Given a presentably symmetric monoidal \(\infty\)-category \(\mathcal C\) and a small \(\infty\)-operad \(\mathcal O\), the \(\infty\)-category \(\mathrm{Alg}_{\mathcal O}(\mathcal C)\) is presentable and carries a symmetric monoidal structure. However, it is not necessarily presentably symmetric monoidal: The symmetric monoidal structure on \(\mathrm{Alg}_{\mathcal O}(\mathcal C)\) is not necessarily compatible with finite coproducts (though it is always compatible with sifted colimits). An easy counterexample is \(\mathcal C= \mathrm{Set}\) and \(\mathcal O=\mathbb E_1\).
A.8 Some basics of \(\infty\)-operads[00IP]
Here we briefly discuss some relevant features of the theory of \(\infty\)-operads introduced in [Lur17, § 2].
A.8.1 Basic notions[00IQ]
The notion of an \(\infty\)-operad is an \(\infty\)-categorical version of the theory of colored operads. A colored operad consists of a set \(\iota_0\underline{\mathcal O}\) of colors along with for every finite set \(\{X_i \in \iota_0 \underline{\mathcal O} \}_{i \in I}\) of colors and every color \(Y \in \iota_0 \underline{\mathcal O}\) a set \(\mathrm{Mul}_\mathcal O(\{X_i\}_{i \in I},Y)\) of multimorphisms from \(\{X_i\}_{i \in I}\) to \(Y\), which altogether must be equipped with a associative and unital composition law.53 In particular, the unary multimorphisms (i.e. those with \(|I| = 1\)) define a category \(\underline{\mathcal O}\) of colors (whose set of objects is \(\iota_0 \underline{\mathcal O}\)).
We now give a hint of the main definition. An \(\infty\)-operad \(\mathcal O\) is an \(\infty\)-category \(\mathcal O^{\otimes}\) (called the \(\infty\)-category of operators of \(\mathcal O\)) equipped with a functor \(\mathcal O^{\otimes} \rightarrow\mathrm{Fin}_*\) to the category of finite pointed sets satisfying certain conditions. We immediately introduce the notation \(\underline{n}_+ \coloneqq \{ 1, 2, \ldots, n \}_+ \in \mathrm{Fin}_*\) for the indicated standard object, as well as the notation \(\underline{\mathcal O} \coloneqq \mathcal O^{\otimes}_{\underline{1}_+}\) for the indicated fiber. We refer to \(\underline{\mathcal O}\) as the \(\infty\)-category of colors of \(\mathcal O\) (or sometimes as its underlying \(\infty\)-category, for reasons that will be explained shortly). We will sometimes abuse notation and denote the underlying \(\infty\)-category \(\underline{\mathcal O}\) of an \(\infty\)-operad \(\mathcal O\) simply also by \(\mathcal O\). The crux of the definition of an \(\infty\)-operad is that \(\mathcal O^{\otimes}\) satisfies a sort of “fiberwise” Segal condition which implies that for every \(n \geq 0\) there is a natural equivalence \(\mathcal O^{\otimes}_{\underline{n}_+} \simeq \underline{\mathcal O}^{\times n}\), as well as an “internal” Segal condition which implies that for every pair of objects \(X \coloneqq (X_1,\ldots,X_m) \in \underline{\mathcal O}^{\times m} \simeq \mathcal O^{\otimes}_{\underline{m}_+}\) and \(Y \coloneqq (Y_1,\ldots,Y_n) \in \underline{\mathcal O}^{\times n} \simeq \mathcal O^{\otimes}_{\underline{n}_+}\), we have a natural equivalence \[\mathrm{Hom}_{\mathcal O^{\otimes}}(X,Y) \simeq \bigsqcup_{f \in \mathrm{Hom}_{\mathrm{Fin}_*}(\underline{m}_+,\underline{n}_+)} \prod_{i = 1}^n \mathrm{Hom}_{\mathcal O^{\otimes}}(\{X_j\}_{j \in f^{-1}(i)}, Y_i) ~.\] An ordinary colored operad \(\mathcal O'\) defines an \(\infty\)-operad \(\mathcal O\) with \(\mathrm{Hom}_{\mathcal O^{\otimes}}(\{X_i\}_{i \in I},Y) \coloneqq \mathrm{Mul}_{\mathcal O'}(\{X_i\}_{i \in I},Y)\). As a result, we also write \(\mathrm{Mul}_\mathcal O(\{X_i\}_{i \in I},Y) \coloneqq \mathrm{Hom}_{\mathcal O^{\otimes}}(\{X_i\}_{i \in I},Y)\) for the hom-spaces in an \(\infty\)-operad \(\mathcal O\) whose targets lies in \(\underline{\mathcal O}\), and refer to their points as multimorphisms. Altogether, \(\infty\)-operads assemble into a (non-full) subcategory \(\mathrm{Op}\subset (\mathrm{Cat}_\infty)_{/\mathrm{Fin}_*}\): in essence, morphisms of \(\infty\)-operads are required to respect the Segal condition equivalences. In fact, allowing all 2-morphisms in \((\mathrm{Cat}_\infty)_{/\mathrm{Fin}_*}\) endows \(\mathrm{Op}\) with the structure of an \((\infty,2)\)-category, whose hom-\((\infty,1)\)-categories we denote by \(\underline{\mathrm{Hom}}_\mathrm{Op}(-,-)\).
We say that an \(\infty\)-operad \(\mathcal O\) is single-colored if its \(\infty\)-category of colors \(\underline{\mathcal O}\) is contractible. In this case, we may write \(\ast \in \underline{\mathcal O}\) for the unique point, and we write \(\mathcal O(n) \coloneqq \mathrm{Mul}_\mathcal O(\{\ast\}_{i \in \{1,\ldots,n\}} , \ast)\) for the unique space of \(n\)-ary multimorphisms in \(\mathcal O\).
A.8.2 Key examples[00IR]
Perhaps the most important family of examples of \(\infty\)-operads is the sequence \(\mathbb E_0 \rightarrow\mathbb E_1 \rightarrow\cdots \rightarrow\mathbb E_\infty\). These are single-colored, with the space \(\mathbb E_k(n)\) of \(n\)-ary operations given by (the underlying space of) the topological space of configurations of \(n\) disjoint points in \(\mathbb R^k\).54 The above maps are induced by the standard embeddings \(\mathbb R^0 \hookrightarrow\mathbb R^1 \hookrightarrow\cdots \hookrightarrow\mathbb R^\infty\). We note that \(\mathbb E_1\) and \(\mathbb E_\infty\) are respectively the \(\infty\)-operads underlying the colored operads that parametrize associative and commutative algebras (and in particular, their spaces of multimorphisms are discrete). Hence, we also write \({\textup{Assoc}}\coloneqq \mathbb E_1\) and \(\text{Comm}\coloneqq \mathbb E_\infty\) and respectively refer to these as the associative and commutative \(\infty\)-operads. In fact, \(\text{Comm}\) is simply the identity functor \(\text{Comm}\coloneqq \mathbb E_\infty \simeq \mathrm{Fin}_* \xrightarrow{\mathrm{id}} \mathrm{Fin}_*\), and defines a terminal object of \(\mathrm{Op}\).
Another illustrative example is the \(\infty\)-operad \(\mathrm{LM}\) associated to the two-colored operad parametrizing pairs of an associative algebra object along with a left module over it. We will return to \(\mathrm{LM}\) in subsection A.9.
It will occasionally be useful for us to refer to the single-colored \(\infty\)-operad \(\mathrm{Triv}\), which has no \(n\)-ary multimorphisms for \(n \not= 1\) and the only \(1\)-ary morphism is the identity morphism. Given an \(\infty\)-operad \(\mathcal O\), we write \(\mathcal O_\mathrm{Triv}\coloneqq \mathcal O\times_\text{Comm}\mathrm{Triv}\).
A.8.3 \(\mathcal O\)-monoidal \(\infty\)-categories[00IS]
Given an \(\infty\)-operad \(\mathcal O\), an \(\mathcal O\)-monoidal \(\infty\)-category \(\mathcal C\) is a coCartesian fibration \(\mathcal C^{\otimes} \rightarrow\mathcal O^{\otimes}\) satisfying analogous Segal conditions, which are equivalent to the condition that the composite \(\mathcal C^{\otimes} \rightarrow\mathcal O^{\otimes}\rightarrow\mathrm{Fin}_*\) is also an \(\infty\)-operad. In particular, an \(\mathcal O\)-monoidal \(\infty\)-category can be equivalently specified by the straightening \(\mathcal O^{\otimes} \rightarrow\mathrm{Cat}_\infty\) of this coCartesian fibration. 55 We often abuse the notation by denoting an \(\mathcal O\)-monoidal \(\infty\)-category by its source operad \(\mathcal C\). Altogether, \(\mathcal O\)-monoidal \(\infty\)-categories define a full subcategory \(\mathrm{Alg}_\mathcal O(\mathrm{Cat}_\infty) \subseteq {\textup{coCart}}_{\mathcal O^{\otimes}} \simeq \mathrm{Fun}(\mathcal O^{\otimes},\mathrm{Cat}_\infty)\). As special cases, we write \(\mathrm{Alg}(\mathrm{Cat}_\infty) \coloneqq \mathrm{Alg}_{\textup{Assoc}}(\mathrm{Cat}_\infty)\) for the \(\infty\)-category of monoidal \(\infty\)-categories and \(\mathrm{CAlg}(\mathrm{Cat}_\infty) \coloneqq \mathrm{Alg}_\text{Comm}(\mathrm{Cat}_\infty)\) for the \(\infty\)-category of symmetric monoidal \(\infty\)-categories. The restricted coCartesian fibration \(\underline{\mathcal C} \rightarrow\underline{\mathcal O}\) (or simply its source) may be thought of as the “underlying \(\infty\)-category” of \(\mathcal C\), although this is most immediately meaningful when \(\mathcal O\) is single-colored.
An \(\infty\)-category that admits finite products canonically upgrades to a Cartesian symmetric monoidal \(\infty\)-category. We note that it is merely a condition for a symmetric monoidal \(\infty\)-category to be Cartesian symmetric monoidal. Dual remarks apply in the case of finite coproducts.
A.8.4 \(\mathcal O\)-algebra objects[00IT]
Given an \(\mathcal O\)-monoidal \(\infty\)-category \(\mathcal C\), an \(\mathcal O\)-algebra object in \(\mathcal C\) is a section of the structure map \(\mathcal C\rightarrow\mathcal O\) in \(\mathrm{Op}\). These assemble into an \(\infty\)-category \(\mathrm{Alg}_\mathcal O(\mathcal C) \coloneqq \underline{\mathrm{Hom}}_{\mathrm{Op}_{/\mathcal O}}(\mathcal O,\mathcal C)\). As special cases, we write \(\mathrm{Alg}(\mathcal C) \coloneqq \mathrm{Alg}_{\textup{Assoc}}(\mathcal C)\) for the \(\infty\)-category of (associative) algebra objects in \(\mathcal C\) and \(\mathrm{CAlg}(\mathcal C) \coloneqq \mathrm{Alg}_\text{Comm}(\mathcal C)\) for the \(\infty\)-category of commutative algebra objects in \(\mathcal C\).
More generally, given a morphism \(\mathcal P\xrightarrow{p} \mathcal O\) in \(\mathrm{Op}\), we analogously define the \(\infty\)-category \(\mathrm{Alg}_{\mathcal P/\mathcal O}(\mathcal C) \coloneqq \underline{\mathrm{Hom}}_{\mathrm{Op}_{/\mathcal O}}(\mathcal P,\mathcal C)\) of \(\mathcal P\)-algebras in \(\mathcal C\) (relative to \(p\)). Equivalently, the base change \(p^*\mathcal C\rightarrow\mathcal P\) defines the underlying \(\mathcal P\)-monoidal \(\infty\)-category of \(\mathcal C\in \mathrm{Alg}_\mathcal O(\mathrm{Cat}_\infty)\), and we have \(\mathrm{Alg}_{\mathcal P/\mathcal O}(\mathcal C) \simeq \mathrm{Alg}_\mathcal P(p^*\mathcal C)\). For example, there is a natural morphism \(\mathrm{LM} \rightarrow{\textup{Assoc}}\), and so we can contemplate \(\mathrm{LM}\)-algebras in any monoidal \(\infty\)-category \(\mathcal C\in \mathrm{Alg}(\mathrm{Cat}_\infty)\). Note that when \(\mathcal O= \mathrm{Fin}_*\) we also write this as \(\mathrm{Alg}_{\mathcal P}(\mathcal C)\). 56 Altogether, for an \(\mathcal O\)-monoidal \(\infty\)-category \(\mathcal C\) we obtain a functor \[(\mathrm{Op}_{/\mathcal O})^\mathrm{op} \xrightarrow{\mathrm{Alg}_{(-)/\mathcal O}(\mathcal C)} \mathrm{Cat}_\infty ~,\] whose functoriality is given by precomposition.
As a matter of terminology, it is common to refer to \(\mathcal O\)-algebra objects in a Cartesian symmetric monoidal \(\infty\)-category as \(\mathcal O\)-monoids (e.g. in \(\mathcal S\) or \(\mathrm{Cat}_\infty\)). In particular, \(\mathcal O\)-monoidal \(\infty\)-categories are indeed \(\mathcal O\)-monoids in \(\mathrm{Cat}_\infty\). When referring to notions in spaces, one generally simply prepends “\(\infty\)-” to the classical terms, so e.g. the objects of \(\mathrm{Alg}(\mathcal S)\) may be referred to as “\(\infty\)-monoids”.
Of particular relevance to this paper is the case \(\mathcal O= \mathbb E_2\), and we generally use the term braided in place of the prefix “\(\mathbb E_2\)-”: in particular, a braided monoidal \((\infty,2)\)-category is an \(\mathbb E_2\)-algebra in \(\mathrm{Cat}_{(\infty,2)}\). Indeed, a braided monoidal \(\infty\)-category in the classical sense defines an \(\mathbb E_2\)-monoid in \(\mathrm{Cat}_\infty\).
A.8.5 \(\mathcal O\)-algebras of symmetric monoidal \(\infty\)-categories[00IU]
Let \(\mathcal O, \mathcal C\) be \(\infty\)-operads, then the \(\infty\)-category \(\mathrm{Alg}_{\mathcal O}(\mathcal C)\) has the structure of an \(\infty\)-operad [Lur17, Ex. 3.2.4.4]. From now on, we will denote the \(\infty\)-operad of \(\mathcal O\)-algebras in \(\mathcal C\) by \(\mathrm{Alg}_{\mathcal O}(\mathcal C)\), and the underlying \(\infty\)-category of \(\mathcal O\)-algebra by \(\underline{\mathrm{Alg}}_{\mathcal O}(\mathcal C)\), or just \(\mathrm{Alg}_{\mathcal O}(\mathcal C)\) if clear from context. When the \(\infty\)-operad \(\mathcal C\) is in fact a symmetric monoidal \(\infty\)-category, i.e. \(\mathcal C^{\otimes} \rightarrow\mathrm{Fin}_*\) is a coCartesian fibration, then so is the \(\infty\)-operad \(\mathrm{Alg}_{\mathcal O}(\mathcal C)\). Furthermore, let \(X \in \underline{\mathcal O}\) be a color, the evaluation functor \(e_X \colon \underline{\mathrm{Alg}}_{\mathcal O}(\mathcal C) \rightarrow\underline{\mathcal C}\), which takes an \(\mathcal O\)-algebra to its underlying \(X\)-object, is symmeric monoidal [Lur17, Prop. 3.2.4.3]. More generally, for any map of \(\infty\)-operads \(\mathcal O' \rightarrow\mathcal O\), the pullback functor on algebras \(\mathrm{Alg}_{\mathcal O}(\mathcal C) \rightarrow\mathrm{Alg}_{\mathcal O'}(\mathcal C)\) is a symmetric monoidal functor.
A.8.6 Symmetric monoidal structure on overcategories[00IV]
Let \(\mathcal C\) be a symmetric monoidal \(\infty\)-category and \(A \in \mathrm{CAlg}(\mathcal C)\) be a commutative algebra object therein. Then, there exists a symmetric monoidal structure on the overcategory \(\mathcal C_{/A}\) ([Lur17, Thm. 2.2.2.4]), universally characterized (cf. [Lur17, Def. 2.2.2.1]) by the following equivalence of \(\infty\)-categories for any \(\infty\)-operad \(\mathcal O\) \[\underline{\mathrm{Alg}}_{\mathcal O}(\mathcal C_{/A}) \simeq \underline{\mathrm{Alg}}_{\mathcal O}(\mathcal C)_{/A},\] where on the right hand side we \(A\) is equipped with the \(\mathcal O\)-algebra structure induced by the terminal map of operads \(\mathcal O^{\otimes} \rightarrow\mathrm{Comm}\).
A.8.7 Boardman-Vogt tensor product and Dunn additivity[00IW]
The \(\infty\)-category \(\mathrm{Op}\) of \(\infty\)-operads itself carries a symmetric monoidal structure, called the Boardman-Vogt tensor product uniquely characterized57 by giving rise to an equivalence of \(\infty\)-operads for all \(\infty\)-operads \(\mathcal O, \mathcal O'\) and \(\mathcal P\): \[ \mathrm{Alg}_{\mathcal O}(\mathrm{Alg}_{\mathcal O'}(\mathcal P)) \simeq \mathrm{Alg}_{\mathcal O\otimes \mathcal O'}(\mathcal P).\] Equivalently, the Boardman-Vogt tensor product has \(\mathrm{Alg}_{-}(-)\) as its internal hom.
A fundamental theorem in the theory of \(\infty\)-operad is Dunn’s additivity theorem [Lur17, Thm. 5.1.2.2]: for \(n, m \geq 0\), there is an equivalence of \(\infty\)-operads \(\mathbb E_n \otimes \mathbb E_m \simeq \mathbb E_{n+m}\). In particular, using [00IX], an \(\mathbb E_{n+m}\)-algebra in an \(\infty\)-operad \(\mathcal O\) is equivalent to an \(\mathbb E_n\)-algebra in the \(\infty\)-operad of \(\mathbb E_{m}\)-algebras in \(\mathcal O\).
A.8.8 Laxly \(\mathcal O\)-monoidal functors[00IY]
If \(\mathcal C\) and \(\mathcal D\) are \(\mathcal O\)-monoidal \(\infty\)-categories, then a morphism \(\mathcal C\rightarrow\mathcal D\) in \(\mathrm{Op}_{/\mathcal O}\) is called a laxly \(\mathcal O\)-monoidal functor.58 Let us denote the coCartesian fibrations \(\mathcal C^{\otimes} \rightarrow\mathcal O^{\otimes}\), \(\mathcal D^{\otimes} \rightarrow\mathcal O^{\otimes}\) by \(p\) and \(q\), then an \(\mathcal O\)-monoidal functor is a laxly \(\mathcal O\)-monoidal functor that takes \(p\)-coCartesian morphisms in \(\mathcal C^{\otimes}\) to \(q\)-coCartesian morphisms in \(\mathcal D^{\otimes}\).
Whereas an \(\mathcal O\)-monoidal functor respects the \(\mathcal O\)-monoidal structure up to coherent natural equivalence, a laxly \(\mathcal O\)-monoidal functor \(\mathcal C\rightarrow\mathcal D\) respects it only up to certain (generally noninvertible) coherent natural transformations, which nevertheless suffices to obtain an induced functor \(\mathrm{Alg}_\mathcal O(\mathcal C) \rightarrow\mathrm{Alg}_\mathcal O(\mathcal D)\) on \(\infty\)-categories of \(\mathcal O\)-algebra objects (simply by composition in \(\mathrm{Op}_{/\mathcal O}\)). For instance, given a laxly monoidal functor \(\mathcal C\xrightarrow{F} \mathcal D\) and an algebra object \(A \in \mathrm{Alg}_{\mathbb E_1}(\mathcal C)\), we obtain structure maps \(F(A) \otimes^\mathcal DF(A) \rightarrow F(A \otimes^\mathcal CA) \xrightarrow{F(\mu_A)} F(A)\) and \(\mathbbm{1}_\mathcal D\rightarrow F(\mathbbm{1}_\mathcal C) \xrightarrow{F(\eta_A)} F(A)\) giving the multiplication and unit of \(F(A) \in \mathrm{Alg}(\mathcal D)\).
Furthermore, \(\mathrm{Alg}_{\mathcal O}(-)\) takes (laxly) symmetric monoidal functors between symmetric monoidal \(\infty\)-categories to (laxly) symmetric monoidal functors.
A.8.9 Localizations of \(\mathcal O\)-monoidal \(\infty\)-categories[00IZ]
Given an \(\mathcal O\)-monoidal \(\infty\)-category \(\mathcal C\in \mathrm{Alg}_\mathcal O(\mathrm{Cat}_\infty)\) and a collection \(\mathbf W\) of morphisms in \(\mathcal C\), the \(\mathcal O\)-monoidal localization of \(\mathcal C\) at \(\mathbf W\) is (the target of) the initial object of \(\mathrm{Alg}_\mathcal O(\mathrm{Cat}_\infty)_{\mathcal C/}\) in which the morphisms in \(\mathbf W\) are sent to equivalences. Of course, this generalizes the notion of localization of \(\infty\)-categories discussed in Subsection A.2.5.
As an important special case, we say that a reflective localization ([00IG]) is compatible with a (symmetric) monoidal structure \(\otimes \coloneqq \otimes^\mathcal C\) on \(\mathcal C\) if for all objects \(c,c' \in \mathcal C\) the morphism \(L(c \otimes c') \xrightarrow{L(\eta_c \otimes \eta_{c'})} L(RL(c) \otimes RL(c'))\) in \(\mathcal D\) is an equivalence.59 In this case, \(\mathcal D\) inherits a (resp. symmetric) monoidal structure \(\otimes^\mathcal D\), defined by the formula \(d \otimes^\mathcal Dd' \coloneqq L(R(d) \otimes^\mathcal CR(d'))\) for any \(d,d' \in \mathcal D\) and with unit object \(\mathbbm{1}_\mathcal D\coloneqq L(\mathbbm{1}_\mathcal C)\),60 and the left adjoint \(L\) is canonically (resp. symmetric) monoidal (so that the right adjoint \(R\) is canonically laxly (resp. symmetric) monoidal). In this case, the left adjoint \(L\) witnesses \(\mathcal D\) as not just a localization but also a (resp. symmetric) monoidal localization of \(\mathcal C\).
A.8.10 Presentably \(\mathcal O\)-monoidal \(\infty\)-categories[00J0]
Given an \(\infty\)-operad \(\mathcal O\), a presentably \(\mathcal O\)-monoidal \(\infty\)-category is an \(\mathcal O\)-monoidal \(\infty\)-category \(\mathcal C^{\otimes} \rightarrow\mathcal O^{\otimes}\) such that for every color \(X \in \underline{\mathcal O}\) the \(\infty\)-category \(\mathcal C_X\) is presentable and moreover for every multimorphism \(\{X_i\}_{i \in I} \rightarrow Y\) in \(\mathcal O\) the corresponding multifunctor \(\prod_{i \in I} \mathcal C_{X_i} \rightarrow\mathcal C_Y\) is multi-cocontinuous (i.e. cocontinuous separately in each variable). This is equivalent to the condition that \(\mathcal C\) defines an \(\mathcal O\)-algebra \((\Pr^L,\otimes)\), and we write \(\mathrm{Alg}_\mathcal O(\Pr^L) \subseteq \mathrm{Alg}_\mathcal O(\widehat{\mathrm{Cat}}_\infty)\) for the subcategory whose objects are the presentably \(\mathcal O\)-monoidal \(\infty\)-categories whose morphisms are the \(\mathcal O\)-monoidal left adjoints among them.
Given a presentably \(\mathcal O\)-monoidal \(\infty\)-category \(\mathcal C\in \mathrm{Alg}_\mathcal O(\Pr^L)\), the \(\infty\)-category \(\mathrm{Alg}_\mathcal O(\mathcal C)\) is also presentable. Moreover, the functor \((\mathrm{Op}_{/\mathcal O})^\mathrm{op}\xrightarrow{\mathrm{Alg}_{(-)/\mathcal O}(\mathcal C)} \widehat{\mathrm{Cat}}_\infty\) factors through \(\Pr^R\), i.e. for every morphism \(\mathcal A\rightarrow\mathcal B\) in \(\mathrm{Op}_\mathcal O\) there exists a left adjoint 
the “free \(\mathcal B\)-algebra on an \(\mathcal A\)-algebra” functor [Lur17, Cor. 3.1.3.5]. Of course, these left adjoints then assemble into a functor \(\mathrm{Op}_{/\mathcal O} \xrightarrow{\mathrm{Alg}_{(-)/\mathcal O}} \Pr^L\).
A.8.11 Adjunctions of \(\mathcal O\)-algebras[00J2]
For an \(\infty\)-operad \(\mathcal O\), an \(\mathcal O\)-monoidal left adjoint is an \(\mathcal O\)-monoidal functor \(F \colon \mathcal C\rightarrow\mathcal D\) between \(\mathcal O\)-monoidal \(\infty\)-categories such that for each color \(X \in \mathcal O\), the underlying functor \(F_X \colon \mathcal C\rightarrow\mathcal D\) is a left adjoint.
An important fact which we use repeatedly is that given an \(\mathcal O\)-monoidal left adjoint \(F\), its right adjoint \(G\) is canonically laxly \(\mathcal O\)-monoidal [Lur17, Cor. 7.3.2.7]. Conversely, given a laxly \(\mathcal O\)-monoidal right adjoint, it is merely a condition for its left adjoint to be \(\mathcal O\)-monoidal [Lur17, Cor. 7.3.2.12]. Moreover, such an adjunction determines an adjunction on \(\mathcal O\)-algebra objects [Lur17, Rem 7.3.2.13], whose adjoints both commute with the forgetful functors61, i.e., defines a morphism of adjunction: 
If \(\mathcal C, \mathcal D\) are symmetric monoidal \(\infty\)-categories, and \(F \colon \mathcal C\rightarrow\mathcal D\) is a symmetric monoidal left adjoint, then the symmetric monoidal functor \(\mathrm{Alg}_{\mathcal O}(F) \colon \mathrm{Alg}_{\mathcal O}(\mathcal C) \rightarrow\mathrm{Alg}_{\mathcal O}(\mathcal D)\) is a symmetric monoidal left adjoint.
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2