B) \simeq \pi_0 Alg_P(Q A,B) \,,

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

This appears as (Hinich, section 4.8.10).

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)