Smith Ideals of Operadic Algebras in Monoidal Model Categories

Building upon Hovey's work on Smith ideals for monoids, we develop a homotopy theory of Smith ideals for general operads in a symmetric monoidal category. For a sufficiently nice stable monoidal model category and an operad satisfying a cofibrancy condition, we show that there is a Quillen equivalence between a model structure on Smith ideals and a model structure on algebra maps induced by the cokernel and the kernel. For symmetric spectra, this applies to the commutative operad and all Sigma-cofibrant operads. For chain complexes over a field of characteristic zero and the stable module category, this Quillen equivalence holds for all operads. This paper ends with a comparison between the semi-model category approach and the $\infty$-category approach to encoding the homotopy theory of algebras over Sigma-cofibrant operads that are not necessarily admissible.


INTRODUCTION
A major part of stable homotopy theory is the study of structured ring spectra. These include strict ring spectra, commutative ring spectra, A ∞ -ring spectra, E ∞ring spectra, E n -ring spectra, and so forth. Based on an unpublished talk by Jeff Smith, in [Hov∞] Hovey developed a homotopy theory of Smith ideals for ring spectra and monoids in more general symmetric monoidal model categories.
Let us briefly recall Hovey's work in [Hov∞]. For a symmetric monoidal closed category M, its arrow category → M is the category whose objects are morphisms in M and whose morphisms are commutative squares in M. It has two symmetric monoidal closed structures, namely, the tensor product monoidal structure → M ⊗ and the pushout product monoidal structure → M ◻ . A monoid in → M ◻ is a Smith ideal, and a monoid in → M ⊗ is a monoid morphism. If M is a model category, then → M ⊗ has the injective model structure → M ⊗ inj , where weak equivalences and cofibrations are defined entrywise, and the category of monoid morphisms inherits a model structure from → M ⊗ inj . Likewise, → M ◻ has the projective model structure → M ◻ proj , where weak equivalences and fibrations are defined entrywise, and the category of Smith ideals inherits a model structure from → M ◻ proj . Surprisingly, when M is pointed (resp., stable), the cokernel and the kernel form a Quillen adjunction (resp., Quillen equivalence) between → M ◻ and → M ⊗ and also between Smith ideals and monoid morphisms.
Since monoids are algebras over the associative operad, a natural question is whether there is a satisfactory theory of Smith ideals for algebras over other operads. For the commutative operad, the first author showed in [Whi17] that commutative Smith ideals in symmetric spectra, equipped with either the positive flat (stable) or the positive (stable) model structure, inherit a model structure. The purpose of this paper is to generalize Hovey's work to Smith ideals for general operads in monoidal model categories. For an operad O we define a Smith O-ideal as an algebra over an associated operad → O ◻ in the arrow category → M ◻ . We will prove a precise version of the following result in Theorem 4.4.1. For example, this Theorem holds in the following situations: (1) O is an arbitrary C-colored operad, and M is (i) the category Ch(R) of bounded or unbounded chain complexes over a semi-simple ring containing Q (Corollary 5.2.4), (ii) the stable module category of k[G]-modules for some field k and finite group G (Corollary 6.2.5), or (iii) the category of classical, equivariant, or motivic symmetric spectra with the positive or positive flat stable model structure (Example 4.4.2).
The rest of this paper is organized as follows. In Section 2 we recall some basic facts about model categories and arrow categories. In Section 3 we define Smith ideals for an operad and prove that, when M is pointed, there is an adjunction between Smith O-ideals and O-algebra morphisms given by the cokernel and the kernel. In Section 4 we define the model structures on Smith O-ideals and Oalgebra morphisms and prove the Theorem above. We also include a discussion of what happens when there are only semi-model structures on Smith O-ideals and O-algebra morphisms. In Section 5 we apply the Theorem to the commutative operad and Σ C -cofibrant operads. In Section 6 we apply the Theorem to entrywise cofibrant operads. In Section 7 we include a comparison between various approaches to encoding the homotopy theory of operad-algebras, including model categories, semi-model categories, and ∞-categories. This discussion holds in general, beyond the situation of Smith O-ideals and O-algebra morphisms.

MODEL STRUCTURES ON THE ARROW CATEGORY
In this section we recall a few facts about monoidal model categories and arrow categories. Our main references for model categories are [Hir03,Hov99,SS00]. In this paper, (M, ⊗, 1, Hom) will usually be a bicomplete symmetric monoidal closed category [Mac98] (VII.7) with monoidal unit 1, internal hom Hom, initial object ∅, and terminal object * . Since M is closed, ∅ ⊗ X = ∅ for any X.

Monoidal Model Categories.
A model category is cofibrantly generated if there are a set I of cofibrations and a set J of trivial cofibrations (that is, morphisms that are both cofibrations and weak equivalences) that permit the small object argument (with respect to some cardinal κ), and a morphism is a (trivial) fibration if and only if it satisfies the right lifting property with respect to all morphisms in J (resp. I).
Let I-cell denote the class of transfinite compositions of pushouts of morphisms in I, and let I-cof denote retracts of such [Hov99] (2.1.9). In order to run the small object argument, we will assume the domains K of the morphisms in I (and J) are κ-small relative to I-cell (resp. J-cell). In other words, given a regular cardinal λ ≥ κ and any λ-sequence X 0 / / X 1 / / ⋯ formed of morphisms X β / / X β+1 in I-cell, the map of sets colim β<λ M K, X β / / M K, colim β<λ X β is a bijection. An object is small if there is some κ for which it is κ-small. We will say that a model category is strongly cofibrantly generated if the domains and codomains of I and J are small with respect to the entire category.
In Section 4, we will produce homotopy theories for operad-algebras valued in arrow categories equipped with some model structure. Depending on the colored operad and properties of M, sometimes we will only have a semi-model structure on a category of algebras. However, as shown in Section 7, it still encodes the correct ∞-category. A semi-model category satisfies axioms similar to those of a model category, but one only knows that morphisms with cofibrant domain admit a factorization into a trivial cofibration followed by a fibration, and one only knows that trivial cofibrations with cofibrant domain lift against fibrations. To the authors' knowledge, every result about model categories has a corresponding result for semi-model categories, often obtained by first cofibrantly replacing everything in sight (see, for example, [BW20a]). The following is Definition 2.1 in [BW20a]. i Every morphism in M can be functorially factored into a cofibration followed by a trivial fibration. ii Every morphism whose domain is cofibrant can be functorially factored into a trivial cofibration followed by a fibration.
If, in addition, M is bicomplete, then we call M a semi-model category. M is said to be cofibrantly generated if there are sets of morphisms I and J in M such that the class of (trivial) fibrations is characterized by the right lifting property with respect to J (resp. I), the domains of I are small relative to I-cell, and the domains of J are small relative to morphisms in J-cell whose domain is cofibrant.
An adjunction with left adjoint L and right adjoint R is denoted by L ⊣ R. (1) We call L ⊣ R a Quillen adjunction if the right adjoint R preserves fibrations and trivial fibrations. In this case, we call L a left Quillen functor and R a right Quillen functor.
(2) We call a Quillen adjunction L ⊣ R a Quillen equivalence if, for each morphism f ∶ LX / / Y ∈ N with X cofibrant in M and Y fibrant in N, f is a weak equivalence in N if and only if its adjoint f # ∶ X / / RY is a weak equivalence in M.
Definition 2.1.3. Suppose M is a category with pushouts and pullbacks.
(1) Given a solid-arrow commutative diagram in M in which the square is a pullback, the unique dotted induced morphism is denoted f ⧅ g and called the pullback corner morphism of f and g.
(2) Given a solid-arrow commutative diagram in M in which the square is a pushout, the unique dotted induced morphism is denoted f ⊛ g and called the pushout corner morphism of f and g.
In the next definition, we follow simplicial notation 0 / / 1 so the reader can distinguish source and target at a glance.
Definition 2.1.4. Suppose (M, ⊗, 1) is a monoidal category with pushouts. Suppose f ∶ X 0 / / X 1 and g ∶ Y 0 / / Y 1 are morphisms in M. The pushout corner morphism f ◻g of f ⊗ 1 and 1 ⊗ g is denoted f ◻ g and called the pushout product of f and g.
Definition 2.1.5. A symmetric monoidal closed category M equipped with a model structure is called a monoidal model category if it satisfies the following pushout product axiom [SS00] (3.1): • Given any cofibrations f ∶ X 0 / / X 1 and g ∶ Y 0 / / Y 1 , the pushout product morphism is a cofibration. If, in addition, either f or g is a weak equivalence, then f ◻ g is a trivial cofibration.
Additionally, in order to guarantee that the unit 1 descends to the unit in the homotopy category, it is sometimes convenient to assume the unit axiom [Hov99] (4.2.6): if Q1 / / 1 is a cofibrant replacement, then for any cofibrant object X, the induced morphism Q1 ⊗ X / / 1 ⊗ X ≅ X is a weak equivalence. Since (−) ⊗ X is a left Quillen functor, if the unit axiom holds for one cofibrant replacement of 1, then it holds for any cofibrant replacement of 1.

Arrow Categories.
Definition 2.2.1. A lax monoidal functor F ∶ M / / N between two monoidal categories is a functor equipped with structure morphisms for X and Y in M that are associative and unital in a suitable sense, as discussed in [Mac98] (XI.2), where this notion is referred to simply as a monoidal functor. If, furthermore, M and N are symmetric monoidal categories, and F 2 is compatible with the symmetry isomorphisms, then F is called a lax symmetric monoidal functor. If the structure morphisms F 2 and F 0 are isomorphisms (resp., identity morphisms), then F is called a strong monoidal functor (resp., strict monoidal functor).
property with respect to i, and β has the right lifting property with respect to α i if and only if Ev 0 f / / Ev 1 f × Ev 1 g Ev 0 g has the right lifting property with respect to i. Thus these sets generate the injective model structure. The pushout product axiom and the unit axiom on → M ⊗ inj follows from the same on M [Bar10] (4.51).

Projective Model Structure.
The following result about the projective model structure is from [Hov∞] (3.1).
Theorem 2.4.1. Suppose M is a model category.
(1) There is a model structure on → M, called the projective model structure, in which a morphism α ∶ f / / g as in (2.2.3) is a weak equivalence (resp., fibration) if and only if α 0 and α 1 are weak equivalences (resp., fibrations) in M. A morphism α is a (trivial) cofibration if and only if α 0 and the pushout corner morphism Note that this implies that α 1 is also a (trivial) cofibration. The arrow category equipped with the projective model structure is denoted by → M proj .
(2) If M is cofibrantly generated, then so is Proof.
For a category M with all small limits and colimits, recall from [Hov99] (Sections 1.1, 6.1) that M is pointed if the unique morphism ∅ / / * is an isomorphism. In such a category, we define the cokernel of a morphism f ∶ X 0 / / X 1 to be the morphism coker f ∶ X 1 / / Z defined by the following pushout: / / X 1 For the left adjoints L 0 and L 1 in (2.2.4), we note the following equalities for each object X.
Most of the observations in Proposition 2.4.3 are from [Hov∞] (1.4, 4.1, 4.3). We provide proofs here for completeness. (1) The cokernel is a strictly unital strong symmetric monoidal functor from → M ◻ to → M ⊗ whose right adjoint is the kernel.
(2) The strong symmetric monoidality of the cokernel induces a strictly unital lax symmetric monoidal structure on the kernel such that the adjunction (coker, ker) is monoidal.
(3) If M is also a model category, then (coker, ker) is a Quillen adjunction.
Proof. For (1), first note that coker preserves the units since the cokernel of ∅ / / 1 is Id 1 . Next, it is strong monoidal because, given f ∶ X 0 / / X 1 and g ∶ Y 0 / / Y 1 we can form the following commutative diagram: Vertical pushouts yield a span whose pushout is coker( f ◻ g). Horizontal pushouts yield a span whose pushout is coker f ⊗ coker g. Since pushouts commute, we obtain the natural isomorphism .
We take this isomorphism as the ( f , g)-component of the monoidal constraint for coker. Using similar reasoning and the universal property of pushouts, one can show that the symmetric monoidal coherence diagrams commute.
morphisms f and g.
This diagram commutes because the adjoint of each composite is the identity morphism of coker( f ◻ g). For the long composite, this uses (i) the naturality of (coker 2 ) −1 and (ii) one of the triangle identities for the adjunction (coker, ker) [Mac98] (IV.1 Theorem 1).
To prove that the counit ε ∶ coker ○ ker / / Id is a monoidal natural transformation, it remains to show that the following diagram commutes.
This diagram commutes because, starting from the lower-left corner to f ⊗ g, each composite is adjoint to ker 2 f ,g . For (3), let α be a (trivial) cofibration and note that coker α is the colimit of a morphism of pushout diagrams. That morphism of pushout diagrams is a Reedy (trivial) cofibration. The colimit functor is left Quillen as a functor from the Reedy model structure to the underlying category [Hov99] (Section 5.2). Hence, coker α is again a (trivial) cofibration, so coker is a left Quillen functor. See Lemma 6.1.8 for an analogous proof.
For (4), we must prove that, if f is cofibrant in → M ◻ (so, a cofibration of cofibrant objects) and g is fibrant in → M ⊗ (so, a fibration of fibrant objects), then α ∶ coker f / / g is a weak equivalence if and only if its adjoint β ∶ f / / ker g is a weak equivalence [Hov99] (1.3.12). We display both morphisms: In the homotopy category, these data give rise to fiber and cofiber sequences. Since M is stable, every fiber sequence is canonically isomorphic to a cofiber sequence [Hov99] (Chapter 7). We can extend to the right and realize α and β as giving a morphism of cofiber sequences in the homotopy category: If either α or β is a weak equivalence, then so is the other, by the two out of three property. Hence, coker and ker form a Quillen equivalence.

SMITH IDEALS FOR OPERADS
Suppose (M, ⊗, 1) is a cocomplete symmetric monoidal category in which the monoidal product commutes with colimits on both sides, which is automatically true if M is a closed symmetric monoidal category. In this section we define Smith ideals for an arbitrary colored operad O in M. When M is pointed, we observe in Theorem 3.4.2 that the cokernel and the kernel induce an adjunction between the categories of Smith O-ideals and of O-algebra morphisms. This will set the stage for the study of the homotopy theory of Smith O-ideals in the next several sections.
Definition 3.1.1. Suppose C is a set, whose elements will be called colors.
(1) A C-profile is a finite, possibly empty sequence c = (c 1 , . . . , c n ) with each c i ∈ C.
(2) When permutations act on C-profiles from the left (resp., right), the resulting groupoid is denoted by Σ C (resp., Σ op C ). (3) The category of C-colored symmetric sequences in M is the diagram category M Σ op C ×C . For a C-colored symmetric sequence X, we think of Σ op C (resp., C) as parametrizing the inputs (resp., outputs). For (c; d) ∈ Σ op C × C, the corresponding entry of a C-colored symmetric sequence X is denoted by X d c .
(4) A C-colored operad (O, γ, 1) in M consists of: • objects X c ∈ M for c ∈ C and • structure morphisms in M for all 1 ≤ i ≤ n with n ≥ 1, d ∈ C, and c = (c 1 , . . . , c n ) ∈ Σ C . These data are required to satisfy associativity, unity, and equivariant conditions similar to those of an O-algebra but with one input entry A and the output entry replaced by X. A morphism of A-bimodules is required to preserve the structure morphisms. As a consequence of (2.4.2) and (3.1.2), we have the following equalities. Example 3.1.5. Every strongly cofibrantly generated model category is operadically cofibrantly generated. The category of compactly generated topological spaces is not strongly cofibrantly generated. However, it is operadically cofibrantly generated. Indeed, the domains and codomains of I ∪ J are small relative to inclusions [Hov99] M ⊗ for all d ∈ C and c = (c 1 , . . . , c n ) ∈ Σ C . This structure morphism is equivalent to the commutative square The associativity, unity, and equivariance of λ translate into those of λ 0 and λ 1 , making (X, λ 0 ) and (Y, λ 1 ) into O-algebras in M. The commutativity of the previous square means that f ∶ (X, Remark 3.2.3. For the associative operad As, whose algebras are monoids, the identification of → As ⊗ -algebras (that is, monoids in → M ⊗ ) with monoid morphisms in M is [Hov∞] (1.5).
Propositions 3.3.3 and 3.3.11 below unpack Definition 3.3.1. They should be compared with Proposition 3.2.2. For objects or morphisms A cs , . . . , A c t with s ≤ t, we use the abbreviation that are associative, unital, and equivariant. Since For n ≥ 1, the structure morphism λ is equivalent to the commutative diagram The domain of the iterated pushout product f c 1 ◻ ⋯ ◻ f cn is the colimit . The morphisms that define the colimit are given by the f c i 's. For each n-tuple of indices ǫ = (ǫ 1 , . . . , ǫ n ) ∈ {0, 1} n ∖ {(1, . . . , 1)}, we denote by the morphism that comes with the colimit. For each i ∈ {1, . . . , n}, we denote by the n-tuple with 0 in the ith entry and 1 in every other entry.
The upper left quadrilateral is commutative because D is the colimit in (3.3.6). The other two triangles are commutative by the definition of λ ǫ i 0 and λ ǫ j 0 in (3.3.8). The argument above can be reversed. In particular, to see that the commutative diagram (3.3.4), which is the boundary of (3.3.9), yields the top horizontal morphism λ 0 in (3.3.5), observe that the full subcategory of the punctured n-cube {0, 1} n ∖ {(1, . . . , 1)} consisting of (ǫ 1 , . . . , ǫ n ) with at most two 0's is a final subcat- where X ′ becomes an A-bimodule via the restriction along h 1 , such that the square Proof. Following the proof of Proposition 3.3.3, we unravel the given morphism in M such that the square (3.3.12) commutes.
The compatibility of h with the → O ◻ -algebra structure means the following diagram commutes in → M for all d, c 1 , . . . , c n ∈ C. (3.3.14) If n = 0, then (3.3.14) is the commutative diagram below.
For n ≥ 1, using the abbreviation 3.14) becomes the following commutative cube.
The six commutative faces of (3.3.16) are as follows.
(1) The back face is (3) The right face is the square (3.3.12) for d ∈ C.
(4) The bottom face and the n = 0 case (3.3.15) together express the fact that The left face imposes no extra condition because D is the colimit in (3.3.6) and similarly for D ′ . In more detail, for each n-tuple (ǫ 1 , . . . , is commutative because it is a tensor product of n commutative squares corresponding to the n tensor factors of the upper left corner.
• For a tensor factor with ǫ i = 0, by definition In this case, we have the commutative square (3.3.12) for c i ∈ C.
• For a tensor factor with ǫ i = 1, by definition Both f * and f ′ * are given by the identity in the respective tensor factors, while both h * and h 1 * are given by h 1 c i .
Pre-composing the top face of the commutative cube (3.3.16) with the morphism Id ⊗ ι ǫ i in (3.3.8) yields the following commutative diagram.
This commutative diagram expresses the fact that h 0 ∶ X / / X ′ is a morphism of A-bimodules, where X ′ becomes an A-bimodule via the restriction along h 1 .
Thus, pre-composing the top face of (3.3.16) with the morphism Id ⊗ ι ǫ yields a diagram that factors into two sub-diagrams, one of which is (3.3.18). The other subdiagram commutes and imposes no extra condition by the same argument above for (3.3.17).
The description of Smith O-ideals and their morphisms in Propositions 3.3.3 and 3.3.11 imply the following result.
Proof. Denote the first and the second copies of C in C ⊔ C by, respectively, C 0 and C 1 . For an element c ∈ C, we write c ǫ ∈ C ǫ for the same element for ǫ ∈ {0, 1}. The entries of O s are defined as follows for d, c 1 , . . . , c n ∈ C and ǫ 1 , . . . , ǫ n ∈ {0, 1}.
The operad structure morphisms of O s are either those of O or the unique morphism from the initial object ∅.
The identification of O s -algebra morphisms and Smith O-ideal morphisms follows similarly from Proposition 3.3.11. More explicitly, a morphism h of O s -algebras consists of a To see that these component morphisms make the diagram (3.3.12) commute, we use the fact that the components of f are the composites in (3.3.22) and similarly for f ′ . The desired diagram (3.3.12) is the boundary of the following diagram. This shows that the diagram (3.3.12) is commutative.
The other two conditions in Proposition 3.3.11 are the following: This finishes the proof.
The colored operad O s is somewhat similar to the two-colored operad for monoid morphisms in [Yau16] (Section 14.3).

Operadic Smith Ideals and Morphisms of Operadic Algebras.
In Proposition 2.4.3 we observe that, if M is a pointed symmetric monoidal category with all small limits and colimits, then there is an adjunction with cokernel as the left adjoint and kernel as the right adjoint. Since cokernel is a strictly unital strong symmetric monoidal functor, the kernel is a strictly unital lax symmetric monoidal functor, and the adjunction is monoidal. If M is a pointed model category, then (coker, ker) is a Quillen adjunction. If M is a stable model category, then (coker, ker) is a Quillen equivalence.
in which the left adjoint, the right adjoint, the unit, and the counit are defined entrywise.
Proof. To simplify the notation, in this proof we write C = coker and K = ker. First we lift the functors C and K. Then we lift the unit and the counit for the adjunction.
Step 1: Lifting the Kernel and the Cokernel to Algebra Categories The functors in (3.4.1) lifts entrywise to the functors in (3.4.3) for the following reasons.
• The functor becomes an → O ⊗ -algebra with structure morphism λ # given by the following composite for all d, c 1 , . . . , c n ∈ C, with C 2 = coker 2 the monoidal constraint of the cokernel in (2.4.4).
The → O ⊗ -algebra axioms for (C f , λ # ) follow from the → O ◻ -algebra axiom for ( f , λ) and the symmetric monoidal axioms for the cokernel. The same reasoning also applies to the kernel.
Thus there is a diagram of functors with both U forgetful functors and To see that this equality holds, suppose ( f , λ) is an → O ⊗ -algebra as in the proof of the → O ◻ -algebra structure morphism λ ′ is constructed from the monoidal constraint K 2 and Kλ. Since each U forgets the operad algebra structure morphism, we obtain the equalities The equality UK = KU holds on → O ⊗ -algebra morphisms because (i) both K apply entrywise to morphisms and (ii) both U do not change the morphisms.
Next we show that the unit and the counit,

.5) lift to the top between algebra categories.
Step 2: Lifting the Unit To show that η defines a natural transformation for the top functors in (3.4.5), first we need to show that, for each 3), and K 2 = ker 2 the monoidal constraint defined in (2.4.5). (3.4.6) Kλ # To see that (3.4.6) is commutative, we consider the adjoint of each composite, which yields the boundary of the following diagram in → M. (3.4.7) The three sub-regions in (3.4.7) are commutative for the following reasons.
• The left triangle is commutative by the naturality of the monoidal constraint C 2 = coker 2 of the cokernel. • The upper right region is commutative by the definition of λ # in (3.4.4).
• To see that the lower right triangle is commutative, first note that the counit component morphism For each of the other n tensor factors in the lower right triangle, the composite ε C fc i ○ Cη i is the identity morphism by one of the triangle identities for the adjunction C ⊣ K [Mac98] (IV.1 Theorem 1).
This proves that / / KC is a natural transformation for the top horizontal functors in (3.4.5) between algebra categories.
Step 3: Lifting the Counit Next we show that the counit ε ∶ CK / / Id of the bottom adjunction C ⊣ K in (3.4.5) lifts to the top between algebra categories. First we need to show that, for each the → O ◻ -algebra obtained by applying the top functor K in (3.4.5). The → O ◻ -algebra structure morphism λ is the analogue of (3.4.4) for the kernel. In other words, it is the composite (3.4.10) The four sub-regions in (3.4.10) are commutative for the following reasons.
• The top triangle is commutative by the definition of (−) # in (3.4.4).
• The triangle to its lower right is commutative by the definition of λ in (3.4.9) and the functoriality of C. • The lower right quadrilateral is commutative by the naturality of the counit ε ∶ CK / / Id.
Using the inverse of C 2 = coker 2 , the left triangle in (3.4.10) is equivalent to the following diagram. (3.4.11) The diagram (3.4.11) is commutative because the adjoint of each composite is K 2 = ker 2 defined in (2.4.5). This shows that (3.4.10) is commutative, and ε g is an   (1) Pointed or unpointed simplicial sets [Qui67] and all of their left Bousfield localizations [Hir03]. (2) Bounded or unbounded chain complexes over a commutative ring containing the rationals Q [Qui67]. (3) Symmetric spectra built on either simplicial sets or compactly generated topological spaces, motivic symmetric spectra, and G-equivariant symmetric spectra with either the positive stable model structure or the positive flat stable model structure [PS18]. (4) The category of small categories with the folk model structure [Rez∞].
(5) Simplicial modules over a field of characteristic zero [Qui67]. (6) The stable module category of k[G]-modules [Hov99] (2.2), where k is a field and G is a finite group. We recall that the homotopy category of this example is trivial unless the characteristic of k divides the order of G (the setting for modular representation theory).
The condition (♠) for (1)-(2) is proved in [WY18] (Section 8, which also handles symmetric spectra built on simplicial sets), and (4)-(5) can be proved using similar arguments. The condition (♠) for the stable module category is proved by the argument in [WY20] (12.2). For symmetric spectra built on topological spaces, motivic symmetric spectra, and equivariant symmetric spectra, we refer to [PS18] (Section 2, and the references therein) starting with C = Top, sSet G , Top G , and the A 1 -localization of simplicial presheaves with the injective model structure.
In each of these examples except those built from Top, the domains and the codomains of the generating (trivial) cofibrations are small with respect to the entire category. So Proposition 2.4.6 applies to show that, in each case, the arrow category with either the injective or the projective model structure is strongly cofibrantly generated. The category of (equivariant) symmetric spectra built on topological spaces is operadically cofibrantly generated by an argument analogous to that of Example 3.1.5, as are the arrow categories, by the remark below.  Proof. Suppose M satisfies (♠) with respect to a subclass C of weak equivalences that is closed under transfinite composition and pushout. We write C ′ for the subclass of weak equivalences β in → M ⊗ inj such that β 0 , β 1 ∈ C. Then C ′ is closed under transfinite composition and pushout. Suppose We will show that f X ⊗ Σn α ◻n belongs to C ′ . The morphism f X ⊗ Σn α ◻n in → M ⊗ is the commutative square Since α 0 and α 1 are trivial cofibrations in M and since X 0 , X 1 ∈ M Σ op n , the condition (♠) in M implies that the two horizontal morphisms in the previous diagram are both in C. This shows that → M ⊗ inj satisfies (♠) with respect to the subclass C ′ of weak equivalences.
The second assertion is now a consequence of Proposition 2.4.6, Example 3.1.5, and Theorem 4.1.1.  When (♠) is not satisfied but the classes of morphisms above still define semimodel structures (e.g., Remark 5.1.6, Corollary 5.2.3, and Theorem 6.2.1), we still denote those semi-model structures by Alg Since there is an equality (3.4.5) U ker α = ker Uα

right Quillen functor by Proposition 2.4.3 (3), we finish the proof by observing that
Recall that a pointed (semi-)model category is stable if its homotopy category is a triangulated category [Hov99] (7.1.1). → M ◻ is a weak equivalence. So ker α is entrywise a weak equivalence in M, or equivalently U ker α ∈ ( → M ◻ proj ) C is a weak equivalence. We must show that α is a weak equivalence, that is, that Uα ∈ ( → M ⊗ inj ) C is a weak equivalence. The morphism Uα is still a morphism between fibrant objects, and In other words, we must show that Uη is a weak equivalence in Here the left vertical morphism is a trivial cofibration and is a fibrant replacement of U coker f X . The top horizontal morphism is a weak equivalence and is U applied to a fibrant replacement of coker f X . The other two morphisms are fibrations. So there is a dotted morphism α that makes the whole diagram commutative. By the 2-out-of-3 property, α is a weak equivalence between fibrant objects in ( C is a right Quillen functor, by Ken Brown's Lemma [Hov99] (1.1.12) ker α is a weak equivalence in ( → M ◻ proj ) C . We now have a commutative diagram where ε is the derived unit of U f X . To show that Uη is a weak equivalence, it suffices to show that ε is a weak equivalence. By assumption U f X is a cofibrant object in ( → M ◻ proj ) C . Since (coker, ker) is a Quillen equivalence between ( M ◻ proj is more subtle. We will consider this issue in the next two sections, proving this condition for (1) in Corollary 5.2.4 and for (2) in Corollary 6.2.5.
For classical, equivariant, or motivic symmetric spectra, we must tweak the proof of Theorem 4.4.1. Let ( → M ◻ proj ) C refer to the projective model structure on the arrow category where M is the injective stable model structure on the relevant category of symmetric spectra. Since the weak equivalences of the injective stable model structure coincide with those of the positive (flat) stable model structure, in the last paragraph of the proof, it is enough to prove that ǫ is a weak equivalence with respect to the injective stable model structure on spectra. Hence, it suffices for U f X to be a cofibrant object in ( → M ◻ proj ) C , which follows from the proof of [WY18] (8.3.3), using our filtrations and the fact that the cofibrations of the injective stable model structure are the monomorphisms.
We note that we cannot add the injective stable model structure on symmetric spectra to the list in Example 4.4.2 because it is not true that every operad is admissible. A famous obstruction due to Gaunce Lewis prevents the Com operad from being admissible, for example.

SMITH IDEALS FOR COMMUTATIVE AND SIGMA-COFIBRANT OPERADS
In this section we apply Theorem 4.4.1 and consider Smith ideals for the commutative operad and Σ C -cofibrant operads (Definition 5.2.1). In particular, in Corollary 5.2.3 we will show that Theorem 4.4.1 is applicable to all Σ C -cofibrant operads. On the other hand, the commutative operad is usually not Σ-cofibrant. However, as we will see in Example 5.1.3, Theorem 4.4.1 is applicable to the commutative operad in symmetric spectra with the positive flat stable model structure.

Commutative Smith Ideals.
For the commutative operad, which is entrywise the monoidal unit and whose algebras are commutative monoids, we use the following definition from [Whi17] (3.4). The notation ? Σ n means taking the Σ n -coinvariants.
Definition 5.1.1. A monoidal model category M is said to satisfy the strong commutative monoid axiom if, whenever f ∶ K / / L is a (trivial) cofibration, then so is f ◻n Σ n , where f ◻n is the n-fold pushout product (which can be viewed as the unique morphism from the colimit Q n of a punctured n-dimensional cube to L ⊗n ), and the Σ n -action is given by permuting the vertices of the cube.
The following result says that, under suitable conditions, commutative Smith ideals and commutative monoid morphisms have equivalent homotopy theories.
Corollary 5.1.2. Suppose M is a cofibrantly generated stable monoidal model category that satisfies the strong commutative monoid axiom, the monoid axiom, and in which cofibrant → Com ◻ -algebras are also underlying cofibrant in → M ◻ proj (this occurs, for example, if the monoidal unit is cofibrant). Then there is a Quillen equivalence in which Com is the commutative operad in M.
Proof. First, [Whi17] (5.12 and 5.14) ensures that → M ⊗ and → M ◻ satisfy the strong commutative monoid axiom, and [Hov∞] (2.2 and 3.2) (also Theorems 2.3.1 and 2.4.1) ensures that they satisfy the monoid axiom. Hence, by [Whi17] For the commutative operad, it is proved in [Whi17] (3.6 and 5.14) that, with the strong commutative monoid axiom and a cofibrant monoidal unit, cofibrant → Com ◻ -algebras are also underlying cofibrant in → M ◻ proj . So Theorem 4.4.1 applies. Example 5.1.3 (Commutative Smith Ideals in Symmetric Spectra). Example 4.4.2 shows that the category of symmetric spectra with the positive flat stable model structure satisfies the hypotheses in Theorem 4.4.1. It also satisfies the strong commutative monoid axiom [Whi17] (5.7) and the monoid axiom [SS00]. While the monoidal unit is not cofibrant, nevertheless, [Whi17] (5.15) shows that cofibrant commutative Smith ideals forget to cofibrant objects of → M ◻ . Therefore, Corollary 5.1.2 applies to the commutative operad Com in symmetric spectra with the positive flat stable model structure.
Example 5.1.4 (Commutative Smith Ideals in Algebraic Settings). Let R be a commutative ring containing the ring of rational numbers Q. Corollary 5.2.4 shows that the category of (bounded or unbounded) chain complexes of R-modules satisfies the conditions of Theorem 4.4.1. They also satisfy the strong commutative monoid axiom and the monoid axiom [Whi17] (5.1). Hence, Corollary 5.1.2 applies, to give a homotopy theory of ideals of CDGAs. The same is true of the stable module category of R = k[G] where k is a field and G is a finite group, using Corollary 6.2.5. The result is a homotopy theory of ideals of commutative R-algebras. Of course, taking G trivial in Example 5.1.5, one obtains that Corollary 5.1.2 applies to orthogonal spectra with the positive flat stable model structure [Whi22] (Section 8).  6)).

Smith Ideals for Sigma-Cofibrant Operads.
For a cofibrantly generated model category M and a small category D, recall that the diagram category M D inherits a projective model structure with weak equivalences and fibrations defined entrywise in M [Hir03] (11.6.1). We use this below when D = Σ op C × C is the groupoid in Definition 3.1.1. In this case, the category M D is the category of C-colored symmetric sequences.
Proof. The Quillen adjunction

lifts to a Quillen adjunction of D-diagram categories
by [Hir03] (11.6.5(1)), and similarly for (L 0 , Ev 0 ). If X ∈ M D is cofibrant, then L 1 X and L 0 X are cofibrant since L 1 and L 0 are left Quillen functors. The following provides one source of applications of Corollary 5.2.3, and answers a question Pavel Safranov asked the first author. This result generalizes [Whi17] (5.1) and [WY18] (8.1), as it applies in particular to fields of characteristic zero.
Corollary 5.2.4. Suppose R is a commutative ring with unit and M is the category of bounded or unbounded chain complexes of R-modules, with the projective model structure. The following are equivalent: (1) R is a semi-simple ring containing the rational numbers Q.
In particular, for such rings R, every C-colored operad in M is Σ C -cofibrant, so Corollary 5.2.3 is applicable for all colored operads in M. If R contains Q (but is not necessarily semi-simple) then every entrywise cofibrant C-colored operad in M is Σ C -cofibrant and admissible.
Proof. Assume (1). Maschke's Theorem [MS02] (3.4.7) guarantees that each group ring R[Σ n ] is semi-simple (since 1 n! exists in R, making n! invertible). This means every module M over R[Σ n ] is projective. In particular, M is a direct summand of a module induced from the trivial subgroup, and has a free Σ n -action. Hence, (2) follows.
Conversely, if (2) is true, then it implies that, for every n, every module in R[Σ n ] is projective. This means each R[Σ n ] is a semi-simple ring. By [MS02] (3.4.7), this implies that R is semi-simple and n! is invertible in R for every n. It follows that Q is contained in R.
For such R, the projective model structure on (bounded or unbounded) chain complexes of R-modules has every object cofibrant (so, automatically, cofibrant operad-algebras forget to cofibrant chain complexes). Hence, any C-colored operad is entrywise cofibrant, and hence Σ C -cofibrant. Furthermore, Theorem 4.1.1 implies that all operads are admissible, since every X ∈ M Σ op n is Σ n -projectively cofibrant.
If R contains Q but is not semi-simple, then there can be non-projective R-modules, but the argument of [MS02] (3.4.7) shows that an R[Σ n ]-module that is projective as an R-module is projective as a R[Σ n ]-module. It follows that Corollary 5.2.3 holds for entrywise cofibrant operads, including the operad Com. Indeed, all operads are admissible thanks to Theorem 4.1.1, since for any trivial cofibration f and any X ∈ M Σ op n , maps of the form X ⊗ Σn f ◻n are trivial h-cofibrations and this class of morphisms is closed under pushout and transfinite composition [Whi22] (Section 8). Smith Ideals: The associative operad As, which has As(n) = ∐ Σn 1 as the nth entry and which has monoids as algebras, is Σ-cofibrant. In this case, Corollary 5.2.3 is Hovey's Corollary 4.4 (1) in [Hov∞]. Smith A ∞ -Ideals: Any A ∞ -operad, defined as a Σ-cofibrant resolution of As, is Σ-cofibrant. In this case, Corollary 5.2.3 says that Smith A ∞ -ideals and A ∞ -algebra morphisms have equivalent homotopy theories. For instance, one can take the standard differential graded A ∞ -operad [Mar96] and, for symmetric spectra, the Stasheff associahedra operad [Sta63]. Smith E ∞ -Ideals: Any E ∞ -operad, defined as a Σ-cofibrant resolution of the commutative operad Com, is Σ-cofibrant. In this case, Corollary 5.2.3 says that Smith E ∞ -ideals and E ∞ -algebra morphisms have equivalent homotopy theories. For example, for symmetric spectra, one can take the Barratt-Eccles E ∞ -operad EΣ * [BE74]. An elementary discussion of the Barratt-Eccles operad is in [JY∞] (Section 11.4).
Smith E n -Ideals: For each n ≥ 1, the little n-cubes operad C n [BV73, May72] is Σ-cofibrant and is an E n -operad by definition [Fre17] (4.1.13). In this case, with M being symmetric spectra with the positive (flat) stable model structure, Corollary 5.2.3 says that Smith C n -ideals and C n -algebra morphisms have equivalent homotopy theories. One may also use other Σ-cofibrant E noperads [Fie∞], such as the Fulton-MacPherson operad ( [GJ∞] and [Fre17] (4.3)), which is actually a cofibrant E n -operad. An elementary discussion of a categorical E n -operad is in [JY∞] (Chapter 13). (1) S-modules with the model structure from [EKMM97].
(3) Mandell's model structure on G-equivariant symmetric spectra built on simplicial sets or topological spaces, where G is a finite group in the former case and a compact Lie group in the latter case [Man04]. (4) Model structures for (equivariant) stable homotopy theory based on Lydakis's theory of enriched functors [DR∅03]. For example, this includes the model category of G-enriched functors from finite G-simplicial sets to G-simplicial sets, where G is a finite group, from [DR∅03] (Theorem 2). (5) Any model structure M on symmetric spectra built on (C, G) where C is a model category and G is an endofunctor, as long as M is an operadically cofibrantly generated, monoidal, stable model structure. For example, taking C to be the canonical model structure on small categories, and using the suspension discussed in [WY20] (Section 13), one obtains by [Hov01] (7.3) a combinatorial, stable, monoidal model structure on symmetric spectra of small categories with applications to Goodwillie calculus. Using [PS18] (Section 2) one may obtain positive and positive flat variants. Another example is taking C to be the I-spaces or J-spaces of Sagave and Schlichtkrull, and building projective, positive, or positive flat spectra on them as in [PS18] (Section 2). (6) The projective model structure on bounded or unbounded chain complexes over a commutative ring R [WY20] (Section 11). (7) The stable module category of k[G] where G is a finite group and k is a principal ideal domain [WY20] (Section 12).
All of these examples are stable monoidal model categories, so Corollary 5.2.3 applies, once the requisite smallness hypothesis for the generating (trivial) cofibrations is checked. Symmetric spectra, motivic symmetric spectra, examples (6) and (7), and Mandell's model (3) of G-equivariant symmetric spectra built on simplicial sets are all combinatorial, as is the model structure on enriched functors (4) in simplicial contexts. Symmetric spectra as in (5) are combinatorial if C is combinatorial. S-modules, G-equivariant orthogonal spectra, Mandell's model (3) in topological contexts, and symmetric spectra built on topological spaces (another example of (5)) are operadically cofibrantly generated just as in Example 3.1.5, since they are built from compactly generated spaces. We recall that spaces are small relative to inclusions, and the morphisms in (O ○ (I ∪ J))-cell are inclusions [WY20] (5.10).

SMITH IDEALS FOR ENTRYWISE COFIBRANT OPERADS
In this section we apply Theorem 4.4.1 to operads that are not necessarily Σ Ccofibrant. To do that, we need to redistribute some of the cofibrancy assumptionsthat cofibrant Smith O-ideals are underlying cofibrant in the arrow category-from the colored operad to the underlying category. We will show in Theorem 6.2.1 that Theorem 4.4.1 is applicable to all entrywise cofibrant operads, provided that M satisfies the cofibrancy condition (♡) below. This implies that, over the stable module category [Hov99] (2.2), Theorem 4.4.1 is always applicable. Proof. For simplicial sets with either model structure, a cofibration is precisely an injection, and the pushout product of two injections is again an injection. Dividing an injection by a Σ n -action is still an injection. The other cases are proved similarly. Proof. The condition (♡) only refers to cofibrations, which remain the same in any left Bousfield localization.
The next observation is the key that connects the cofibrancy condition (♡) in M to the arrow category. Proof. Suppose f X ∶ X 0 / / X 1 is an object in ( This means that f X is a morphism in M Σ op n that is an underlying cofibration between cofibrant objects in M. The condition (♣) cof for the pushout of the bottom row in the commutative diagram (6.1.9) (X 1 ⊗ Z) Σn Here the left square is commutative by definition, and the right square is X 0 ⊗ Σn (−) applied to α ◻ 2 n in (6.1.6).
We consider the Reedy category D with three objects {−1, 0, 1}, a morphism 0 / / − 1 that lowers the degree, a morphism 0 / / 1 that raises the degree, and no other non-identity morphisms. Using the Quillen adjunction (1) The left and the middle vertical arrows are cofibrations in M.
(2) The pushout corner morphism of the right square is a cofibration in M.
The objects X 0 and X 1 in M Σ op n are cofibrant in M. The morphism ζ 1 = Ev 0 (α ◻ 2 n ) ∈ M Σn is an underlying cofibration in M. Indeed, since α ∈ → M ◻ proj is a cofibration, so is the iterated pushout product α ◻ 2 n by the pushout product axiom [WY19b]. In particular, Ev 0 (α ◻ 2 n ) is a cofibration in M. The condition (♡) in M (for the morphism ∅ / / X i ) now implies that the left and the middle vertical morphisms X i ⊗ Σn ζ 1 in (6.1.9) are cofibrations in M.
Finally, since X 0 ∈ M Σ op n is cofibrant in M and since the pushout corner morphism of α ◻ 2 n ∈ ( → M ◻ proj ) Σn is a cofibration in M, the condition (♡) in M again implies the pushout corner morphism of the right square X 0 ⊗ Σn α ◻ 2 n in (6.1.9) is a cofibration in M.
Lemma 6.1.10. The pushout corner morphism of f X ◻ Σn α ◻ 2 n in (6.1.7) is a cofibration in M.
Proof. The pushout corner morphism of f X ◻ Σn α ◻ 2 n is the morphism f X ◻ Σn (α ◻n 1 ⊛ f ◻n W ). This is the Σ n -coinvariants of the pushout product in the diagram is a cofibration in M.

Underlying Cofibrancy of Cofibrant Smith Ideals for Entrywise Cofibrant
where ∅ M is the initial object in M and the symbol ∅ in d ∅ is the empty C-profile. Since O is assumed entrywise cofibrant, it follows that each entry of the By Proposition 4.2.5, the semi-model structure on Alg where I and J are the generating (trivial) co- The three arrows in this diagram are as follows: • d 0 is induced by the composition of O.
• d 1 is induced by the O-algebra structure on A.
• The common section s is induced by the unit A / / O ○ A.
Lemma 6.2.3. Under the hypotheses of Theorem 6.2.1, suppose α ∶ f / / g is a morphism in (L 0 I ∪ L 1 I) c for some color c ∈ C, and Proof. By the filtration in [WY18] (4.3.16) and the fact that cofibrations are closed under pushouts, to show that Uj ∈ ( → M ◻ proj ) C is a cofibration, it is enough to show that, for each n ≥ 1 and each color d ∈ C, the morphism Proof. The stable module category is a stable model category that satisfies the hypotheses of Theorem 6.2.1 in which every object is cofibrant [Hov99] (2.2.12), [WY20] (Section 12).
There are several more examples where Theorem 4.4.1 likely applies to all entrywise cofibrant operads, but where (♡) has not been checked. For example, the positive flat stable model structure on symmetric spectra built on compactly generated spaces have the property that, for any entrywise cofibrant colored operad O, cofibrant O-algebras forget to cofibrant spectra [PS18] (Section 2), but the authors do not know a reference proving the same for → M ◻ proj . Conjecture 6.2.6. The positive flat stable model structure on symmetric spectra built on compactly generated spaces satisfies the conclusion of Theorem 6.2.1.
Similarly, by analogy with the positive flat model structure on symmetric spectra, one would expect that the positive flat model structure on G-equivariant orthogonal spectra would satisfy this property. (1) Work out a positive complete flat stable model structure on GSp O .
(2) Prove that it satisfies the condition that all colored operads are admissible.
(4) Prove that this model structure satisfies the conclusion of Theorem 6.2.1.
In a related vein, we have the following problem. (1) Prove that the positive injective stable model structure M + i is a monoidal model category.
(2) Prove that all operads are admissible in M + i . If so, then automatically cofibrant O-algebras forget to cofibrant underlying objects.
(3) Prove that M + i satisfies the conclusion of Theorem 6.2.1. (4) Do the same for symmetric spectra valued in a general base model category C, where stabilization is with respect to an endofunctor G. (5) Do the same for orthogonal spectra and equivariant orthogonal spectra, possibly restricting to ∆-generated spaces as is done in [Whi22] (Section 8). (6) Produce a model structure on the category of S-modules, Quillen equivalent to the one in [EKMM97], with the property that cofibrant commutative ring spectra are underlying cofibrant. Do the same for general entrywise cofibrant colored operads, and prove that the conclusion of Theorem 6.2.1 holds in this setting.

SEMI-MODEL CATEGORIES AND ∞-CATEGORIES FOR OPERAD ALGEBRAS
In this paper, we often transferred model structures, using (♠), or semi-model structures, using Def. 6.1.1 or using Σ C -cofibrant operads O, to categories of Oalgebras. The language of ∞-categories could also be used to study the homotopy theory of O-algebras. We work in the model of quasi-categories, i.e., everywhere we write ∞-category we mean quasi-category. The main results of this section, Theorems 7.3.1 and 7.3.3, show that the two approaches-namely, semi-model categories and ∞-categories-are equivalent in a suitable sense for Σ C -cofibrant Ccolored operads that are not necessarily admissible.
Lurie [Lur∞] (4.5.4.12) proves this property for the Com-operad and a restrictive class of model categories M: namely, combinatorial and freely powered (4.5.4.2) monoidal model categories. Lurie then deduces (4.5.4.7) that the underlying ∞- We extend this result in two ways. First, we will show that it holds when O is only semi-admissible instead of admissible (i.e., Alg(O; M) has a transferred semi-model structure). Second, we will show the same thing for the setting of enriched ∞-operads. For the latter, we work in a monoidal model category M (notnecessarily simplicial) and consider a colored operad O valued in M. Note that if M is a V-model category for some monoidal model category V, and O is a colored operad valued in V, then there is a colored operad O ′ valued in M with the same algebras (obtained by tensoring the levels of O with the unit of M), so we focus on the case when O is valued in M. In this case, there is an associated enriched ∞-operad [CH20] as we now describe. First, we must restate [Hau∞] (4.1).
Definition 7.1.1. Let M be a monoidal model category. A subcategory of flat objects is a full symmetric monoidal subcategory M ♭ (which implies the unit is flat) that satisfies the following two conditions: (1) All cofibrant objects are flat (that is, are in M ♭ ).
(2) If X is flat and f is a weak equivalence in M ♭ , then X ⊗ f is a weak equivalence.
If the unit of M is cofibrant, then the subcategory of cofibrant objects is a subcategory of flat objects [Hau∞] (4.2), by Ken Brown's lemma. We note that, if the unit of M is cofibrant, then the same is true for both → M ◻ proj and → M ⊗ inj . The purpose of the definition above is to avoid assuming the monoidal unit is cofibrant, as this would rule out positive (flat) model structures on spectra (which do admit a subcategory of flat objects, namely the cofibrant objects of the flat model structure, by [Hau∞] (4.11)). In [Whi17] and [Whi22], the first author gives many examples of model categories with a subcategory of flat objects (namely, the subcategory of cofibrant objects), including spaces, simplicial sets, chain complexes, diagram categories, simplicial presheaves, and various categories of spectra. It is known that for every cofibrantly generated monoidal model category M, every Σ C -cofibrant colored operad O in M is semi-admissible. In other words, there is a transferred semi-model structure on O-algebras [WY18] (6.3.1). An alternative approach assumes M satisfies (♣) and appeals to [WY18] (6.2.3) for such a semimodel structure. It is also known that there are Σ C -cofibrant colored operads O whose category of O-algebras do not admit a full model structure [BW21] (2.9). Hence, the results in this section really do apply to previously unknown examples, and complete the study of semi-model structures on operad-algebras set out in [WY18,WY20,WY19a,WY16]. For completeness, we handle the case of both symmetric and non-symmetric colored operads [Mur11], noting that for the nonsymmetric case, being Σ C -cofibrant is the same as being entrywise cofibrant. Proof. We follow the proof from [PS18] (7.9), which is itself based on the proof of [Lur∞] (4.5.4.12). First, as pointed out in [Lur∞], the reflection property is implied by the preservation property, and it is sufficient to prove that U preserves homotopy colimits indexed by a small category D such that the nerve N(D) is homotopy sifted.
Consider the projective model structure (M C ) D , the projective semi-model structure Alg(O; M) D guaranteed by [Bar10] (3.4), and the forgetful functor The right hand side is canonically weakly equivalent to U(F Alg(O) (A)) because A is projectively cofibrant, and this is weakly equivalent to F(U D A) via α. At this point, the proof in [Lur∞] (4.5.4.12) requires a detailed analysis of so-called "good" objects and morphisms in (M C ) D . However, when O is Σ C -cofibrant, the situation is much simpler, because U takes cofibrant algebras to cofibrant objects of M C [WY18] (6.3.1) ([Mur11] (9.5) for the non-symmetric case).
Furthermore, the D-constant operad O D , taking value O at every a ∈ D, is Σ Ccofibrant in Alg(O; M) D . This can be seen directly, as Σ C -cofibrancy for an operad P valued in M D is the condition that, for each a ∈ D and each (c; d) ∈ Σ op C × C, the object P a Remark 7.2.2. Following the model of [Lur∞] (or [PS18]), after establishing Proposition 7.2.1, the next step should be to prove that the semi-model category Alg(O; M) describes the ∞-category of N ⊗ O-algebras in the ∞-category associated to M, as discussed above. However, when Alg(O; M) is only a semi-model structure, an additional step is needed. We need to know that homotopy colimits (given by colimits of projectively cofibrant objects in Alg(O; M) D ) agree with ∞-categorical colimits. In the case of full model structures, one knows that the projective model structure on Alg(O; M) D describes the ∞-category of functors, and that a Quillen adjunction gives rise to an adjunction of ∞-categories. For the case of semi-model categories, we invoke [LoM∞] (A.10) for the latter.
Remark 7.2.3. We conjecture that Proposition 7.2.1 remains true for entrywise cofibrant colored operads O, if M satisfies (♣), and if we replace appeals to [WY18] (6.3.1) above by appeals to [WY18] (6.2.3). However, the proof of this would require a detailed analysis of 'good' objects and would take us too far afield. For both cases, we handle the cases where O is a symmetric colored operad and where O is a non-symmetric colored operad simultaneously. We handle the enriched case first. Proof. The proof of [Hau∞] (4.10) goes through directly by replacing the appeal to [PS18] (7.8) with an appeal to Proposition 7.2.1. That is, we consider the forgetful functors from both categories to the ∞-category associated to M C , and appeal to the Barr-Beck theorem for ∞-categories [Lur∞] (4.7.3.16) to see that these forgetful functors are monadic right adjoints (this is where Proposition 7.2.1 is needed). We appeal to [Hau∞] (3.8), which occurs entirely on the ∞-category level, for the usual formula for free O-algebras and the observation that the two associated monads on M C have equivalent underlying endofunctors. This proof works for both symmetric and non-symmetric colored operads O, as both are known to inherit transferred semi-model structures from M C , and as Proposition 7.2.1 applies in both settings. Remark 7.3.2. The proof of [Hau∞] (4.10) relies on the observation that a Quillen adjunction F ∶ M ⇄ N ∶ G induces an adjunction between the underlying ∞categories. We appeal to [LoM∞] (A.10) for the semi-model category analogue of this fact.
We turn now to the unenriched case.  is an equivalence of ∞-categories.
Proof. We deliberately phrased the proof of Theorem 7.3.1, so that word-for-word it proves this result as well (again with the critical step hinging on an appeal to Proposition 7.2.1). We only stated the two theorems separately to highlight the difference between enriched and unenriched ∞-operads, and the connection to where the colored operad O is valued.  [Whi17] (for Com rectifying to E ∞ ) and [WY19a] (for general colored operads) among other places.