nLab model structure on algebras over an operad

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

Existence

Cofibrant operads

Theorem

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

See (Spitzweck 01, Theorem 4).

$G$-objects

Assumption

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

• a closed symmetric monoidal category;

• a monoidal model category;

such that

• the model structure is cofibrantly generated;

• the tensor unit $I$ is cofibrant.

Proposition

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

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

preserves and reflects fibrations and weak equivalences.

This is discussed in the examples at monoidal model category.

Coloured operads

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

$$U_P \;\colon\; Alg_{\mathcal{E}}(P) \to \mathcal{E}^C$$

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

Definition

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

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

exists.

Remark

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

Below we discuss general properties of $P$ 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 $H\in \mathcal{E}$ is

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

• equipped with a factorization

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

  of the codiagonal on $I$ 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

• the model structure on compactly generated topological spaces;

• the model structure on simplicial sets;

• the model structure on symmetric spectra.

In

• the model structure on chain complexes

there is a coalgebra interval.

Theorem

If $\mathcal{E}$ has a symmetric monoidal fibrant replacement functor and a coalgebra interval object $H$ 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.

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 $W(H,P)$ 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 $\phi : P \to Q$ a weak equivalence between admissible $\Sigma$-cofibrant well-pointed $C$-coloured operads in $\mathcal{E}$, the adjunction

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

is a Quillen equivalence.

This is (BergerMoerdijk, theorem 4.1).

Theorem

(rectification of homotopy $T$-algebras)

Let still $\mathcal{E}$ be left proper.

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

Then

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

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

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

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

is an equivalence of quasicategories.

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

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

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

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

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

Examples

Monoids (associative algebras)

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

$A_\infty$-Algebras

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

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

Claim

The relative Boardman-Vogt resolution

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

produces precisely Stasheff‘s A-∞ operad.

This is (BergerMoerdijk, page 13)

Corollary

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

Proof

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

Remark

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

$L_\infty$-algebras and simplicial Lie algebras

Let $Lie$ be the Lie operad.

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

Now (…)

Homotopy coherent diagrams

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

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

whose algebras are enriched functors

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

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

$$HoCoDiag_C := W(H,Diag_C)$$

is the operad for homotopy coherent diagrams over $C$ 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.

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

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

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

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

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

• 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 $A_0, A_1, A_2$ as well as one $A_i$-$A_j$-bimodule $N_{ i j}$ for each $0 \leq i \lt j \leq 2$ and a homomorphism of bimodules

  $$N_{0 1} \otimes_{A_1} N_{1 2} \to N_{0 2}$$

• and so on.

The simplicial category of algebras over $Assoc^\bullet$ 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

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

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).

Related concepts

• model structure on monoids in a monoidal model category

  • model structure on dg-algebras

  • model structure for ring spectra

• model structure on modules in a monoidal model category

• algebra over a monad

  ∞-algebra over an (∞,1)-monad

  • model structure on algebras over a monad

• algebra over an algebraic theory

  ∞-algebra over an (∞,1)-algebraic theory

  • homotopy T-algebra / model structure on simplicial T-algebras

• (∞,1)-operad, model structure on operads

  • algebra over an (∞,1)-operad, model structure on algebras over an operad

    • module over an algebra over an (∞,1)-operad, model structure on modules over an algebra over an operad

References

A general discussion of the model structure on operads is in

• Clemens Berger, Ieke Moerdijk, Axiomatic homotopy theory for operads Comment. Math. Helv. 78 (2003), 805–831. (arXiv:math/0206094)

See also

• Markus Spitzweck, Operads, Algebras and Modules in General Model Categories, arXiv:math/0101102.

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

• Clemens Berger, Ieke Moerdijk, The Boardman-Vogt resolution of operads in monoidal model categories, Topology 45 (2006), 807–849. (pdf)

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

• Clemens Berger, Ieke Moerdijk, Resolution of coloured operads and rectification of homotopy algebras, in Categories in Algebra, Geometry and Mathematical Physics, Contemp. Math. 431 (2007) 31-58 [arXiv:math/0512576, ams:conm/431]

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

• Carles Casacuberta, Javier Gutiérrez, Ieke Moerdijk, Rainer Vogt, Localization of algebras over coloured operads, Proceedings of the London Mathematical Society (3) 101 (2010), no. 1, 105-136 (arXiv:0806.3983)

• Javier Gutiérrez, Transfer of algebras over operads along derived Quillen adjunctions, Journal of the London Mathematical Society 86 (2012), 607-625 (arXiv:1104.0584)

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

• Stefan Schwede, section III.6 of Symmetric spectra, 2012 (pdf)

Discussion with application to homotopical algebraic quantum field theory is in

• Marco Benini, Alexander Schenkel, Lukas Woike, Homotopy theory of algebraic quantum field theories (arXiv:1805.08795)

On rectification of symmetric colored operads:

• Dmitri Pavlov, Jakob Scholbach, Admissibility and rectification of colored symmetric operads, Journal of Topology 11 3 (2018) 559-601 $[$doi:10.1112/topo.12008, arXiv:1410.5675$]$

• Rune Haugseng, Algebras for enriched ∞-operads, arXiv.

These are based on an earlier account by Lurie:

• Jacob Lurie, Higher Algebra.