# nLab model structure on algebras over an operad

Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

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

for $(\infty,1)$-operads

for $(n,r)$-categories

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

#### Higher algebra

higher algebra

universal 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.

## Existence

###### 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

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.

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

In

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 Lurie.HA, Pavlov–Scholbach PavlovScholbach18, and Haugseng 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 PavlovScholbach18, Theorem 4.10 in Haugseng 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 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 $O$ 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 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
$I_* \hookrightarrow I_*[i] \hookrightarrow W([0,1], I_* \to Assoc) \stackrel{\simeq}{\to} Assoc$

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).

## 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

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

Discussion with application to homotopical algebraic quantum field theory is in

These are based on an earlier account by Lurie:

Last revised on August 10, 2022 at 16:02:00. See the history of this page for a list of all contributions to it.