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model structure on dg-algebras over an operad

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

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general

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for (,1)(\infty,1)-categories

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for (,1)(\infty,1)-operads

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for (,1)(\infty,1)-sheaves / \infty-stacks

Contents

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

Definition

An operad PP over Ch (k)Ch_\bullet(k) is called Σ\Sigma-split if (…)

A quasi-isomorphism between such operads P 1P 2P_1 \to P_2 is said to be compatible with Σ\Sigma-splitting if (…)

Proposition

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

The associative operad Assoc kAssoc_k is Σ\Sigma-split for all kk.

This is (Hinich, example 4.2.5).

Theorem

Let PP be a Σ\Sigma-split operad, def. 1, in Ch (k)Ch_\bullet(k). Then the category Alg Ch (k)(P)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 (k)Ch_\bullet(k).

The free-forgetful adjunction

Alg Ch (k)(P)UFCh (k) 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 1P 2P_1 \to P_2 is a quasi-isomorphism of Σ\Sigma-split operads compatible with splittings, then there is an induced Quillen equivalence

Alg(P 1)Alg(P 2) 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 2 in particular provides rectification results for homotopy algebras: if PP is some operad and P˜P\tilde P \stackrel{\simeq}{\to} P a cofibrant resolution in the suitable model structure on operads, then the theorem says that PP-homotopy algebras have the same homotopy theory as the plain PP-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 (k)(P)Alg_{Ch_\bullet(k)}(P) is almost enhanced to a simplicial model category structure.

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

Definition

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

as a graded algebra it is

Ω poly (Δ n):=k[t 0,,t n,dt 0,,dt n]/(t i1,dt 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 Ω poly (Δ n)Ω (Δ n)\Omega^\bullet_{poly}(\Delta^n) \hookrightarrow \Omega^\bullet(\Delta^n).

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

Ω poly (f):Ω poly (Δ l)Ω poly (Δ 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

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

This yields a simplicial commutative dg-algebra

Ω poly (Δ ()):Δ opcdgAlg k \Omega^\bullet_{poly}(\Delta^{(-)}) : \Delta^{op} \to cdgAlg_k

or equivalently a cosimplicial object in the opposite category cdgAlg k opcdgAlg_k^{op}.

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

Ω poly :sSetcdgAlg k op \Omega^\bullet_{poly} : sSet \to cdgAlg_k^{op}

given by

Ω poly (S)= [k]ΔS kΩ poly (Δ 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 cdgAlg k opcdgAlg_k^{op} over Set.

Definition

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

Alg P(A,B):=([n]Hom Alg P(A,BΩ poly (Δ n))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 SS a degreewise finite simplicial set, we have a natural isomorphism

Hom Alg P(A,BΩ poly (S))Hom sSet(S,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 Alg PAlg_P is given by

Ho(Alg P)(A,B)π 0Alg P(QA,B), Ho(Alg_P)(A,B) \simeq \pi_0 Alg_P(Q A,B) \,,

where QAQ A is a cofibrant resolution of AA.

This appears as (Hinich, section 4.8.10).

Examples

References

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

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)
Revised on March 14, 2013 13:15:07 by Urs Schreiber (82.169.65.155)