On functorial (co)localization of algebras and modules over operads

Motivated by calculations of motivic homotopy groups, we give widely attained conditions under which operadic algebras and modules thereof are preserved under (co)localization functors


Introduction
Operads are key mathematical devices for organizing hierarchies of higher homotopies in a variety of settings. The earliest applications were concerned with iterated topological loop spaces. More recent developments have involved derived categories, factorization homology, knot theory, moduli spaces, representation theory, string theory, deformation quantization, and many other topics. This paper is a sequel to our work on operads in the context of the slice filtration in motivic homotopy theory [9].
The problem we address here is that of preservation of algebras over colored operads, and also modules over such algebras, under Bousfield (co)localization functors. For this we only require a few widely attained technical assumptions and notions on the underlying model categories and the operads, e.g., that of strongly admissible operads in a cofibrantly generated symmetric monoidal model category. We refer to [5], [18], and [17] for related results on (co)localization of monadic algebras.
Here, 1 is the motivic sphere spectrum, K M denotes Milnor K-theory, and KQ is the hermitian K-theory spectrum. The solution of Morel's π 1 -conjecture [15] involves an explicit calculation in the slice spectral sequence of the motivic sphere spectrum. One of the precursors for this calculation is the fact that the total slice functor takes E ∞ motivic spectra, in particular the algebraic cobordism spectrum, to graded E ∞ MZ-algebras in a functorial way. Here, MZ denotes the motivic Eilenberg-MacLane spectrum. Theorems 3.8 and 3.14 in this paper coupled with our construction of the slice filtration in [9, §6] verify the mentioned multiplicative property (which in turn is used in the proof of [15,Theorem 2.20]). We envision that future calculations with slice spectral sequences will exploit multiplicative structures to a greater extent, and as such will be relying on the results herein. The paper starts with §2 on model structures on operads and algebras. Our main results on preservation of algebras and modules under Bousfield (co)localization functors are shown in §3 and §4. To make the paper reasonably self-contained we have included two appendices fixing our conventions on model categories and colored operads. In particular, we review tensor-closed sets of objects in a homotopy category, the Reedy model structure, operadic algebras, and modules over such algebras.

Model structures of operads and algebras
Let C be a cocomplete closed symmetric monoidal category with tensor product ⊗, unit I, initial object 0, and internal hom functor Hom(−, −). For a set C we refer to Appendix B for the definitions of C-colored collections and C-colored operads in C.
Recall that a C-colored collection K is pointed if it is equipped with unit maps I → K(c; c) for every c ∈ C. Denote by Coll C (C) and Coll • C (C) the categories of C-colored collections and pointed C-colored collections, respectively. If K is a C-colored collection, we can define a pointed C-colored collection  We denote by Oper C (C) and Oper(C) the categories of C-colored operads and (onecolored) operads in C, respectively. Suppose C is a cofibrantly generated symmetric monoidal model category. Then Coll C (C) and Coll • C (C) have transferred model structures, where weak equivalences and fibrations are defined colorwise. There is a free-forgetful adjoint pair Under suitable conditions, the model structure on (pointed) C-colored collections can be transferred along (2) to a cofibrantly generated model structure on Oper C (C), in which a map of C-colored operads is a fibration or a weak equivalence if its underlying (pointed) C-colored collection is so. This holds for k-spaces, simplicial sets, and symmetric spectra; see [2,  In general, (2) does not furnish a model structure on Oper C (C), but rather the weaker structure of a semi model structure. In a semi model category the axioms of a model category hold with the exceptions of the lifting and factorization axioms, which hold only for maps with cofibrant domains. The trivial fibrations have the right lifting property with respect to cofibrant objects, since the initial object of a semi model category is assumed to be cofibrant. For operads the following result is shown in [16,Theorem 3.2] (cf. [7,Theorem 12.2A]). Our extension to colored operads follows similarly.
Theorem 2.1. If C is a cofibrantly generated symmetric monoidal model category, then the model structure on Coll • C (C) transfers along the free-forgetful adjunction (2) to a cofibrantly generated semi model structure on Oper C (C), in which a map O → O ′ is a fibration or a weak equivalence if O(c 1 , . . . , c n ; c) → O ′ (c 1 , . . . , c n ; c) is a fibration or a weak equivalence in C, respectively, for every tuple of colors (c 1 , . . . , c n , c).
Throughout the paper we will implicitly assume that Oper C (C) always admits a cofibrantly generated transferred model structure, where the weak equivalences and fibrations are defined at the level of the underlying collections.
Let C C denote the product category c∈C C. If O is a C-colored operad, denote by Alg O (C) the category of O-algebras in C; see Appendix B. There is a free-forgetful adjoint pair where the left adjoint is the free O-algebra functor defined by If it is clear from the context we shall write F and U instead of F O and U O , respectively. Let C be a cofibrantly generated symmetric monoidal model category. Recall from [2] that a C-colored operad O is admissible if the product model structure on C C transfers to a cofibrantly generated model structure on As indicated in [16, I.5], if C is a simplicial symmetric monoidal model category and O is an admissible C-colored operad, then Alg O (C) is naturally a simplicial model category. For a simplicial set K and an O-algebra A , the cotensor A K is the object (U O A ) K with O-algebra structure given by the composition O → End(A ) → End(A K ) -for the endomorphism colored operad -induced by the diagonal map K → K × · · · × K. For K fixed, the functor (−) K has a left adjoint defining the tensor. For A fixed, the functor A (−) has a right adjoint defining the simplicial enrichment in Alg O (C).
Definition 2.2. Let C be a cofibrantly generated symmetric monoidal model category.
and O ′ satisfies one of the conditions: (i) It has an underlying cofibrant C-colored collection.
(ii) It has an underlying cofibrant pointed C-colored collection, and C has an additional cofibrantly generated symmetric monoidal model structure with the same weak equivalences and cofibrant unit.
We Definition 2.5. Let C be a cofibrantly generated symmetric monoidal model category. A monoid A in C is strongly admissible if there is another monoid A ′ and a weak equivalence ϕ : The constant simplicial object functor sends an object X to the simplicial object X • with X n = X for all n. If C is symmetric monoidal, this is a symmetric monoidal functor for the objectwise tensor product on sC. Thus, if O is a C-colored operad in C, we can view it as a C-colored operad in the category of simplicial objects sC by applying the constant functor levelwise.
where U and | − | denote the corresponding forgetful and realization functor, respectively.
Proof. In any simplicial model category there are adjoint functors | − | C : sC → C and Sing C : C → sC, where Sing C (X) is the simplicial object with Sing C (X) n = X ∆[n] . Since is also a simplicial model category, we have the adjunction  We refer to Appendix A.1 for the Reedy model structure on simplicial categories.    Proof. The category of O-algebras in sC has a model structure transferred from (sC) C by assumption and Lemma 2.9. Moreover, sC is a symmetric monoidal model category by Lemma A.3 (i). Also every object in sC is small relative to the whole category. To prove part (i) note that the constant operad on O in sC has an underlying cofibrant collection by Lemma 2.8. Thus we can apply Proposition 2.7(i) to sC with the Reedy model structure. Since a Reedy cofibrant object of sAlg(O) is cofibrant for the transferred model structure, by Lemma 2.9, this gives the result.
Part (ii) is proved similarly by reference to [16,Proposition 4.8]. (By assumption the constant operad on O in sC has an underlying cofibrant collection in sC for the Reedy model structure induced by the second model structure on C.)

Colocalization of algebras
In this section we show that tensor-closed K-colocalization functors preserve algebras over cofibrant C-colored operads. More precisely, we prove that if O is a cofibrant Ccolored operad and f is an If K is a set of isomorphism classes of objects of Ho(C), and C is a set of colors, denote by K C the set of objects in Ho(C) C defined as K C = c∈C K. Note that an object in C C is K C -colocal if and only if it is colorwise K-colocal.
is a weak equivalence, where U denotes the corresponding forgetful functor.
Proof. Since C is a simplicial symmetric monoidal model category, Remark 2.4 shows we may assume O is strongly monoidal and satisfies Definition 2.2(i). We may also assume that O has an underlying cofibrant collection, due to the Quillen equivalence between Alg O (C) and Alg O ′ (C) in the definition of a strongly admissible operad.
By homotopy invariance of the homotopy colimit, we may further assume that A • is Reedy cofibrant. By Lemma A.4, sC is cofibrantly generated. Thus by Corollary 2.10(i) U (A • ) is Reedy cofibrant as well. By Lemma A.1, |A • | Alg(O) computes the homotopy colimit of A • , and |U (A • )| C C computes the homotopy colimit of U (A • ). Corollary 2.6 gives an isomorphism |U (A • )| C C ∼ = U (|A • | Alg(O) ), which finishes the proof.
Let C be a symmetric monoidal model category and O a C-colored operad in C. Given any O-algebra A in Alg O (C) we define the standard simplicial object associated to A by setting A n = (F U ) n+1 A with the usual structure maps. Here, F and U denote the free functor and the forgetful functor, respectively. There is a canonical augmentation A • → A obtained by viewing A as a constant simplicial object. Proof. This follows from Lemma 3.1 since U (A • ) → U (A ) is a split augmented simplicial object.
Remark 3.3. In Lemma 3.2 one should be mindful of forming the "correct" derived simplicial object, i.e., in degree n it is weakly equivalent to (F QU ) n+1 A , where Q is a cofibrant replacement functor in C C .
Lemma 3.4. Let O be a strongly admissible C-colored operad in a cofibrantly generated simplicial symmetric monoidal model category C, and K a tensor-closed set of isomorphism classes of objects of Ho(C).
Proof. Since O is strongly admissible and C is a simplicial symmetric monoidal model category, we may assume that O has an underlying cofibrant collection. We note that O(c; c) ⊗ X is cofibrant in C for every c ∈ C and cofibrant X in C. Moreover, we have The result follows now from the fact that K is tensor-closed, K-colocal objects are closed under coproducts, and F (X)(c) is a homotopy quotient of K-colocal objects for every c ∈ C, hence K-colocal.  Proof. We may assume O has an underlying cofibrant collection because C is a simplicial symmetric monoidal model category. Also assume without loss of generality that D takes values in cofibrant objects. For every i ∈ I, let D(i) • → D(i) be the augmented standard simplicial object associated to D(i). Note that by Proposition 2.7(i) and the explicit formula for the free functor F , U (F U ) n D(i) is cofibrant for every i ∈ I and n ≥ 0. For Proposition 3.7. With the same assumptions as in Lemma 3.4 the following holds.
Proof. Again we may assume that O has an underlying cofibrant collection since C is a simplicial symmetric monoidal model category.
To prove part (i) we may assume A is cofibrant. Let A • → A be the associated augmented standard simplicial object. As in the proof of Lemma 3.6 it follows that A n has the correct homotopy type for every n, i.e., each A n is weakly equivalent to For part (ii), note that if X in C C is colorwise K-colocal, then LF (X) is underlying colorwise K-colocal by Lemma 3.4. We conclude from Lemma 3.6 since by assumption the F (K C )-colocal objects are generated under homotopy colimits by F (K C ).
Remark 3.9. Theorem 3.8 implies Proposition 3.7(i) provided Alg O (C) acquires a good LF (K C )-colocalization. If C C has a good K C -colocalization, the theorem states that for a cofibrant replacement Proof. It suffices to check that the fibrations and weak equivalences coincide. For the fibrations, note that the model structures on the algebras are transferred from C and C K , for the same classes of fibrations. For the weak equivalences we use Lemma A.7(ii).
is right proper, e.g., when C is right proper. We also remark that Alg O (C K ) exists if the colocalized model structure C K can be transferred.

Localization of algebras
In this section we show that tensor-closed S-localization functors preserve algebras over cofibrant C-colored operads. More precisely, we prove that if O is a cofibrant C-colored operad and f an LF If S is a set of homotopy classes of maps in C and C is a set of colors, we denote by S C the set c∈C S. Note that a map in C C is an S C -local equivalence if and only if it is colorwise an S-local equivalence.
Lemma 3.12. Let O be a strongly admissible C-colored operad in a cofibrantly generated simplicial symmetric monoidal model category C. Suppose S is set of homotopy classes of maps such that S-equivalences are tensor-closed. If g is colorwise an S-equivalence, then F (g) is underlying colorwise an S-equivalence.
By assumption, the map A (c 1 ) ⊗ · · · ⊗ A (c n ) → B(c 1 ) ⊗ · · · ⊗ B(c n ) is an S-local equivalence for every n-tuple (c 1 , . . . , c n ), and tensoring with O(c 1 , . . . , c n ) preserves this property. The result follows by using that S-local equivalences are closed under homotopy colimits and coproducts.
Remark 3.13. The assumptions of the theorem are automatically satisfied if, for instance, the functor X ⊗ L − preserve S-local equivalences for all X in C.
Theorem 3.14. Let O, C, C, and S be as above and suppose in addition that is an F (S C )-local equivalence. Taking homotopy colimits in the previous diagrams yields the commutative square A The right vertical map is an F (S C )-local equivalence (a homotopy colimit of F (S C )local equivalences). The horizontal maps are weak equivalences by Lemma 3.2. Hence A → B is an F (S C )-local equivalence. By repeating the same construction with A ′ instead of A , we obtain a commutative diagram where all four maps are F (S C )-local equivalences and A ′ , B and B ′ are F (S C )-local.
Hence the left vertical map and the bottom horizontal map are weak equivalences. If we apply the forgetful functor we get a commutative diagram where the left vertical map is an S C -local equivalence and the right vertical and bottom horizontal maps are weak equivalences. It follows that U (A ) → U (A ′ ) is an S C -local equivalence.

(Co)localization of modules over algebras
In the following we shall run similar arguments for modules over a given monoid instead of algebras over a colored operad, culminating in analogous statements of Theorem 3.8 and Theorem 3.14. When colocalizing (resp. localizing) a module over a monoid A with respect to a tensor-closed set of objects K (resp. of morphisms S) for which A is K-colocal (resp. S-local), one can simply apply Theorem 3.8 or Theorem 3.14 because there exists an operad whose algebras are exactly the modules over the given monoid. That is, let O be the operad with O(1) = A and O(i) = ∅ for i = 1. Then the categories of O-algebras and A -modules are equivalent. Furthermore, O is strongly admissible if A is. But in practice, e.g., for the motivic slice filtration, one wants to colocalize or localize a module with respect to a colocalization or localization functor other than the one for which the monoid is colocal or local.

Colocalization of modules
We first address colocalization of modules over monoids, and second colocalization of modules over arbitrary operads. In the latter case we employ enveloping algebras and restrict to monoids. Proof. Since A is strongly admissible, we may assume its underlying object is cofibrant (case (i) in Definition 2.2), or its unit map is a cofibration in C (case (ii)). In the first case, U is a left Quillen functor since its right adjoint given by the internal hom Hom(A , −) preserves fibrations and trivial fibrations, so the result follows. In the second case, the same argument shows that U is a left Quillen functor for the model structure on C furnished by the strong admissibility of A . For the second assertion, we use Lemma 4.1, the fact that A ⊗ L K-colocal A -modules are generated under homotopy colimits by A ⊗ L K, and the assumption that A ⊗ L K is underlying K-colocal. Remark 4.3. Note that since we are dealing with monoids instead of arbitrary operads, we do not assume in Proposition 4.2 that the set K is tensor-closed (cf. Proposition 3.7).
Proof. Proposition 4.2 implies that M ′ is underlying K-colocal. Using Lemma A.7(ii) we conclude that M ′ → M is an underlying K-colocal equivalence.
Next we discuss E ∞ operads, i.e., parameter spaces for multiplication maps that are associative and commutative up to all higher homotopies, and their algebras.  Remark 4.6. In the above theorem we could also assume that O is cofibrant as an operad (the operads in C form a left semi model category over C Σ,• -for notation, see [ It is desirable to have a parallel theory for modules over operad algebras (in the one-colored case). Since we have the equivalence Mod(A ) ≃ Mod(Env O (A )) and the enveloping algebra is always a monoid, we can restrict to the latter case. A key point is to show that Env O (A ) is underlying K-colocal under suitable assumptions, making our proof of Theorem 4.5 for E ∞ operads go through. For this we employ the simplicial resolution A • → A . It is easily seen that Env O (A n ) is underlying K-colocal for each n ≥ 0, so the result follows provided Env O (A ) is weakly equivalent to the homotopy colimit over ∆ op of the diagram Env O (A • ).
For a symmetric monoidal category C, we denote by Pairs(C) the category of pairs (O, A ), where O ∈ Oper(C) and A ∈ Alg O (C). Next we review some facts about the colored operads O and P whose algebras are Oper(C) and Pairs(C), respectively. The set of colors for O is N, while for P it is N ∪ {a}. The operad O is a special case of a colored operad defined in [10, §3] whose algebras are itself colored operads for a fixed set of colors C. We take C to be a one point set and let O = S C C in the notation of [10]. The colored operad O is the image in C of an N-colored operad in sets denoted S C , which we now describe (an explicit description of O can be found in [3, 1.5.6]). Let S C (n 1 , . . . , n k ; n) denote the set of isomorphism classes of certain trees. We consider planar connected directed trees such that each vertex has exactly one outgoing edge. There are two different types of edges, namely inner edges with vertices at both ends, and external edges with a vertex only at one end or no vertices at all. It follows that there is exactly one external edge leaving a vertex, the so-called root. There are n external edges which are input edges to vertices, called leaves. These are numbered by  Note that, as for O, P is the image in C of a colored operad, say P s , in sets. Proof. This uses the explicit description of these colored operads: two isomorphic planar trees of the type we consider are already uniquely isomorphic, the additional numbering of the vertices -and leaves for the case of P s -force the actions to be free.
Proposition 4.9. Let C be a cofibrantly generated simplicial symmetric monoidal model category such that all of its objects are small relative to the whole category. Suppose P is strongly admissible (e.g., the unit in C is cofibrant and Pairs(C) has a transferred model structure by Lemma 4.8). For a simplicial object A • in Pairs(C) there is a canonical weak equivalence Here, U denotes the forgetful functor Pairs(C) → C N∪{a} .
Proof. This follows directly from Lemma 3.1 and Proposition 4.7.
There is an embedding φ :    For X ∈ C the enveloping algebra Env O (F X) ∼ = O F X (1) is given by the formula It follows that Env O (A n ) is underlying K-colocal for each n ≥ 0. Since the augmented simplicial object U A • → U A splits, Proposition 4.9 for Pairs(C) implies there is a canonical weak equivalence Here, the homotopy colimit is computed in C. It follows that O A (1) ∼ = Env O (A ) is underlying K-colocal, as claimed. Remark 4.14. One may ask for other hypothesis such that Theorem 4.12 still holds. With C and K as above, suppose Oper(C) and Pairs(C) have transferred model structures. Suppose C has a second simplicial model structure with the same weak equivalences and cofibrant unit. We wish to conclude that a cofibrant underlying K-colocal (O, A ) yields an underlying K-colocal enveloping algebra Env O (A ). As a replacement for Proposition 4.9 we sketch an alternate argument: Suppose every Reedy cofibrant object X • ∈ sPairs(C) is cofibrant in sC N∪{a} for the Reedy model structure. Now P s -viewed as a colored operad in sSets -has an underlying cofibrant collection. Let us assume the objectwise tensor functor sSets × sC → sC is a Quillen bifunctor. Then ci → X • is a cofibration in sPairs(C), where ci is the constant simplicial object on the initial object i of Pairs(C), see [16,Proposition 4.8]. Since ∆ op has cofibrant constants, it follows that X • is underlying Reedy cofibrant.
The same argument works for Oper(C). Alternatively, one can use that Oper(C) is a left semi model category over C Σ,• .

Localization of modules
As in the previous section, we first discuss localization of modules over monoids and then localization of modules over arbitrary operads.
Given a monoid A , we say that the functor A ⊗ L − preserves S-equivalences if the tensor product of A with any S-equivalence is an underlying S-equivalence. Proof. The proof is basically the same as for Theorem 3.14. We note the assumption of tensor-closedness on the S-local equivalences is not needed since the free A -module functor is defined by F (X) = A ⊗X for every X in C, and therefore A n = (F U ) n+1 A → (F U ) n+1 B = B n is an F (S)-equivalence for every map of monoids A → B. Proof. Let F : C ⇄ Alg(O) : U be the free-forgetful adjunction. Let A • → A be the standard augmented simplicial object with A n = (F U ) n+1 A . Suppose that for every S-local equivalence g the map A ⊗ L g is an S-local equivalence. Then, Env O (A n ) ⊗ L g is also an S-local equivalence for every n ≥ 0. Now, using the same argument as in the proof of Theorem 4.12 with the operad O A , it follows that Env O ⊗ g is an Sequivalence.

A Preliminaries on model categories
If C is a cofibrantly generated model category with set of generating cofibrations I and set of generating trivial cofibrations J, we implicitly assume the (co)domains of the elements of I are small relative to the I-cellular maps and that the (co)domains of the elements of J are small relative to the J-cellular maps. This condition is satisfied if C is a combinatorial model category; that is, C is cofibrantly generated and locally presentable, since in this case every object is λ-small for some cardinal λ. Let sSets denote the category of simplicial sets. Recall that for a simplicial symmetric monoidal model category C there exists a monoidal Quillen adjunction i : sSets ⇄ C : r. Any such C is canonically enriched and (co)tensored over sSets. The tensor, enrichment, and cotensor are defined by the functors i(−) ⊗ −, r(Hom(−, −)), and Hom(i(−), −), respectively, where Hom(−, −) denotes the internal hom in C. These three functors form a Quillen adjunction of two variables.

A.1 The Reedy model structure on simplicial objects
Let C be a model category. The simplicial objects in C is the category sC of ∆ op -diagrams in C, where ∆ denotes the simplicial category. In its Reedy model structure [11, 15.3] the weak equivalences are the levelwise weak equivalences, while the cofibrations and fibrations are defined by means of latching and matching objects, respectively.
Let C be a simplicial model category. The realization |X • | C of a simplicial object X • : ∆ op → C is the coequalizer of the diagram Using coend notation, as in [11, 18.3.2] and [12,IX.6], this can be recast as If the category is clear from the context we write |X • | instead of |X • | C .
Lemma A.1. Let C be a simplicial model category and X • a Reedy cofibrant simplicial object in C. Then the Bousfield-Kan map Proof. See [11,Theorem 18.7.4].
The category s 2 C of bisimplicial objects in C is the category of simplicial objects in sC. There is an obvious diagonal functor diag : s 2 C → sC defined by diag(X •,• ) n = X n,n .
Lemma A.2. Let X •,• be a bisimplicial object in a simplicial model category C. Then there is a natural isomorphism If C has a symmetric monoidal structure, there is a symmetric monoidal tensor product in sC defined by the objectwise tensor product, i.e., (X • ⊗ Y • ) n = X n ⊗ Y n . Lemma A.3. Let C be a symmetric monoidal model category.
(i) Then sC is a symmetric monoidal model category for the Reedy model structure.
(ii) If C is simplicial the realization functor is symmetric monoidal.
Proof. The first part is an application of [1, Theorem 3.51 and Example 3.52]. For the second part, observe that where the last isomorphism follows by applying Lemma A.2 to the bisimplicial object (X ⊗ Y ) n,m = X n ⊗ Y m .
Lemma A.4. Let C be a cofibrantly generated model category. Then the Reedy model structure on sC is cofibrantly generated.
Proof. Here we make use of smallness of the (co)domains of the sets of generating (trivial) cofibrations, see [11,Theorem 15.6.27].

A.2 Bousfield (co)localizations
Let C be a simplicial model category, S a set of homotopy classes of maps in C, and K a set of isomorphism classes of objects of Ho(C). The homotopy type of the derived simplicial mapping space map(X, Y ) can be computed using Map ( is an isomorphism in Ho(sSets). An object Z in C is S-local if its image in Ho(C) is so. The class of S-local objects is closed under homotopy limits. A map g : X → Y in Ho(C) is an S-local equivalence or simply an S-equivalence if for every S-local Z, the induced map is an isomorphism in Ho(sSets). A map X → Y in C is an S-local equivalence if its image in Ho(C) is so. A map f : X → Y in Ho(C) is a K-colocal equivalence if for any representative K of an element of K, the induced map is an isomorphism in Ho(sSets). Likewise, a map in C is a K-colocal equivalence if its image in Ho(C) is so. An object W in Ho(C) is called K-colocal if for every K-colocal equivalence g : X → Y , there is an induced isomorphism in Ho(sSets). An object W in C is K-colocal if its image in Ho(C) is so. The class of K-colocal objects is closed under homotopy colimits.
If X is an object of C, an S-localization is an S-local equivalence X → X ′ for X ′ S-local. Dually, a K-colocalization is a K-colocal equivalence X ′ → X for X ′ K-colocal.
A simplicial symmetric monoidal model category C is tensor-closed if the class of Slocal equivalences is closed under the derived tensor product. Likewise, K is tensor-closed if the class of K-colocal objects is closed under the derived tensor product.
Definition A.5. Let S be a set of maps and K be a set of objects in a simplicial model category C.
(i) C has a good S-localization if the left Bousfield localization with respect to S exists; that is, if the classes of cofibrations in C and S-local equivalences define a model structure on C. This is the S-local model structure denoted by C S .
(ii) C has a good K-colocalization if the right Bousfield localization with respect to K exists; that is, if the classes of fibrations in C and K-colocal equivalences define a model structure on C, and the K-colocal objects are generated under homotopy colimits by the objects of K. This is the K-colocal model structure denoted by C K .
The S-local fibrations are the maps in C with the right lifting property with respect of all maps of C that are cofibrations and S-local equivalences, Similarly, the K-colocal cofibrations are the maps in C with the left lifting property with respect to all maps of C that are fibrations and K-colocal equivalences.
If C has a good C S -localization, then an S-localization of X is just a fibrant replacement of X in the localized model structure C S (also called an S-local replacement). Similarly, if C has a good K-colocalization, then a K-colocalization is a cofibrant replacement in the colocalized model structure C K .
Theorem A.6. Let C be a cellular or combinatorial simplicial model category.
(i) If C is left proper, then C has a good S-localization for every set of maps S.
(ii) If C is right proper, then C has a good K-colocalization for every set of objects K.
Moreover, the K-colocal objects is the smallest class of objects of C that contains K and is closed under homotopy colimits and weak equivalences.
Proof. For C cellular see [11,  If F : C → D is a left Quillen functor, denote by LF : Ho(C) → Ho(D) its left derived functor. If U : D → C is a right Quillen functor, denote by RU its right derived functor.
Lemma A.7. Let F : C ⇄ D : U be a simplicial Quillen adjunction, S a set of homotopy classes of maps in C, and K a set of isomorphism classes of objects of Ho(C).
if g is an S-local equivalence in C, then LF (g) is an LF (S)-local equivalence in D.
Proof. Both statements follow by using derived adjunctions.

B Colored operads
In this appendix we recall the definitions and basic properties of colored operads and their algebras that are used in the paper. Throughout, V denotes a cocomplete closed symmetric monoidal category with tensor product ⊗, initial object 0, unit I, and internal hom Hom V (−, −). The elements in the set C are referred to as colors.
Definition B.1. A C-colored collection K in V consists of a set of objects K(c 1 , . . . , c n ; c) in V for each (n + 1)-tuple of colors (c 1 , . . . , c n , c) equipped with a right action of the symmetric group Σ n given by maps where α ∈ Σ n (by default, Σ n is the trivial group if n = 0 or n = 1).
A map of C-colored operads is a map of the underlying C-colored collections that is compatible with the unit and composition product maps.
Denote by V C the product category of copies of V indexed by the set of colors C; that is, V C = c∈C V. For every object X = (X(c)) c∈C in V C , the endomorphism colored operad End(X) of X is the C-colored operad defined by End(X)(c 1 , . . . , c n ; c) := Hom V (X(c 1 ) ⊗ · · · ⊗ X(c n ), X(c)).
Here, X(c 1 )⊗· · · ⊗X(c n ) is the unit I when n = 0. The composition product is ordinary composition and the Σ n -action is defined by permutation of the factors. Definition B.3. Let O be any C-colored operad in V. An O-algebra (or an algebra over O) A is an object X = (X(c)) c∈C of V C together with a map O −→ End(X) of C-colored operads.
Equivalently, since the monoidal category V is closed, an O-algebra is a family of objects X(c) in V for every c ∈ C together with maps O(c 1 , . . . , c n ; c) ⊗ X(c 1 ) ⊗ · · · ⊗ X(c n ) −→ X(c), for every (n + 1)-tuple (c 1 , . . . , c n , c), that are compatible with the symmetric group action, the unit maps of O, and subject to the usual associativity isomorphisms. A Note that End(f ) inherits a C-colored operad structure from the C-colored operads End(X) and End(Y). We denote the category of O-algebras by Alg O (V).
There is a canonical inclusion of C-colored operads End O (X) −→ End(X), and thus every map O −→ End(X) of C-colored operad factors uniquely through the restricted endomorphism operad End O (X). Hence an O-algebra structure on X is given by a map of C-colored operads O −→ End O (X).
When C = {c}, a C-colored operad O is an operad, where O(n) is short for O(c, . . . , c; c) with n ≥ 0 inputs. The associative operad Ass is the one-color operad with Ass(n) = I[Σ n ] for n ≥ 0. Here, I[Σ n ] is the coproduct of copies of the unit I indexed by Σ n , on which Σ n acts freely by permutations. The commutative operad Com is the one-color operad with Com(n) = I for n ≥ 0. Algebras over Ass are the associative monoids in V, while algebras over Com are the commutative monoids in V. For