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model structure on 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

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

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

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

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

for (n,r)(n,r)-categories

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

Higher algebra

Contents

Idea

A model category structure on a category of algebras over an operad enriched in some suitable homotopical category \mathcal{E} is supposed to be a presentation of the (∞,1)-category of ∞-algebras over an (∞,1)-operad.

Definition

Assumption

Let \mathcal{E} be a category equipped with the structure of

such that

Proposition

Under these conditions there is for each finite group GG the structure of a monoidal model category on the category BG\mathcal{E}^{\mathbf{B}G} of objects in \mathcal{E} equipped with a GG-action, for which the forgetful functor

BG \mathcal{E}^{\mathbf{B}G} \to \mathcal{E}

preserves and reflects fibrations and weak equivalences.

This is discussed in the examples at monoidal model category.

For CC \in Set a set of colours and PP a CC-coloured operad in \mathcal{E} we write Alg (P)Alg_{\mathcal{E}}(P) for the category of PP-algebras over an operad. There is a forgetful functor

U P:Alg (P) C U_P \;\colon\; Alg_{\mathcal{E}}(P) \to \mathcal{E}^C

from the category of algebras over the operad in \mathcal{E} to the underlying CC-colored objects of \mathcal{E}.

Definition

A CC-coloured operad PP is called admissible if the transferred model structure on Alg (P)Alg_{\mathcal{E}}(P) along the forgetful functor

U P:Alg (P) C U_P : Alg_{\mathcal{E}}(P) \to \mathcal{E}^{C}

exists.

Remark

So if PP is admissible, then Alg (P)Alg_{\mathcal{E}}(P) carries the model structure where a morphism of PP algebras f:ABf : A \to B is a fibration or weak equivalence if the underlying morphism in \mathcal{E} is, respectively.

Below we discuss general properties of PP under which this model structure indeed exists.

Properties

Existence by coalgebra intervals

The above transferred model structure on algebras over an operad exists if there is a suitable interval object in \mathcal{E}.

Definition

A cocommutative coalgebra interval object HH\in \mathcal{E} is

  • a cocommutative co-unital comonoid in \mathcal{E}

  • equipped with a factorization

    :IIHI \nabla : I \coprod I \hookrightarrow H \to I

    of the codiagonal on II into two homomorphisms of comonoids with the first a cofibration and the second a weak equivalence in \mathcal{E}.

Examples

Such cocommutative coalgebra intervals exist in

In

there is a coalgebra interval.

Theorem

If \mathcal{E} has a symmetric monoidal fibrant replacement functor and a coalgebra interval object HH then every non-symmetric coloured operad in \mathcal{E} is admissible, def. 2: the transferred model structure on algebras exists.

If the interval is moreover cocommutative, then the same is true for every symmetric coloured operad.

This is (BergerMoerdijk, theorem 2.1), following (BergerMoerdijk-Homotopy, theorem 3.2). For more details see at model structure on operads.

Remark

Since the coalgebra interval in the category of chain complexes is not cocommutative, this case requires special discussion, as some of the statements below will not apply to it. For more on this case see model structure on dg-algebras over an operad.

Rectification of algebras

Recall the notion of resolutions of operads and of the Boardman-Vogt resolution W(H,P)W(H,P) from model structure on operads.

We now discuss conditions under which model categories of algebras over a resolved operad is Quillen equivalent to that over the original operad. This yields general rectification results for homotopy-algebras over an operad (see also the Examples below.)

Theorem

Let \mathcal{E} be in addition a left proper model category.

Then for ϕ:PQ\phi : P \to Q a weak equivalence between admissible Σ\Sigma-cofibrant well-pointed CC-coloured operads in \mathcal{E}, the adjunction

(ϕ !ϕ *):Alg (P)Alg (Q) (\phi_! \dashv \phi^*) : Alg_\mathcal{E}(P) \stackrel{\leftarrow}{\to} Alg_\mathcal{E}(Q)

is a Quillen equivalence.

This is (BergerMoerdijk, theorem 4.1).

Theorem

(rectification of homotopy TT-algebras)

Let still \mathcal{E} be left proper.

Let PP be an admissible Σ\Sigma-cofibrant operad in \mathcal{E} such that also W(H,P)W(H,P) is admissible.

Then

(ϵ !ϵ *):Alg (P)Alg (W(H,P)) (\epsilon_! \dashv \epsilon^*) : Alg_\mathcal{E}(P) \stackrel{\leftarrow}{\to} Alg_\mathcal{E}(W(H,P))

is a Quillen equivalence.

Examples

Monoids (associative algebras)

For P=AssocP = Assoc the associative operad it category of algebras Alg PAlg_{\mathcal{E}} P is the category of monoids in \mathcal{E}. The above model structure on Alg PAlg_{\mathcal{E}} P is the standard model structure on monoids in a monoidal model category.

A A_\infty-Algebras

Let AssocAssoc be the associative operad in Set regarded as an operad in Top under the discrete space embedding Disc:SetTopDisc : Set \to Top.

Let I *I_* be the operad whose algebras are pointed objects. There is a canonical morphism i:I *Associ : I_* \to Assoc.

Claim

The relative Boardman-Vogt resolution

I *I *[i]W([0,1],I *Assoc)Assoc I_* \hookrightarrow I_*[i] \hookrightarrow W([0,1], I_* \to Assoc) \stackrel{\simeq}{\to} Assoc

produces precisely Stasheff’s A-∞ operad.

This is (BergerMoerdijk, page 13)

Corollary

Every A-∞ space is equivalent as an A A_\infty-space to a topological monoid.

Proof

This follows from the rectification theorem, using that by the above algebras over W([0,1],I *Assoc)W([0,1], I_* \to Assoc) are precisely A-∞ spaces.

Remark

This is a classical statement. See A-∞ algebra for background and references.

L L_\infty-algebras and simplicial Lie algebras

Let LieLie be the Lie operad.

A cofibrant resolution is L L_\infty, the operad whose algebras in chain complexes are L-infinity algebras.

Now (…)

Homotopy coherent diagrams

Let CC be a small \mathcal{E}-enriched category with set of objects Obj(C)Obj(C). There is an operad Diag CDiag_{C}

Diag C(c 1,,c n;c)={C(c 1,c) ifn=1 otherwise Diag_C(c_1, \cdots, c_n;c) = \left\{ \array{ C(c_1, c) & if n = 1 \\ \emptyset & otherwise } \right.

whose algebras are enriched functors

F:C, F : C \to \mathcal{E} \,,

hence diagrams in \mathcal{E}. Then the Boardman-Vogt resolution

HoCoDiag C:=W(H,Diag C) HoCoDiag_C := W(H,Diag_C)

is the operad for homotopy coherent diagrams over CC in \mathcal{E}.

The rectification theorem above now says that every homotopy coherent diagram is equivalent to an ordinary \mathcal{E}-diagram. For =\mathcal{E} = Top this is known as Vogt's theorem.

(,1)(\infty,1)-Categories of algebras and bimodules over an operad

The constuction Alg (P)Alg_{\mathcal{E}}(P) of a category of algebras over an operad is contravariantly functorial in PP. Therefore if P P^\bullet is a cosimplicial object in the category of operads, we have that Alg (P )Alg_{\mathcal{E}}(P^\bullet) is a (large) simplicial category of algebras. Moreover, the Boardman-Vogt resolution W(P)W(P) is functorial in PP.

These two facts together allow us to construct simplicial categories of homotopy algebras.

Specifically, there is a cosimplicial operad Assoc Assoc^\bullet which

  • in degree 0 is the usual associative operad Assoc 0=AssocAssoc^0 = Assoc,

  • in degree 1 is the operad whose algebras are triples consisting of two associative monoids and one bimodule between them;

  • in degree 2 it is the operad whose algebras are tuples consisting of three associative algebras A 0,A 1,A 2A_0, A_1, A_2 as well as one A iA_i-A jA_j-bimodule N ijN_{ i j} for each 0i<j20 \leq i \lt j \leq 2 and a homomorphism of bimodules

    N 01 A 1N 12N 02 N_{0 1} \otimes_{A_1} N_{1 2} \to N_{0 2}
  • and so on.

The simplicial category of algebras over Assoc Assoc^\bullet is one incarnation of the 2-category of algebras, bimodules and bimodules homomorphisms.

We can pass to the corresponding \infty-algebras by applying the Boardman-Vogt resolution to the entire cosimplicial diagram of operads, to obtain the cosimplicial A-∞ operad

A :=W(Assoc ). A_\infty^\bullet := W(Assoc^\bullet) \,.

The simplicial category of algebras over this has as objects A-∞ algebras, as morphism bimodules between these, and so on.

This is discussed in (BergerMoerdijkAlgebras, section 6).

References

A general discussion of the model structure on operads is in

The concrete construction of the specific cofibrant resolutions in these structures going by the name Boardman-Vogt resolution is in

The discussion of the model structure on algebras over a suitable operad is in

More discussion on the transport of operad algebra structures along Quillen adjunctions/Bousfield localizations between the underlying model categories is in

Revised on September 4, 2013 13:52:47 by Urs Schreiber (212.238.84.235)