nLab model structure on algebras over an operad



Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

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



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.



Cofibrant operads


Let CC be a cofibrantly generated symmetric monoidal model category. Let OO be a cofibrant operad. If CC satisfies the monoid axiom in a monoidal model category, then there is an induced model structure on the category Alg C(O)Alg_C(O) of algebras over an operad.

See (Spitzweck 01, Theorem 4).



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

such that


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.

Coloured operads

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}.


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}



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.


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}.


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}.


Such cocommutative coalgebra intervals exist in


there is a coalgebra interval.


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. : 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.


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.)


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).


(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.


(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.

Equivalence of model-categorical algebras and quasicategorical algebras

The results of Lurie, Pavlov–Scholbach, and Haugseng establish an equivalence of quasicategories between the underlying quasicategory of the model category of algebras over an operad and the quasicategory of quasicategorical algebras over the underlying quasicategorical operad, provided some mild conditions are met.


(Theorem 7.11 in Pavlov–Scholbach, Theorem 4.10 in Haugseng.) Suppose VV is a symmetric monoidal model category equipped with a subcategory V V^\flat of flat objects. Given a flat admissible Σ-cofibrant VV-operad OO, the canonical comparison functor

Alg O(V) c[W O 1]Alg O(V[W 1])Alg_O(V)^c[W_O^{-1}]\to Alg_O(V[W^{-1}])

is an equivalence of quasicategories.

Here a full subcategory V VV^\flat\subset V is a subcategory of flat objects (Haugseng, Definition 4.1) if it contains all cofibrant objects of VV, is closed under monoidal products, and tensoring a weak equivalence with an object produce a weak equivalence (in V V^\flat).

Here a VV-operad is flat if it is enriched in the subcategory V VV^\flat\subset V.

An operad is admissible if the category of algebras admits a transferred model structure.

An operad OO is Σ-cofibrant if the unit map 1O(1)1\to O(1) is a cofibration and the object O(n)O(n) is cofibrant in the projective model structure on Σ n\Sigma_n-objects in VV.

By Remark 4.9 in Haugseng, a Σ-cofibrant operad is flat whenever the objects of unary endomorphisms O(x,x)O(x,x) are flat.


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.


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)


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


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.


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).


A general discussion of the model structure on operads is in

See also

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:

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

Discussion with an eye towards ring spectra realized as symmetric spectra is in

Discussion with application to homotopical algebraic quantum field theory is in

On rectification of symmetric colored operads:

These are based on an earlier account by Lurie:

Last revised on July 22, 2023 at 15:11:04. See the history of this page for a list of all contributions to it.